電路學(一)
Chapter 7
First-Order Circuits
1
First-Order Circuits
Chapter 7
7.1
7.2
7.3
7.4
7.5
7.6
The Source-Free RC Circuit
The Source-Free RL Circuit
Singularity Functions
Step Response of an RC Circuit
Step Response of an RL Circuit
Applications
2
7.1 The Source-Free
RC Circuit (1)
• A first-order circuit is characterized by a firstorder differential equation.
By KCL
iR  iC  0
Ohms law
v
dv
C
0
R
dt
Capacitor law
• Apply Kirchhoff’s laws to purely resistive circuit results in
algebraic equations.
• Apply the laws to RC and RL circuits produces differential
equations.
3
7.1 The Source-Free
RC Circuit (2)
• The natural response of a circuit refers to the behavior
(in terms of voltages and currents) of the circuit itself,
with no external sources of excitation.
Time constant
RC
Decays more slowly
Decays faster
• 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.
• v decays faster for small  and slower for large .
4
7.1 The Source-Free
RC Circuit (3)
The key to working with a source-free RC circuit is
finding:
v (t )  V0 e  t / 
where
RC
1. The initial voltage v(0) = V0 across the capacitor.
2. The time constant  = RC.
5
7.1 The Source-Free
RC Circuit (4)
Example 1
In the figure, let vC(0) = 15 V. Find vC, vx, and ix
for t > 0.
8
ix
+
5
0.1 F
vC
–
+
12 
vx
–
6
7.1 The Source-Free
RC Circuit (5)
Example 2
Refer to the circuit below, determine vC, vx, and
io for t ≥ 0.
Assume that vC(0) = 30 V.
• Please refer to lecture or textbook for more detail elaboration.
Answer: vC = 30e–0.25t V ; vx = 10e–0.25t ; io = –2.5e–0.25t A
7
7.1 The Source-Free
RC Circuit (6)
Example 3
The switch in circuit has been closed for a long
time, and it is opened at t = 0. Find v(t) for t ≥ 0.
Calculate the initial energy stored in the capacitor.
8
7.1 The Source-Free
RC Circuit (7)
Example 4
The switch in circuit below is opened at t = 0,
find v(t) for t ≥ 0.
• Please refer to lecture or textbook for more detail elaboration.
Answer: V(t) = 8e–2t V
9
7.2 The Source-Free
RL Circuit (1)
• A first-order RL circuit consists of a inductor
L (or its equivalent) and a resistor (or its
equivalent)
By KVL
vL  vR  0
di
L
 iR  0
dt
Inductors law
Ohms law
di
R
  dt
i
L
i (t )  I 0 e
Rt / L
10
7.2 The Source-Free
RL Circuit (2)
A general form representing a RL
i (t )  I 0 e
where
•
•
•
t / 
L

R
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.
i(t) decays faster for small  and slower for large .
The general form is very similar to a RC source-free circuit.
11
7.2 The Source-Free
RL Circuit (3)
Comparison between a RL and RC circuit
A RL source-free circuit
i (t )  I 0 e
t / 
where
L

R
A RC source-free circuit
v(t )  V0 e  t /
where
  RC
12
7.2 The Source-Free
RL Circuit (4)
The key to working with a source-free
circuit is finding:
i(t )  I 0 e
 t /
where
RL
L

R
1. The initial voltage i(0) = I0 through the
inductor.
2. The time constant  = L/R.
13
7.2 The Source-Free
RL Circuit (5)
Example 5
Assume that i(0) = 10 A, calculate i(t) and ix(t) in the circuit.
.
1. Equivalent circuit
2. Loop analysis
14
7.2 The Source-Free
RL Circuit (6)
Example 6
Find i and vx in the circuit.
Assume that i(0) = 5 A.
• Please refer to lecture or textbook for more detail elaboration.
Answer: i(t) = 5e–53t A
15
7.2 The Source-Free
RL Circuit (7)
Example 7
The switch in the circuit has been closed for a long time.
At t = 0, the switch is opened. Calculate i(t) for t > 0.
16
7.2 The Source-Free
RL Circuit (8)
Example 8
For the circuit, find i(t) for t > 0.
• Please refer to lecture or textbook for more detail elaboration.
Answer: i(t) = 2e–2t A
17
7.2 The Source-Free
RL Circuit (9)
Example 9
Assume the switch in the circuit was open for a long time,
find io, vo, and i for all time.
18
7.3 Singularity Functions (1)
• The singularity functions 奇 異 函 數 are
functions that either are discontinuous or have
discontinuous derivatives.
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7.3 Singularity Functions (2)
• The unit step function單位步階函數 u(t) is 0
for negative values of t and 1 for positive
values of t.
 0,
u (t )  
1,
t0
 0,
u (t  to )  
1,
t  to
 0,
u(t  to )  
1,
t   to
t0
t  to
t   to
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7.3 Singularity Functions (3)
Represent an abrupt change for:
1. voltage source.
2. for current source:
21
7.3 Singularity Functions (4)
• The unit impulse function單位脈衝函數  (t) is
zero everywhere except at t = 0, where is
undefined.
0,
d

