revision lecture 4: transients & lines · e1.1 analysis of circuits (2014-4649) revision...
TRANSCRIPT
![Page 1: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/1.jpg)
Revision Lecture 4: Transients &Lines
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 1 / 9
![Page 2: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/2.jpg)
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 2 / 9
• Transients happen in response to a sudden change
◦ Input voltage/current abruptly changes its magnitude,frequency or phase
◦ A switch alters the circuit
![Page 3: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/3.jpg)
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 2 / 9
• Transients happen in response to a sudden change
◦ Input voltage/current abruptly changes its magnitude,frequency or phase
◦ A switch alters the circuit
• 1st order circuits only: one capacitor/inductor
![Page 4: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/4.jpg)
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 2 / 9
• Transients happen in response to a sudden change
◦ Input voltage/current abruptly changes its magnitude,frequency or phase
◦ A switch alters the circuit
• 1st order circuits only: one capacitor/inductor
• All voltage/current waveforms are: Steady State + Transient
◦ Steady States: find with nodal analysis or transfer function
• Note: Steady State is not the same as DC Level• Need steady states before and after the sudden change
![Page 5: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/5.jpg)
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 2 / 9
• Transients happen in response to a sudden change
◦ Input voltage/current abruptly changes its magnitude,frequency or phase
◦ A switch alters the circuit
• 1st order circuits only: one capacitor/inductor
• All voltage/current waveforms are: Steady State + Transient
◦ Steady States: find with nodal analysis or transfer function
• Note: Steady State is not the same as DC Level• Need steady states before and after the sudden change
◦ Transient: Always a negative exponential: Ae−tτ
![Page 6: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/6.jpg)
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 2 / 9
• Transients happen in response to a sudden change
◦ Input voltage/current abruptly changes its magnitude,frequency or phase
◦ A switch alters the circuit
• 1st order circuits only: one capacitor/inductor
• All voltage/current waveforms are: Steady State + Transient
◦ Steady States: find with nodal analysis or transfer function
• Note: Steady State is not the same as DC Level• Need steady states before and after the sudden change
◦ Transient: Always a negative exponential: Ae−tτ
• Time Constant: τ = RC or LR where R is the Thévenin
resistance at the terminals of C or L
![Page 7: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/7.jpg)
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 2 / 9
• Transients happen in response to a sudden change
◦ Input voltage/current abruptly changes its magnitude,frequency or phase
◦ A switch alters the circuit
• 1st order circuits only: one capacitor/inductor
• All voltage/current waveforms are: Steady State + Transient
◦ Steady States: find with nodal analysis or transfer function
• Note: Steady State is not the same as DC Level• Need steady states before and after the sudden change
◦ Transient: Always a negative exponential: Ae−tτ
• Time Constant: τ = RC or LR where R is the Thévenin
resistance at the terminals of C or L• Find transient amplitude, A, from continuity since VC orIL cannot change instantly.
![Page 8: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/8.jpg)
Transients: Basic Ideas
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 2 / 9
• Transients happen in response to a sudden change
◦ Input voltage/current abruptly changes its magnitude,frequency or phase
◦ A switch alters the circuit
• 1st order circuits only: one capacitor/inductor
• All voltage/current waveforms are: Steady State + Transient
◦ Steady States: find with nodal analysis or transfer function
• Note: Steady State is not the same as DC Level• Need steady states before and after the sudden change
◦ Transient: Always a negative exponential: Ae−tτ
• Time Constant: τ = RC or LR where R is the Thévenin
resistance at the terminals of C or L• Find transient amplitude, A, from continuity since VC orIL cannot change instantly.
• τ and A can also be found from the transfer function.
![Page 9: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/9.jpg)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).
