bme 372 new - new jersey institute of technologyjoelsd/electronicsnew/classnotes/lecture 1.pdfbme...
TRANSCRIPT
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BME372New
Lecture#1ElectricalBasics
BME372NewSchesser 2
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3
Circuit Analysis • Circuit Elements
– Passive Devices – Active Devices
• Circuit Analysis Tools – Ohms Law – Kirchhoff’s Law – Impedances – Mesh and Nodal Analysis – Superposition
• Examples
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4
Characterize Circuit Elements
• Passive Devices: dissipates or stores energy – Linear – Non-linear
• Active Devices: Provider of energy or supports power gain – Linear – Non-linear
BME372NewSchesser
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5
Circuit Elements – Linear Passive Devices
• Linear: supports a linear relationship between the voltage across the device and the current through it. – Resistor: supports a voltage and current which
are proportional, device dissipates heat, and is governed by Ohm’s Law, units: resistance or ohms Ω
BME372NewSchesser
resistor with theassociated resistance theof value theis R where)( (t)RItV RR =
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6
Circuit Elements – Linear Passive Devices
– Capacitor: supports a current which is proportional to its changing voltage, device stores an electric field between its plates, and is governed by Gauss’ Law, units: capacitance or farads, f
capacitor with theassociated ecapacitanc theof value theis C where)( dt(t)dVCtI C
C =
BME372NewSchesserBME372NewSchesser
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7
Circuit Elements – Linear Passive Devices
– Inductor: supports a voltage which is proportional to its changing current, device stores a magnetic field through its coils and is governed by Faraday’s Law, units: inductance or henries, h
inductor with theassociated inductance theof value theis L where)( dt(t)dILtV L
L =
BME372NewSchesser
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8
Circuit Elements - Passive Devices Continued
• Non-linear: supports a non-linear relationship among the currents and voltages associated with it – Diodes: supports current flowing through it in
only one direction
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9
Circuit Elements - Active Devices • Linear
– Sources • Voltage Source: a device which supplies a voltage
as a function of time at its terminals which is independent of the current flowing through it, units: Volts – DC, AC, Pulse Trains, Square Waves, Triangular
Waves
• Current Source: a device which supplies a current as a function of time out of its terminals which is independent of the voltage across it, units: Amperes – DC, AC, Pulse Trains, Square Waves, Triangular
Waves
+
-- Vab(t)
a
b
Iab(t)
a
b
BME372NewSchesser
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10
Circuit Elements - Active Devices Continued
– Ideal Sources vs Practical Sources • An ideal source is one which only depends on the
type of source (i.e., current or voltage) • A practical source is one where other circuit
elements are associated with it (e.g., resistance, inductance, etc. ) – A practical voltage source consists of an ideal voltage
source connected in series with passive circuit elements such as a resistor
– A practical current source consists of an ideal current source connected in parallel with passive circuit elements such as a resistor
+
--
Vab(t)
a
b
Rs
Vs(t)
Is(t)
a
b
Rs Iab(t)
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11
Circuit Elements - Active Devices Continued
– Independent vs Dependent Sources • An independent source is one where the output
voltage or current is not dependent on other voltages or currents in the device • A dependent source is one where the output voltage
or current is a function of another voltage or current in the device (e.g., a BJT transistor may be viewed as having an output current source which is dependent on the input current)
+
-- Vab(t)
a
b
vs vo vi
Ro
Ri
Rs
RL Avovi
io ii
+ +
- -
+
-
BME372NewSchesser
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12
Circuit Elements - Active Devices Continued
• Non-Linear – Transistors: three or more terminal devices
where its output voltage and current characteristics are a function on its input voltage and/or current characteristics, several types BJT, FETs, etc.
