Introduction to Circuit Theory
First-Order Circuits
2012-10-12
Jieh-Tsorng Wu
National Chiao-Tung University
Department of Electronics Engineering
Outline
1.
2.
3.
4.
5.
6.
The Source-Free RC Circuit
The Source-Free RL Circuit
Singularity Functions
Step Response of an RC Circuit
Step Response of an RL Circuit
First-Order OPAMP Circuits
7. First-Order Circuits
2
Circuit Theory; Jieh-Tsorng Wu
1
The Source-Free RC Circuit
dv v
dv 1
 0 

v0
dt R
dt RC
  RC  Time Constant
 V0 e t 
iC  iR  0  C
v(t )  V0 e t RC
7. First-Order Circuits
Circuit Theory; Jieh-Tsorng Wu
3
Natural Response and Time Constant
 The natural response of a circuit refers to the behavior of the circuit itself, with no
external sources of excitation.
 The time constant of a circuit is the time required for the response to decay by a
factor of 1/e or 36.8% of its initial value.
t

2
3
4
5
7. First-Order Circuits
4
v(t ) V0
0.36788
0.13534
0.04979  5%
0.01832  2%
0.00674  1%
Circuit Theory; Jieh-Tsorng Wu
2
Effect of Time Constant
7. First-Order Circuits
Circuit Theory; Jieh-Tsorng Wu
5
Power and Energy of a RC Circuit
v  t   V0 e t 
  RC
v  t  V0 t 
 e
R
R
V2
p  t   viR  0 e2t 
R
iR 
t
V2
V2  
1

wR  t    pdt   0 e 2t  dt  0    e 2t    CV02 1  e 2t  
R
R  2
0 2
0
0
t
t
1
As t  , wR     CV02
2
7. First-Order Circuits
6
Circuit Theory; Jieh-Tsorng Wu
3
The Key for RC Circuit Analysis
 Find the initial voltage v(0)=V0 across the capacitor.
 Find the time constant =RC.
 R is the resistance as seen by the C.
7. First-Order Circuits
Circuit Theory; Jieh-Tsorng Wu
7
RC Circuit Example 1
Req  5‖20  4 
  Req C  4  (0.1)  0.4 s
vC (t )  vC (0)e  t   15e t 0.4 V
vx  vC 
12
 9e 2.5t V
8  12
7. First-Order Circuits
ix 
8
vx
 0.75e 2.5t A
12
Circuit Theory; Jieh-Tsorng Wu
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RC Circuit Example 2
For t<0
9
 15 V
39
vC (0)  20 
1
wC (0)  CvC 2 (0)
2
1
wC (0)   (20m) 152  2.25 J
2
For t  0
Req  1  9  10 
t<0
t>0
  Req C  10   20m   0.2 s
v(t )  vC (0)e  t   15e  t 0.2 V
7. First-Order Circuits
9
Circuit Theory; Jieh-Tsorng Wu
The Source-Free RL Circuit
di
di R
 Ri  0 
 v0
dt
dt L
L
   Time Constant
 V0 e t 
R
vL  vR  0  L
i (t )  I 0 e tR L
7. First-Order Circuits
10
Circuit Theory; Jieh-Tsorng Wu
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Power and Energy of a RL Circuit
L
R
vR  i (t )  R  I 0 Re t 
i  t   I 0 e t 

p  t   vR i  I 02 Re 2t 
t
t
wR  t    pdt   I Re
2
0
0
0
As t  , wR    
7. First-Order Circuits
t
2 t 
1

 
dt  I R    e 2t    LI 02 1  e 2t  
 2
0 2
2
0
1 2
LI 0
2
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Circuit Theory; Jieh-Tsorng Wu
The Key for RL Circuit Analysis
 Find the initial voltage i(0)=I0 through the inductor.
 Find the time constant =L/R.
 R is the resistance as seen by the L.
7. First-Order Circuits
12
Circuit Theory; Jieh-Tsorng Wu
6
RL Circuit Example 1
2(i1  i2 )  vo  0  3i1  4i2  2(i2  i1 )  0  vo  2i1  2i2
1
vo   i1
3
Req 
vo vo 1
L 0.5 3

   
 s
io i1 3
R 13 2
i  i (0)e t   10e  (2/3)t A v  L
ix 
5
i2  i1
6
10
di
 2
 0.5      10e  (2/3)t   e  (2/3)t V
3
dt
 3
5
v
  e  (2/3)t A
2
3
7. First-Order Circuits
13
Circuit Theory; Jieh-Tsorng Wu
RL Circuit Example 2
For t  0
t<0
40
12