-5 0 5 10 15 200
2
4
6
t (µs)
![Page 10: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/10.jpg)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
-5 0 5 10 15 200
2
4
6
t (µs)
![Page 11: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/11.jpg)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
-5 0 5 10 15 200
2
4
6
t (µs)
![Page 12: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/12.jpg)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
-5 0 5 10 15 200
2
4
6
t (µs)
![Page 13: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/13.jpg)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
ySS(0−) = 1, ySS(0+) = 3
-5 0 5 10 15 200
2
4
6
t (µs)
![Page 14: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/14.jpg)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
ySS(0−) = 1, ySS(0+) = 3
Method 2: Transfer functionYX (jω) = R+jωL
2R+jωL
-5 0 5 10 15 200
2
4
6
t (µs)
![Page 15: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/15.jpg)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
ySS(0−) = 1, ySS(0+) = 3
Method 2: Transfer functionYX (jω) = R+jωL
2R+jωL
set ω = 0: YX (0) = 1
2
-5 0 5 10 15 200
2
4
6
t (µs)
![Page 16: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/16.jpg)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
ySS(0−) = 1, ySS(0+) = 3
Method 2: Transfer functionYX (jω) = R+jωL
2R+jωL
set ω = 0: YX (0) = 1
2ySS(0−) = 1, ySS(0+) = 3
-5 0 5 10 15 200
2
4
6
t (µs)
![Page 17: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/17.jpg)
Steady States
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9
A steady-state output assumes the input frequency, phase and amplitudeare constant forever. You need to determine two ySS(t) steady stateoutputs: one for before the transient (t < 0) and one after (t ≥ 0).At t = 0, ySS(0−) means the first one and ySS(0+) means the second.
Method 1: Nodal analysisInput voltage is DC (ω = 0)⇒ ZL = 0 (for capacitor: ZC =∞)So L acts as a short citcuit
Potential divider: ySS = 12x
ySS(0−) = 1, ySS(0+) = 3
Method 2: Transfer functionYX (jω) = R+jωL
2R+jωL
set ω = 0: YX (0) = 1
2ySS(0−) = 1, ySS(0+) = 3
-5 0 5 10 15 200
2
4
6
t (µs)
Sinusoidal input⇒ Sinusoidal steady state⇒ use phasors.Then convert phasors to time waveforms to calculate the actual outputvoltages ySS(0−) and ySS(0+) at t = 0.
![Page 18: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/18.jpg)
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin
![Page 19: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/19.jpg)
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor
![Page 20: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/20.jpg)
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.
![Page 21: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/21.jpg)
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
![Page 22: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/22.jpg)
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
![Page 23: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/23.jpg)
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function
![Page 24: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/24.jpg)
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function(a) Calculate transfer function using nodal analysis
![Page 25: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/25.jpg)
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function(a) Calculate transfer function using nodal analysis
KCL @ V: V−X4R + V
8R + jωCV + V−Y2R = 0
KCL @ Y: Y−V2R + Y−X
6R = 0
![Page 26: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/26.jpg)
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function(a) Calculate transfer function using nodal analysis
KCL @ V: V−X4R + V
8R + jωCV + V−Y2R = 0
KCL @ Y: Y−V2R + Y−X
6R = 0
→ Eliminate V to get transfer Function: YX = 8jωRC+13
32jωRC+16
![Page 27: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/27.jpg)
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function(a) Calculate transfer function using nodal analysis
KCL @ V: V−X4R + V
8R + jωCV + V−Y2R = 0
KCL @ Y: Y−V2R + Y−X
6R = 0
→ Eliminate V to get transfer Function: YX = 8jωRC+13
32jωRC+16
(b) Time Constant = 1Denominator corner frequency
![Page 28: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/28.jpg)
Determining Time Constant
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 4 / 9
Method 1: Thévenin(a) Remove the capacitor/inductor(b) Set all sources to zero (including the inputvoltage source). Leave output unconnected.(c) Calculate the Thévenin resistancebetween the capacitor/inductor terminals:RTh = 8R||4R||(6R + 2R) = 2R
(d) Time constant: = RThC or LRTh
τ = RThC = 2RC
Method 2: Transfer function(a) Calculate transfer function using nodal analysis
KCL @ V: V−X4R + V
8R + jωCV + V−Y2R = 0
KCL @ Y: Y−V2R + Y−X
6R = 0
→ Eliminate V to get transfer Function: YX = 8jωRC+13
32jωRC+16
(b) Time Constant = 1Denominator corner frequency
ωd = 1632RC ⇒ τ = 1
ωd= 2RC