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13
Circuits • A circuit is a grouping of passive and active
elements • Elements may be connecting is series,
parallel or combinations of both
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14
Circuits Continued • Series Connection: Same current through the devices
– The resultant resistance of two or more Resistors connected in series is the sum of the resistance
– The resultant inductance of two or more Inductors connected in series is the sum of the inductances
– The resultant capacitance of two or more Capacitors connected in series is the inverse of the sum of the inverse capacitances
– The resultant voltage of two or more Ideal Voltage Sources connected in series is the sum of the voltages
– Two of more Ideal Current sources can not be connected in series
BME372NewSchesser
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15
Series Circuits
• Resistors
• Inductors
• Capacitors
R1 R2
RT = R1+ R2
L1 L2
LT = L1+ L2
C1 C2
21
21
21
111
CCCC
CC
CT +=+
=
T
bcabac
IRRRIIRIRVVV
=+=+=+=
)( 21
21
dtdILdt
dILL
dtdILdt
dILVVV
T
bcabac
=+=
+=+=
)( 21
21
∫∫
∫∫=+=
+=+=
IdtCIdtCC
IdtCIdtCVVV
T
bcabac
1)11(
11
21
21
a b c
a b c
a b c
BME372NewSchesser
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16
Series Circuits
• Resistors 20Ω 50Ω
RT = 20+ 50 = 70Ω
70)5020(5020
IIIIVVV bcabac
=+=+=+=
a b c
BME372NewSchesser
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17
Series Circuits
• Inductors 25h 100h
LT =25+ 100 = 125h
dtdI
dtdI
dtdI
dtdIVVV bcabac
125)10025(
10025
=+=
+=+=
a b c
BME372NewSchesser
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18
Series Circuits
• Capacitors 5f 10f
fCT 33.3310
1550
105105
101
511 ===
+×=
+=
∫∫
∫∫=+=
+=+=
IdtIdt
IdtIdtVVV bcabac
103)
101
51(
101
51
a b c
BME372NewSchesser
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19
Series Circuits
• Capacitors 10f 10f
fCT 5210
20100
10101010
101
1011 ===
+×=
+=
∫∫∫
∫∫==+=
+=+=
IdtIdtIdt
IdtIdtVVV bcabac
51
102)
101
101(
101
101
a b c
BME372NewSchesser
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20
Circuits Continued • Parallel Connection: Same Voltage across the devices
– The resultant resistance of two or more Resistors connected in parallel is the inverse of the sum of the inverse resistances
– The resultant inductance of two or more Inductors connected in parallel is the inverse of the sum of the inverse inductances
– The resultant capacitance of two or more Capacitors connected in parallel is the sum of the capacitances
– The resultant current of two or more Ideal Current Sources connected in parallel is the sum of the currents
– Two of more Ideal Voltage sources can not be connected in parallel
BME372NewSchesser
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21
Parallel Circuits
• Resistors
• Inductors
• Capacitors
R1 R2
CT = C1+ C2
L1 L2
C1 C2
21
21
21
111
LLLL
LL
LT +=+
=
21
21
21
111
RRRR
RR
RT +=+
=
T
ab
RVVRR
RV
RVIII
=+=
+=+=
)11(21
2121
∫∫
∫∫=+=
+=+=
VdtLVdtLL
VdtLVdtLIII
T
ab
1)11(
11
21
2121
dtdVCdt
dVCC
dtdVCdt
dVCIII
T
ab
=+=
+=+=
)( 21
2121
a
b
Iab
I1 I2
Iab
I1 I2
a
b
a
b
Iab
I1 I2
BME372NewSchesser
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22
Combining Circuit Elements Kirchhoff’s Laws
• Kirchhoff Voltage Law: The sum of the voltages around a loop must equal zero
• Kirchhoff Current Law: The sum of the currents leaving (entering) a node must equal zero
BME372NewSchesser
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23
Combining Rs, Ls and Cs
• We can use KVL or KCL to write and solve an equation associated with the circuit. – Example: a series Resistive Circuit
V(t) = I(t)R1 + I(t)R2
V(t) = I(t)(R1 + R2)
R1+
-- V(t)
I(t) R2
BME372NewSchesser
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Combining Rs, Ls, and Cs
• We can use KVL or KCL to write and solve an equation associated with the circuit. – Example: a series Resistive Circuit
I1(t)+I2(t)+I3(t)=0 I2(t)=-V(t)/R1; I3(t)=-V(t)/R2; I1(t) - V(t)/R1 - V(t)/R2 =0 I1(t)=V(t)/R1 +V(t)/R2=V(t)[1/R1 +1/R2]
24
R1V(t) R2+
-
I1(t)
I2(t)
I3(t)
BME372NewSchesser
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25
Impedances • Our special case, signals of the form: V(t) or I(t) =Aest
where s can be a real or complex number
• This is only one portion of the solution and does not include the transient response.
t
t
tttt
tt
tt
t
etIA
eA
eeeAe
dtAedtdAeAee
dttIdttdItIe
5
5
5555
55
55
5
3610)(;
3610
)36(
)5
55510(10
551010
)(2.1)(510)(10
==
=
+×+=
++=
++=
∫
∫
t
t
AetI
fChLRetV
dttICdt
tdILRtItV
5
1115
111
)(: trysLet'
2.;5;10;10)(:assume sLet'
)(1)()()(
=
==Ω==
++= ∫
BME372NewSchesser
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26
Impedances • Since the derivative [and integral] of Aest =
sAest [=(1/s)Aest], we can define the impedance of a circuit element as Z(s)=V/I where Z is only a function of s since the time dependency drops out.
