Ch3 Basic RL and RC Circuits
3.1 First-Order RC Circuits
3.2 First-Order RL Circuits
3.3 Examples
ReferencesReferences: Hayt-Ch5, 6; Gao-Ch5;
Circuits and Analog ElectronicsCircuits and Analog Electronics
3.1 First-Order RC Circuits
Key WordsKey Words:
Transient Response of RC Circuits, Time constant
Ch3 Basic RL and RC Circuits
Ch3 Basic RL and RC Circuits
3.1 First-Order RC Circuits
• Used for filtering signal by blocking certain frequencies and passing others. e.g. low-pass filter
• Any circuit with a single energy storage element, an arbitrary number of sources and an arbitrary number of resistors is a circuit of order 1.
• Any voltage or current in such a circuit is the solution to a 1st order differential equation.
Ideal Linear Capacitor
dtdq
i t dtdvc
2
21 cvcvdvpdtwEnergy stored
A capacitor is an energy storage device memory device.
tCtC vv
Ch3 Basic RL and RC Circuits
3.1 First-Order RC Circuits
• One capacitor and one resistor• The source and resistor may be equivalent to a circuit with
many resistors and sources.
R+
-Cvs(t)
+
-
vc(t)
+ -vr(t)
Ch3 Basic RL and RC Circuits
3.1 First-Order RC Circuits
R
1
C
2
K
E
RvEi c
c
KVL around the loop: EvRi Cc
EvRdtdvC c
c
EAev RCt
C
Initial condition 000 CC vv
)1()1( t
RCt
C eEeEv
dtdvCi c
c t
eRE
Switch is thrown to 1
RCCalled time constant
Transient Response of RC Circuits
EA
Ch3 Basic RL and RC Circuits
3.1 First-Order RC Circuits
)1( t
C eEv
/tc eE
dtdv
0
0
t
ct
c
dtdv
EEdtdv
RCTime Constant
R
1
C
2
K
E
Time
0s 1ms 2ms 3ms 4ms 5ms 6ms 7ms 8ms 9ms 10msV(2)
0V
5V
10V
SEL>>
RC
R=2k
C=0.1F
Ch3 Basic RL and RC Circuits
3.1 First-Order RC Circuits
Switch to 2
R
1
C
2
K
E
RCt
c Aev
Initial condition Evv CC 00
0 Riv cc
0dtdvRCv c
c
// tRCtc EeEev
/tc e
REi
Transient Response of RC Circuits
cc
dvi Cdt
Ch3 Basic RL and RC Circuits
3.1 First-Order RC Circuits
RCTime Constant /t
c eREi
/tc e
RE
dtdi
0
0/
t
ct
c
dtdi
RERE
dtdi
R
1
C
2
K
E
R=2k
C=0.1F
Time
0s 1.0ms 2.0ms 3.0ms 4.0ms 5.0ms 6.0ms 7.0ms 8.0msV(2)
0V
5V
10V
SEL>>
Ch3 Basic RL and RC Circuits
3.1 First-Order RC Circuits
Time
0s 0.5ms 1.0ms 1.5ms 2.0ms 2.5ms 3.0ms 3.5ms 4.0ms 4.5ms 5.0ms 5.5ms 6.0msV(2) V(1)
0V
2.0V
4.0V
6.0V
Ch3 Basic RL and RC Circuits
3.2 First-Order RL Circuits
Key WordsKey Words:
Transient Response of RL Circuits, Time constant
Ch3 Basic RL and RC Circuits
3.2 First-Order RL Circuits
Ideal Linear Inductor
i(t) +
-
v(t)
Therestofthe
circuit
Ldt
tdiLdt
dtv )()(
t
dxxvL
ti )(1)(
tLtL iidtdiLiivP
)(21)( 2 tLitwL Energy stored:
• One inductor and one resistor• The source and resistor may be equivalent to a circuit with many
resistors and sources.
Ch3 Basic RL and RC Circuits
3.2 First-Order RL Circuits
Switch to 1
R
1
L
2
K
E
dtdiLvL
KVL around the loop: EviR L
iRdtdiLE
Initial condition 0)0()0(,0 iit
Called time constant RL /
Transient Response of RL Circuits
/
/
/
1
)1(
)1()1(
ttLRt
LR
L
tR
ttLR
EeeRELe
RE
dtdL
dtdiLv
eEiRv
eREe
REi
Ch3 Basic RL and RC Circuits
3.2 First-Order RL Circuits
Time constant
• Indicate new fast i (t) will drop to zero precisely.• It is the amount of time for i (t) to drop zero if it is dropping at
the initial rate.
t
i (t)
0
.