![Page 29: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/29.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 30: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/30.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 31: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/31.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 32: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/32.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuity
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 33: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/33.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 34: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/34.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mA
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 35: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/35.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 36: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/36.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current source
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 37: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/37.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 38: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/38.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer function
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 39: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/39.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 40: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/40.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
Input step, ∆x = x(0+)− x(0−) = +4-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 41: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/41.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
Input step, ∆x = x(0+)− x(0−) = +4y(0+) = y(0−) +H(j∞)×∆x
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 42: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/42.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
Input step, ∆x = x(0+)− x(0−) = +4y(0+) = y(0−) +H(j∞)×∆x
= 1 + 1× 4 = 5
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
ySS
![Page 43: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/43.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
Input step, ∆x = x(0+)− x(0−) = +4y(0+) = y(0−) +H(j∞)×∆x
= 1 + 1× 4 = 5
(ii) A = y(0+)− ySS(0+) = 5− 3 = 2
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
yTr
ySS
![Page 44: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/44.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
Input step, ∆x = x(0+)− x(0−) = +4y(0+) = y(0−) +H(j∞)×∆x
= 1 + 1× 4 = 5
(ii) A = y(0+)− ySS(0+) = 5− 3 = 2
(iii) y(t) = ySS(t) +Ae−t/τ
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
yTr
ySS
![Page 45: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/45.jpg)
Determining Transient Amplitude
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 5 / 9
After an input change at t = 0, y(t) = ySS(t) +Ae−tτ .
⇒ y(0+) = ySS(0+) +A⇒ A = y(0+)− ySS(0+)Method: (a) calculate true output y(0+), (b) subtract ySS(0+) to get A
(i) Version 1: vC or iL continuityx(0−) = 2⇒ iL(0−) = 1mAContinuity⇒ iL(0+) = iL(0−)Replace L with a 1mA current sourcey(0+) = x(0+)− iR = 6− 1 = 5
(i) Version 2: Transfer functionH(jω) = Y
X (jω) = R+jωL2R+jωL
Input step, ∆x = x(0+)− x(0−) = +4y(0+) = y(0−) +H(j∞)×∆x
= 1 + 1× 4 = 5
(ii) A = y(0+)− ySS(0+) = 5− 3 = 2
(iii) y(t) = ySS(t) +Ae−t/τ
= 3 + 2e−t/2µ
-5 0 5 10 15 200
2
4
6
t (µs)
-5 0 5 10 15 200
2
4
6
t (µs)
yTr
ySS
y
![Page 46: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/46.jpg)
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
![Page 47: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/47.jpg)
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
![Page 48: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/48.jpg)
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
0 2 4 6 8 10Time (ns)
f(t-0/u)
![Page 49: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/49.jpg)
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u)
![Page 50: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/50.jpg)
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
Knowing fx(t) forx = x0 fixes it for allother x.
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u) f(t-90/u)
![Page 51: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/51.jpg)
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
Knowing fx(t) forx = x0 fixes it for allother x.
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u) f(t-90/u)
Backward wave: gx(t) is the same but travelling←: gx(t) = g0(
t+ xu
)
.
![Page 52: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/52.jpg)
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
Knowing fx(t) forx = x0 fixes it for allother x.
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u) f(t-90/u)
Backward wave: gx(t) is the same but travelling←: gx(t) = g0(
t+ xu
)
.