6A
i (0) 
2  (4‖12) 12  4
For t  0
Req  16‖(12  4)  8 

2 1
L
  s
Req 8 4
t>0
i (t )  i (0)e t   6e4t A
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
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RL Circuit Example 3
For t  0
10
 2 A vo (0)  3  i (0)  6 V
i (0) 
23
For t  0
2
L
Req  3‖6  2   
 1 s
Req 2
t<0
i (t )  i (0)e t   2e t A
vL  L
v
2
di
 4e t V io  L   e t A
6
3
dt
7. First-Order Circuits
15
t>0
Circuit Theory; Jieh-Tsorng Wu
Singularity Functions
 Singularity functions are functions that either are discontinuous or have
discontinuous derivatives.
 The three most widely used singularity functions in circuit analysis are
 Unit step function, u(t).
 Unit impulse function, (t).
 Unit ramp function, r(t).
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
8
Unit Step Functions
0, t  0
u (t )  
1, t  0
7. First-Order Circuits
0, t  t0
u (t  t0 )  
1, t  t0
17
0, t  t0
u (t  t0 )  
1, t  t0
Circuit Theory; Jieh-Tsorng Wu
Unit Step Function and Equivalent Circuits
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
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Unit Impulse (Delta) Function
t0
0,
d

  t   u  t    Undefined, t  0
dt
0,
t 0


7. First-Order Circuits
0
t
0

  (t )dt  1   (t ')dt '  u (t )
19
Circuit Theory; Jieh-Tsorng Wu
Unit Impulse (Delta) Functions
f (t )  5 (t  2)  10 (t )  4 (t  3)
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
10
Unit Impulse Function and Sampling
y (t )  f (t ) (t  t0 )  f (t0 ) (t  t0 ) 
f (t ) is sampled at t  t0
Let a  t0  b,

b
a
f  t    t  t0  dt   f  t0    t  t0  dt  f  t0     t  t0  dt  f  t0 
b
b
a
a
7. First-Order Circuits
Circuit Theory; Jieh-Tsorng Wu
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Unit Ramp Functions
r t  
t
 u  t ' dt '  tu  t 
r  t  t0 
t0
0,

 t  t0 ,

0,
r t   
 t,
t0
7. First-Order Circuits
22
r  t  t0 
t  t0
t  t0
0,

 t  t0 ,
t  t0
t  t 0
Circuit Theory; Jieh-Tsorng Wu
11
Singularity Function Example 1
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
Singularity Function Example 2
i (t )  10u (t )  20u (t  2)  10u (t  4)
 idt  10r (t )  20r (t  2)  10r (t  4)
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
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Singularity Function Example 3
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
Singularity Function Example 4
u (t )  1  u (t )
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
13
Step Response of an RC Circuit
At t  0, v(0)  V0
For t  0, v(0 )  V0 , and from KCL,
iC  iR  0
dv v  VS

0
dt
R
dv
 RC  v  VS
dt
Let v(t )  a1e  t /  a2
C
For t > 0
a1  t /
e  a1e  t /  a2  VS a1  a2  V0

   RC a2  VS a1  V0  VS
RC
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
Step Response of an RC Circuit
t0
 V0 ,
v(t )  
t 
VS  V0  VS  e , t  0
If V0  0, then, for t  0
v(t )  VS 1  e t  
i (t )  C
V
dv C
 Vs e t   S e  t 
dt 
R
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
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Natural Response and Forced Response
Complete Response = Natural Response + Forced Response
(Stored Energy) (Independent Source)
For t  0
v(t )  VS  V0  VS  e  t 
 V0 e  t   VS 1  e t  
 vn (t )  v f (t )
vn (t )  V0 e  t 
v f (t )  VS 1  e  t  
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
Transient Response and Steady-State Response
Complete Response = Transient Response + Steady-State Response
(Permanent)
(Temporary)
For t  0
v(t )  VS  V0  VS  e  t 
 vss  vt (t )
vss  VS
vt (t )  V0  VS  e  t 
 Transient response is the circuit’s temporary response that will die out with time.
 Steady-state response is the behavior of the circuit a long time after the an
external excitation is applied.
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
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Transient Response and Steady-State Response
v(t )  v()   v(0)  v()  e  t 
v(0)
or
v(t )  v()   v(t0 )  v()  e
  t  t0  
v()
v(0) is the initial value
v() is the final steady-state value
 is the time constant of the circuit
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
RC Circuit Step Response Example 1
For t  0, v(t )  v()   v(0)  v()  e t /
v(0)  v(0 )  24 
5k
 15 V
3k  5k
v()  30 V
  4k  0.5m  2 sec
v(t )  30  [15  30]e 0.5t  30  15e 0.5t V
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
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RC Circuit Step Response Example 2
For t  0, v(t )  v()   v(0)  v()  e t /
t<0
v(0)  v(0 )  10 V
20
 20 V
10  20
1 20 1 5
  (10‖20)     sec
4 3 4 3
v(t )  20  [10  20]e 0.6t  20  10e 0.6t V
v()  30 
7. First-Order Circuits
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t>0
Circuit Theory; Jieh-Tsorng Wu
Step Response of an RL Circuit
For t  0, i (0)  I 0
For t  0, i (0 )  VS / R, and from KVL,
vL  vR  VS
di
 Ri  VS
dt
V
L dv