Ch3 Basic RL and RC Circuits
3.2 First-Order RL Circuits
Switch to 2
tLR
Aei
dtLR
idi
iRdtdiL
0
Initial condition REit 0,0
/ttLR
eREe
REi
Transient Response of RL Circuits
R
1
L
2
K
E
0
0
0
0
0
: 0:
1
ln
i t t
I
i t tI
t ti I i t
Rdi dti L
Ri tL
Ch3 Basic RL and RC Circuits
3.2 First-Order RL Circuits
R
1
L
2
K
E
Transient Response of RL Circuits
Time
0s 1ms 2ms 3ms 4ms 5ms 6ms 7ms 8ms 9ms 10msI(L1)
0A
2.0mA
4.0mA
SEL>>
Time
0s 1ms 2ms 3ms 4ms 5ms 6ms 7ms 8ms 9ms 10msI(L1)
0A
2.0mA
4.0mA
SEL>>
Input energy to L
L export its energy and it is dissipated by R
Ch3 Basic RL and RC Circuits
Initial Value ( t=0)
Steady Value
( t)time
constant
RL Circuits
Source(0 state)
Source-
free(0 input)
RC
Circuits
Source(0 state)
Source-free
(0 input)
00 iREiL
REi 0 0i
00 v Ev
Ev 0 0v
RL /
RL /
RC
RC
Summary
Ch3 Basic RL and RC Circuits
Summary
The Time Constant
• For an RC circuit, = RC• For an RL circuit, = L/R• -1/ is the initial slope of an exponential with an initial value of 1• Also, is the amount of time necessary for an exponential to decay
to 36.7% of its initial value
Ch3 Basic RL and RC Circuits
Summary
• How to determine initial conditions for a transient circuit. When a sudden change occurs, only two types of quantities will remain the same as before the change. – IL(t), inductor current– Vc(t), capacitor voltage
• Find these two types of the values before the change and use them as the initial conditions of the circuit after change.
Ch3 Basic RL and RC Circuits
About Calculation for The Initial Value
iC iL
i
t=0
+
_
1A
+
-vL(0+)
vC(0+)=4V
i(0+)
iC(0+) iL(0+)
3.3 Examples
1 3 2R R
0
28V 4V2 2Cv
0
8V 2A2 2
i
042A 1A
4 4Li
0 0C Cv v
0 0L Li i
Ch3 Basic RL and RC Circuits
3.3 Examples
Method 1
(Analyzing an RC circuit or RL circuit)
Simplify the circuit
2) Find Leq(Ceq), and = Leq/Req ( = CeqReq)
1) Thévenin Equivalent.(Draw out C or L)
Veq , Req
3) Substituting Leq(Ceq) and to the previous solution of differential equation for RC (RL) circuit .
Ch3 Basic RL and RC Circuits
3.3 Examples
Method 2
(Analyzing an RC circuit or RL circuit)
3) Find the particular solution.
1) KVL around the loop the differential equation
4) The total solution is the sum of the particular and homogeneous solutions.
2) Find the homogeneous solution.
3.3 Examples
Method 3 (step-by-step)
(Analyzing an RC circuit or RL circuit)
1) Draw the circuit for t = 0- and find v(0-) or i(0-)
2) Use the continuity of the capacitor voltage, or inductor current, draw the circuit for t = 0+ to find v(0+) or i(0+)
3) Find v(), or i() at steady state4) Find the time constant
– For an RC circuit, = RC– For an RL circuit, = L/R
5) The solution is:/)]()0([)()( teffftf
Given f ( 0 +), thus )()0( ffA
t
t effff
][ )()0()()(
Initial Steady
t
t Aeff
)()(In general,
Ch3 Basic RL and RC Circuits
Ch3 Basic RL and RC Circuits
3.3 ExamplesP3.1 vC(0) = 0, Find vC(t) for t 0. i1
6k
R1
R2 3k +
_ E
C=1000PF pf
i2 i3 t=0
9V
Method 3:
0
3K0 0, 9V 3V6K 3K
t
c c c c
c c
v t v v v e
v v
Apply Thevenin theoren :
6
1
6
2 10
1 1 2K6K 3K
2K 100pF 2 10
3 3 V
Th
Th
t
c
R
R C
v t e
Ch3 Basic RL and RC Circuits
3.3 Examples
vC
R2 3k
+
_ v
U
I
t=0
6V
C=1000PF
R1=10k
R1=20k
+ - P3.2 vC(0)=0, Find vo, vC(t) for t 0.
Apply Thevenin theoren :
6
1
6
2.31 10
1 1 30 K10K 3K 13
30 K 100pF 2.31 1013
4.615 4.615 V
Th
Th
t
c
R
R C
v t e
0 0
10K6V 4.615V10K 3Kc
v t
v
Top Related