Voltage and current are: vx = fx + gx and ix = fx−gxZ0
where ix is
positive in the +x direction (→) and Z0 =√
L0
C0
![Page 53: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/53.jpg)
Transmission Lines Basics
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 6 / 9
Transmission Line: constant L0 and C0 : inductance/capacitance permetre.Forward wave travels along the line: fx(t) = f0
(
t− xu
)
.
Velocity u =√
1L0C0
≈ 12c = 15 cm/ns
fx(t) equals f0 (t)but delayed by x
u .
Knowing fx(t) forx = x0 fixes it for allother x.
0 2 4 6 8 10Time (ns)
f(t-0/u) f(t-45/u) f(t-90/u)
Backward wave: gx(t) is the same but travelling←: gx(t) = g0(
t+ xu
)
.
Voltage and current are: vx = fx + gx and ix = fx−gxZ0
where ix is
positive in the +x direction (→) and Z0 =√
L0
C0
Waveforms of fx and gx are determined by the connections at both ends.
![Page 54: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/54.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
![Page 55: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/55.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
![Page 56: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/56.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
![Page 57: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/57.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
![Page 58: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/58.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
![Page 59: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/59.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t)
![Page 60: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/60.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
![Page 61: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/61.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
![Page 62: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/62.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
![Page 63: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/63.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
![Page 64: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/64.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
×ρ0
−→ f0(t+2Lu )
![Page 65: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/65.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
×ρ0
−→ f0(t+2Lu )
×ρL−→ g0(t+
4Lu )
![Page 66: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/66.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
×ρ0
−→ f0(t+2Lu )
×ρL−→ g0(t+
4Lu )
×ρ0
−→ · · ·
![Page 67: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/67.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
×ρ0
−→ f0(t+2Lu )
×ρL−→ g0(t+
4Lu )
×ρ0
−→ · · ·
Infinite sum:f0(t) = τ0vS(t)+τ0ρLρ0vS(t−
2Lu )+. . .
![Page 68: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/68.jpg)
Reflections
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 7 / 9
vx = fx + gxix = fx−gx
Z0
At x = L, Ohm’s law⇒ vL(t)iL(t) = RL ⇒ gL (t) = RL−Z0
RL+Z0
× fL (t).
Reflection coefficient: ρL = gL(t)fL(t) =
RL−Z0
RL+Z0
ρL ∈ [−1, +1] and increases with RL
Knowing fx(t) for x = x0 now tells you fx, gx, vx, ix ∀x
At x = 0: f0(t) =Z0
RS+Z0
vS(t) +RS−Z0
RS+Z0
g0(t) = τ0vS(t) + ρ0g0(t)
Wave bounces back and forth getting smaller with each reflection:
vS(t)×τ0−→ f0(t)
×ρL−→ g0(t+
2Lu )
×ρ0
−→ f0(t+2Lu )
×ρL−→ g0(t+
4Lu )
×ρ0
−→ · · ·
Infinite sum:f0(t) = τ0vS(t)+τ0ρLρ0vS(t−
2Lu )+. . .=
∑
∞
i=0 τ0ρiLρ
i0vS
(
t− 2Liu
)
![Page 69: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/69.jpg)
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminate t:
2. Time delays are just phase shifts:
![Page 70: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/70.jpg)
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
![Page 71: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/71.jpg)
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
![Page 72: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/72.jpg)
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
![Page 73: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/73.jpg)
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
As before: Vx = Fx +Gx and Ix = Fx−Gx
Z0
![Page 74: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/74.jpg)
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
As before: Vx = Fx +Gx and Ix = Fx−Gx
Z0
As before:GL = ρLFL
F0 = τ0VS + ρ0G0
![Page 75: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/75.jpg)
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
As before: Vx = Fx +Gx and Ix = Fx−Gx
Z0
As before:GL = ρLFL
F0 = τ0VS + ρ0G0
But G0 = F0ρLe−2jkL : roundtrip delay of 2L
u + reflection at x = L.