v  S
R dt
R
 t /
Let i (t )  a1e  a2
L
For t > 0
V
L a1  t /
e  a1e  t /  a2  S a1  a2  I 0
R 
R
V
V
L
 
a2  S a1  I 0  S
R
R
R
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
17
Step Response of an RL Circuit
If I 0  0, then, for t  0
v(t ) 
VS 
V
  I0  S
R 
R
 t 
e

VS
1  et  

R
di L VS  t 
v(t )  L 
e  VS e  t 
dt  R
t0
i (t ) 
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
Step Response of an RL Circuit
i (t )  i ()  i (0)  i ()  e t 
or
i (t )  i ()  i (t0 )  i () e
  t  t0  
i (0) is the initial value
i () is the final steady-state value
 is the time constant of the circuit
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
18
RL Circuit Step Response Example 1
For t  0,
i (t )  i ()  i (0)  i ()  e t /
i (0)  i (0 ) 
10
5 A
2
10
2A
5
1 1 1
  
sec
3 5 15
i (t )  2  [5  2]e 15t  2  3e 15t A
i ( ) 
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
RL Circuit Step Response Example 2
(I) For t  0, i (t )  0
(II) For 0<t  4, i (t )  i ()  i (0)  i ()  e  t /
i (0 )  i (0 )  0 A i () 
40
L
5 1
 4 A RTh  10   
  sec
46
RTh 10 2
i (t )  4  [0  4]e 2t  4 1  e2t  A
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
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RL Circuit Step Response Example 2
(III) For 4<t ,
i (t )  i ()  i (4)  i ()  e  (t  4)/
i (4)  i (4 )  4(1  e 2t )  4(1  e 8 )  4 A
2
4
 10 
 20 V
42
42
22

RTh  (4‖2)  6 
3
V
20
30

 2.272 A
i ( )  S 
RTh 22 / 3 11
VS  40 

5
15
L


sec
RTh 22 / 3 22
i (t )  2.727  [4  2.727]e1.467(t  4)
 2.727  1.273e 1.467(t  4) A
7. First-Order Circuits
Circuit Theory; Jieh-Tsorng Wu
39
First-Order Opamp Circuit Example 1
v(0)  V0
v ( )  0
RTh  R1   R1C
v(t )  v()   v()  v(0)  e  t /  V0 e  t /
i (t )  C
V
dv
C
  V0 e  t /   0 e  t /
dt

R1
v0 (t )   R f i (t ) 
7. First-Order Circuits
40
Rf
R1
 V0 e  t /
Circuit Theory; Jieh-Tsorng Wu
20
First-Order Opamp Circuit Example 2
v(0)  V0
v ( )  0
RTh  R f
  Rf C
v(t )  v()   v()  v(0)  e  t /  V0 e  t /
v0 (t )  v(t )  V0 e  t /
7. First-Order Circuits
Circuit Theory; Jieh-Tsorng Wu
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First-Order Opamp Circuit Example 3
v(0)  V0
v ( )  0
  50k 1  0.05
v(0)  0
20k
2V
10k  20k
 50k 
vo ()  v1  1 
7 V
 20k 
v()  v1  vo ()  5 V
v1  3 
v(t )  v()   v(0)  v()  e  t /
 5  [0  5]e 20t  5  5e 20t V
v0 (t )  v1  v(t )  7  5e 20t V
7. First-Order Circuits
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Circuit Theory; Jieh-Tsorng Wu
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First-Order Opamp Circuit Example 4
vi (t )  2u (t ) V Vab (t )  
vo (0)  0 vo ()  2
Rf
R1
Ro  0 RTh  R2‖R3 
vo (t )  vo () 1  e  t / 
7. First-Order Circuits
Rf

R1
vi (t )  2
Rf
R1
u (t ) V
R3
R2  R3
R2 R3
  RThC
R2  R3
43
Circuit Theory; Jieh-Tsorng Wu
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