151001 1 EECS2200 Electric Circuits Chapter 5 Capacitance, Inductance, and RC/RL circuits Objectives (1) Understand how an inductor or a capacitor behaves in the presence of a constant current or constant voltage Know and being able to use the equations of current, voltage power and energy in an inductor and capacitor Be able to combine inductors (capacitors) with an initial condition in series and parallel to form a single equivalent conductor (capacitor). Understand the basic concept of mutual inductance and being able to write mesh-current equations for a circuit containing magnetically coupled coils using the dot convention correctly.
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EECS2200 Electric Circuits
Chapter 5
Capacitance, Inductance, and RC/RL circuits
Objectives (1) n Understand how an inductor or a capacitor
behaves in the presence of a constant current or constant voltage
n Know and being able to use the equations of current, voltage power and energy in an inductor and capacitor
n Be able to combine inductors (capacitors) with an initial condition in series and parallel to form a single equivalent conductor (capacitor).
n Understand the basic concept of mutual inductance and being able to write mesh-current equations for a circuit containing magnetically coupled coils using the dot convention correctly.
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Objective (2) n Be able to determine the natural response
of RL and RC circuits. n Be able to determine the step response of
RL and RC circuits.
EECS2200 Electric Circuits
Chapter 5 Part 1
Capacitance and RC circuits
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EECS2200 Electric Circuits
Capacitor
Capacitor
n Parallel plates, separated by an insulator, so no charge flows between the plates. Impose a time-varying voltage drop.
n Capacitor equation:
n Units: v(t) is volts, i(t) is amps, and C is farads [F]
dttdvCti )()( =
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Activity 1 Look at the capacitor equation: Suppose v(t) is constant, what is the value of i(t)? A. 0 B. ∞ C. a constant
dttdvCti )()( =
Activity 2 if the voltage drop across the capacitor is constant, its current is 0, so the capacitor can be replaced by: A. A short circuit B. An open circuit C. a constant
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Capacitor n If the voltage drop across a capacitor is constant, the
current is 0, so the capacitor can be replaced by an OPEN CIRCUIT.
n Look at the capacitor equation again:
n Suppose there is a discontinuity in v(t) – that is, at some value of t, the voltage jumps instantaneously. At this value of t, the derivative of the voltage is infinite. Therefore the current is infinite! NOT POSSIBLE.
n Thus, the voltage drop across a capacitor is continuous for all time.
dttdvCti )()( =
Voltage n The equation for voltage in terms of current:
∫
∫∫
+=⇒
=⇒
=⇒=
t
t o
tv
tv
t
t
o
oo
tvdiC
tv
dxCdi
tCdvdttidttdvCti
)()(1)(
)(
)()()()(
)(
)(
ττ
ττ
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Power and Energy n Power and energy
p(t) = v(t)i(t) =Cv(t) dv(t)dt
p(t) = dw(t)dt
=Cv(t) dv(t)dt
⇒ dw(τ ) =Cv(τ )dv(τ )
⇒ dx =C y(τ )dτ0
v(t )∫0
w(t )∫
⇒ w(t) = 12Cv(t)
2
Activity 3 Find the voltage, power, and energy
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Capacitor in series
Capacitor in parallel
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EECS2200 Electric Circuits
RC Natural Response
Introduction n Natural Response: The current and
voltages that arise when the stored energy in the L of C is released.
n Step Response: The current and voltage that arise when energy is being acquired by the L or C when a sudden application of voltage or current is applied to the circuit.
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RC Natural Response n Switch in position A for a long time n The capacitor charged to Vg
RC Natural Response To evaluate the voltage drop across the capacitor for t < 0, put the switch in the “a” position and replace the capacitor with an open circuit. The voltage drop across the open circuit, positive at the top, is A. Vg
B. -Vg
C. RVg / (R + R1)
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RC Natural Response
:0<t
0, <= tVv g
There is an open circuit here, so the current in this circuit is 0, and the voltage drop across R1 is 0. Thus,
Remember the continuity requirement for capacitors – the voltage drop across the capacitor must be continuous everywhere, so that the current through the capacitor remains finite. Thus,
gVv =)0(
RC Natural Response
:0≥tKCL at b: C dv(t)
dt+v(t)R
= 0
0,)(so)0(But1Solving,
01so0)1(g,Rearrangin0ng,Substituti
)(and)(Let
)1( ≥==
=
=−=−
=+−
−==
−
−
−−
−−
teVtvVvRCa
aCReaCRRe)aeC(
aedttdvetv
tRCgg
at
atat
atat
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Activity 4 Find Vc(t) for t>=0
EECS2200 Electric Circuits
RC Step Response
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Step Response of RC Circuits
Step Response of RC Circuit
( ) RCtss
RCt
s
s
s
s
ttv
V s
s
ss
s
eRIVRItv
eRIVRItv
tRCRIV
RItv
dtRCRIv
dv
dtRCRIv
dvRC
RIvC
IRvdtdv
IRv
dtdvC
/0
/
0
0
0
)(
)(
)()()(
1)()()(ln
1
1
/
0
−
−
−+=
=−
−
−=
−
−
−=
−
−=
−
+−=
+−=
=+
∫∫
( )
RCts
RCts
eRVIi
eRC
RIVCi
dtdvCi
/0
/0
1
−
−
⎟⎠
⎞⎜⎝
⎛ −=
⎟⎠
⎞⎜⎝
⎛ −−=
=
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v i
RC Step Response n R=10 KΩ, C=1 µF n Is=5A, V0=0
Activity 5 The switch has been in position 1 for a long time. At t=0, the switch moves to position 2. Find v0(t) for t>=0 and io(t) for t>=0+
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