BME372NewSchesser
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27
Impedances
RAe
RAeIVsZ
RAetRItV Ae tI
sCsCAeAe
IVsZ
CsAedt
tdVCtI Ae tV
sLAe
sLAeIVsZ
LsAedt
tdILtV Ae tI
st
st
stst
st
st
stst
st
st
stst
===
===
===
===
===
===
)(
;)()( then ;)( assume slet' resistor, aFor
1)(
;)()( then ;)( assume slet' capacitor, aFor
)(
;)()( then ;)( assume slet' inductor,an For
BME372NewSchesser
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28
Impedances • What about signals of the type: cos(ωt+θ);
• Recall Euler’s formula e jθ = cos θ +j sin θ where j is the imaginary number =
• A special case of our special case is for sinusoidal inputs, where s=jω
1−
BME372NewSchesser
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29
Sinusoidal Steady State Continued
Real Numbers
Imaginary Numbers
θ
Inductor Imaginary Numbers
θ
Capacitor Imaginary Numbers
θ
Resistor
− For an inductor, ZL = jωL⇒ωL∠π2
− For a capacitor, ZC = 1jωC
⇒ 1ωC
∠− π2
.
− For a resistor, ZR = R⇒ R∠0
BME372NewSchesser
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30
Sinusoidal Steady State Continued For V(t)=Acosωt, using phasor notation for V(t) è and I(t) è I, our equation can be re-written:
IIIV1
11
111
10
tionrepresentaPhasor toConverting
)(1)()()(
CjLjRA
dttICdt
tdILRtItV
ωω ++=∠=
++= ∫
V = A∠0
BME372NewSchesser
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31
Sinusoidal Steady State Continued
]))1(
[tancos()1(
)(
tion,representa time theback to Converting
])1(
[tan)1()1(
01
0
1
11
1
2
11
21
1
11
1
2
11
21
111
111
RC
Lt
CLR
AtI
RC
L
CLR
A
CLjR
A
CjLjR
A
ωω
ω
ωω
ωω
ωωω
ωω
ω
−−
−+=
−−∠
−+=
−+
∠=++
∠=
−
−I
BME372NewSchesser
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Bode Plots
BME372NewSchesser 32
I = A
R12 + (ωL1 −
1ωC1
)2
∠− tan−1[(ωL1 −
1ωC1
)
R1
]
Magnitude of I ⇒ I = A
R12 + (ωL1 −
1ωC1
)2
Angle of I ⇒∠I = ∠− tan−1[(ωL1 −
1ωC1
)
R1
]
• Plotting the magnitude and phase versus frequency.
0
0.2
0.4
0.6
0.8
1
1.E-02 1.E+00 1.E+02 1.E+04 1.E+06Radians
Magnitude
-2-1.5-1
-0.50
0.51
1.52
1.E-02 1.E+00 1.E+02 1.E+04 1.E+06
Radians
Angle
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Bode Plots
BME372NewSchesser 33
• Easy way to plot the magnitude. I = A
R1 + jωL1 +1jωC1
First evaluate at the initial condition ω=0
I |ω=0 = A
R1 + jωL1 +1jωC1
|ω=0→A
R1 + j0L1 +1j0C1
→Determine the dominate terms
A1j0C1
→ j A∞→ 0∠π
2
Next evaluate at the final condition ω →∞
I |ω→∞ = A
R1 + jωL1 +1jωC1
|ω→∞→A
R1 + j∞L1 +1j∞C1
→Determine the dominate terms
Aj∞L1
→− j A∞→ 0∠− π
2
Find an interesting point; for this example choose ω = ω0 =1
LC since this is when the imaginary part
of the deminator is zero.