![Page 76: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/76.jpg)
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
As before: Vx = Fx +Gx and Ix = Fx−Gx
Z0
As before:GL = ρLFL
F0 = τ0VS + ρ0G0
But G0 = F0ρLe−2jkL : roundtrip delay of 2L
u + reflection at x = L.Substituting for G0 in source end equation: F0 = τ0VS + ρ0F0ρLe
−2jkL
![Page 77: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/77.jpg)
Sinewaves and Phasors
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 8 / 9
Sinewaves are easier because:
1. Use phasors to eliminate t: f0(t) = A cos (ωt+ φ)⇔ F0 = Aejφ
2. Time delays are just phase shifts:
fx(t) = A cos(
ω(
t− xu
)
+ φ)
⇔ Fx = Aej(φ−ωux)= F0e
−jkx
k is the wavenumber: radians per metre (c.f. ω in rad per second)
As before: Vx = Fx +Gx and Ix = Fx−Gx
Z0
As before:GL = ρLFL
F0 = τ0VS + ρ0G0
But G0 = F0ρLe−2jkL : roundtrip delay of 2L
u + reflection at x = L.Substituting for G0 in source end equation: F0 = τ0VS + ρ0F0ρLe
−2jkL
⇒ F0 = τ01−ρ0ρL exp(−2jkL)VS so no infinite sums needed ,
![Page 78: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/78.jpg)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
![Page 79: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/79.jpg)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
![Page 80: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/80.jpg)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x)
![Page 81: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/81.jpg)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
![Page 82: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/82.jpg)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx
![Page 83: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/83.jpg)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
![Page 84: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/84.jpg)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣
![Page 85: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/85.jpg)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣: depends on x
![Page 86: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/86.jpg)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣: depends on x
If ρL ≥ 0, max magnitude is (1 + ρL) |F0| whenever e−2jk(L−x) = +1
![Page 87: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/87.jpg)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣: depends on x
If ρL ≥ 0, max magnitude is (1 + ρL) |F0| whenever e−2jk(L−x) = +1⇒ x = L or x = L− π
k or x = L− 2πk or . . .
![Page 88: Revision Lecture 4: Transients & Lines · E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 3 / 9 A steady-stateoutput assumes the input frequency, phase and amplitude](https://reader034.vdocument.in/reader034/viewer/2022050505/5f971879579c3b3dd5206c6a/html5/thumbnails/88.jpg)
Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣: depends on x
If ρL ≥ 0, max magnitude is (1 + ρL) |F0| whenever e−2jk(L−x) = +1⇒ x = L or x = L− π
k or x = L− 2πk or . . .
Min magnitude is (1− ρL) |F0| whenever e−2jk(L−x) = −1
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Standing Waves
Revision Lecture 4:Transients & Lines
• Transients: Basic Ideas
• Steady States
• Determining TimeConstant• Determining TransientAmplitude
• Transmission Lines Basics
• Reflections
• Sinewaves and Phasors
• Standing Waves
E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 – 9 / 9
Standing waves arise whenever a wave meets its reflection:at places where the two waves are in phase their amplitudes addbut where they are anti-phase their amplitudes subtract.
At any point x,delay of x
u ⇒Fx = F0e
−jkx
Backward wave: Gx = ρLFxe−2jk(L−x): reflection + delay of 2L−x
u
Voltage at x: Vx = Fx +Gx = F0e−jkx
(
1 + ρLe−2jk(L−x)
)
Voltage Magnitude : |Vx| = |F0|∣
∣1 + ρLe−2jk(L−x)
∣
∣: depends on x
If ρL ≥ 0, max magnitude is (1 + ρL) |F0| whenever e−2jk(L−x) = +1⇒ x = L or x = L− π
k or x = L− 2πk or . . .
Min magnitude is (1− ρL) |F0| whenever e−2jk(L−x) = −1⇒ x = L− π
2k or x = L− 3π2k or x = L− 5π
2k or . . .