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BME372New
Lecture#1ElectricalBasics
BME372NewSchesser 2
3
Circuit Analysis • Circuit Elements
– Passive Devices – Active Devices
• Circuit Analysis Tools – Ohms Law – Kirchhoff’s Law – Impedances – Mesh and Nodal Analysis – Superposition
• Examples
BME372NewSchesser
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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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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 =
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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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 =
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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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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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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)
BME372NewSchesser
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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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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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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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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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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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Series Circuits
• Resistors 20Ω 50Ω
RT = 20+ 50 = 70Ω
70)5020(5020
IIIIVVV bcabac
=+=+=+=
a b c
BME372NewSchesser
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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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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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Series Circuits
• Capacitors 10f 10f
fCT 5210
20100
10101010
101
1011 ===
+×=
+=
∫∫∫
∫∫==+=
+=+=
IdtIdtIdt
IdtIdtVVV bcabac
51
102)
101
101(
101
101
a b c
BME372NewSchesser
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
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
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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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
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
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
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
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
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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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
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
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
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
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
Bode Plots
• With the three point, plots can be made
BME372NewSchesser 34
ω
Magnitude
0
AR1
0
ω0ω
Phase π2
0
−π2
ω0
35
Homework R31k
R11k a
b
Find the total resistance
R2R45k5k
BME372NewSchesser
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
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
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
39
Homework C3
1f
C1
1f a
b
Find the total capacitance
C2C45f5f
BME372NewSchesser
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
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
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