I |ω0 =
1
LC
= A
R1 + jωL1 +1jωC1
|ω0=
1
L1C1
= A
R1 + j(1L1C1
L1 −1
1L1C1
C1
)= A
R1 + j(L1
C1
− 1C1
L1
)
= AR1
= AR1
∠0
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Bode Plots
• With the three point, plots can be made
BME372NewSchesser 34
ω
Magnitude
0
AR1
0
ω0ω
Phase π2
0
−π2
ω0
![Page 34: BME 372 New - New Jersey Institute of Technologyjoelsd/electronicsnew/Classnotes/Lecture 1.pdfBME 372 New Schesser V R (t) = I R ... =Aest where s can be a real or complex number •](https://reader035.vdocument.in/reader035/viewer/2022071504/61247401f2e0c766183594d4/html5/thumbnails/34.jpg)
35
Homework R31k
R11k a
b
Find the total resistance
R2R45k5k
BME372NewSchesser
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36
Homework Find the total resistance Rab where R1 = 3Ω, R2 = 6Ω, R3 = 12Ω, R4 = 4Ω, R5 = 2Ω, R6 = 2Ω, R7 = 4Ω, R8 = 4Ω
R1
R7
R2
R8
R3
R4
R6
R5
a
b
BME372NewSchesser
![Page 36: BME 372 New - New Jersey Institute of Technologyjoelsd/electronicsnew/Classnotes/Lecture 1.pdfBME 372 New Schesser V R (t) = I R ... =Aest where s can be a real or complex number •](https://reader035.vdocument.in/reader035/viewer/2022071504/61247401f2e0c766183594d4/html5/thumbnails/36.jpg)
37
Homework Find the total resistance Rab where R1 = 2Ω, R2 = 4Ω, R3 = 2Ω, R4 = 2Ω, R5 = 2Ω, R6 = 4Ω,
R6
R1
R2
R3
R4
R5
a
b
BME372NewSchesser
![Page 37: BME 372 New - New Jersey Institute of Technologyjoelsd/electronicsnew/Classnotes/Lecture 1.pdfBME 372 New Schesser V R (t) = I R ... =Aest where s can be a real or complex number •](https://reader035.vdocument.in/reader035/viewer/2022071504/61247401f2e0c766183594d4/html5/thumbnails/37.jpg)
38
Homework Find the total resistance Rab for this infinite resistive network
R
R
R
R
R
R
a
b
R
R
R
R
R
R
R
R
R
R
R
continues to infinity
BME372NewSchesser
![Page 38: BME 372 New - New Jersey Institute of Technologyjoelsd/electronicsnew/Classnotes/Lecture 1.pdfBME 372 New Schesser V R (t) = I R ... =Aest where s can be a real or complex number •](https://reader035.vdocument.in/reader035/viewer/2022071504/61247401f2e0c766183594d4/html5/thumbnails/38.jpg)
39
Homework C3
1f
C1
1f a
b
Find the total capacitance
C2C45f5f
BME372NewSchesser
![Page 39: BME 372 New - New Jersey Institute of Technologyjoelsd/electronicsnew/Classnotes/Lecture 1.pdfBME 372 New Schesser V R (t) = I R ... =Aest where s can be a real or complex number •](https://reader035.vdocument.in/reader035/viewer/2022071504/61247401f2e0c766183594d4/html5/thumbnails/39.jpg)
40
Homework
Find and plot the impedance Zab(jω) as a function of frequency. Use Matlab to calculate the Bode plot.
R=1 C=1
L=1
b
a
BME372NewSchesser
![Page 40: BME 372 New - New Jersey Institute of Technologyjoelsd/electronicsnew/Classnotes/Lecture 1.pdfBME 372 New Schesser V R (t) = I R ... =Aest where s can be a real or complex number •](https://reader035.vdocument.in/reader035/viewer/2022071504/61247401f2e0c766183594d4/html5/thumbnails/40.jpg)
41
Homework
R=1 C=1 L=1
b
a
Find and plot the impedance Zab(jω) as a function of frequency. Use Matlab to calculate the Bode plot.
BME372NewSchesser
![Page 41: BME 372 New - New Jersey Institute of Technologyjoelsd/electronicsnew/Classnotes/Lecture 1.pdfBME 372 New Schesser V R (t) = I R ... =Aest where s can be a real or complex number •](https://reader035.vdocument.in/reader035/viewer/2022071504/61247401f2e0c766183594d4/html5/thumbnails/41.jpg)
42
Homework R25k
R11k
C11n
C25n
a
b
Find and plot the impedance Zab(jω) as function of ω. Use Matlab to calculate the Bode plot.
BME372NewSchesser