 (t )  u (t )  Undefined
dt
0,

t0
t 0
t 0
0
0  (t ) dt  1

b
a f (t ) (t  t0 ) dt  f (t0 )
where a  t0  b
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7.3 Singularity Functions (5)
• The unit ramp function單位斜率函數 r(t) is zero
for negative t and has a unit slope for positive
values of t.
 0,
r (t )  
t ,
t0
t0
 0,
r (t  t0 )  
t  t0 ,
du (t )
dr (t )
 (t ) 
, u (t ) 
dt
dt
t  t0
t  t0
t
t


u(t )    (t )dt , r (t )   u(t )dt
23
7.3 Singularity Functions (6)
Example 10
Express the voltage pulse in the figure in terms of the unit
step. Calculate its derivative and sketch it.
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7.3 Singularity Functions (7)
Example 11
Express the current pulse in the figure in terms of the unit
step. Find its integral and sketch it.
25
7.3 Singularity Functions (8)
Example 12
Express the sawtooth function in the figure in terms of the
singularity functions.
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7.3 Singularity Functions (9)
Example 13
Given the signal
t0
 3,

g(t )   2, 0  t  1
 2t  4, t  1

express g(t) in terms of step and ramp functions.
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7.3 Singularity Functions (10)
Example 14
Evaluate the following integrals involving the impulse function:


10
0


(t 2  4t  2) (t  2)dt
[ (t  1)e  t cos t   (t  1)e  t sin t ]dt
28
7.4 The Step-Response
of a RC Circuit (1)
• The step response of a circuit is its behavior when the
excitation is the step function, which may be a voltage
or a current source.
• Initial condition:
v(0-) = v(0+) = V0
• Applying KCL,
dv v  Vs u (t )
c 
0
dt
R
or
v  Vs u (t )
dv

dt
RC
• Where u(t) is the unit-step function
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7.4 The Step-Response
of a RC Circuit (2)
• Integrating both sides and considering the initial
conditions, the solution of the equation is:
t0
V0
v(t )  
t / 
V

(
V

V
)
e
0
s
 s
Final value
at t -> ∞
Initial value
at t = 0
t 0
Source-free
Response
完整響應
自然響應
激勵響應
Complete Response = Natural response + Forced Response
(stored energy)
(independent source)
=
V0e–t/τ
+
Vs(1–e–t/τ)
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7.4 The Step-Response
of a RC Circuit (3)
Three steps to find out the step response
of an RC circuit:
1. The initial capacitor voltage v(0).
2. The final capacitor voltage v() — DC
voltage across C.
3. The time constant .

v (t )  v ()  [v (0 )  v ()] e
steady-state
response
穩態響應
t / 
transient response
暫態響應
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7.4 The Step-Response
of a RC Circuit (4)
Example 15
The switch has been in position A for a long time.
At t = 0, the switch move to B. Find v(t) for t > 0
in the circuit and calculate v(t) at t = 1 s and 4 s.
32
7.4 The Step-Response
of a RC Circuit (5)
Example 16
Find v(t) for t > 0 in the circuit in below.
Assume the switch has been open for a long
time and is closed at t = 0.
Calculate v(t) at t = 0.5.
• Please refer to lecture or textbook for more detail elaboration.
Answer:
v(t )  15e 2t  5 and v(0.5) = 0.5182V
33
7.4 The Step-Response
of a RC Circuit (6)
Example 17
In the figure, switch has been close for a long
time and is opened at t = 0. Find i and v for all
time.
34
7.5 The Step-response
of a RL Circuit (1)
• The step response of a circuit is its behavior when the
excitation is the step function, which may be a voltage or
a current source.
•
Initial current
i(0-) = i(0+) = Io
•
Final inductor current
i(∞) = Vs/R
•
Time constant  = L/R
Vs
Vs
i( t ) 
 ( I o  )e
R
R

t

35
7.5 The Step-Response
of a RL Circuit (2)
Three steps to find out the step response
of an RL circuit:
1. The initial inductor current i(0) at t = 0+.
2. The final inductor current i().
3. The time constant .

i (t )  i ()  [i (0 )  i ()] e
t / 
Note: The above method is a short-cut method. You may also
determine the solution by setting up the circuit formula directly
using KCL, KVL , ohms law, capacitor and inductor VI laws.
36
7.5 The Step-Response
of a RL Circuit (4)
Example 18
Find i(t) in the circuit for t > 0. Assume that the
switch has been closed for a long time.
37
7.5 The Step-Response
of a RL Circuit (5)
Example 19
The switch in the circuit shown below has been
closed for a long time. It opens at t = 0.
Find i(t) for t > 0.
• Please refer to lecture or textbook for more detail elaboration.
Answer:
i(t )  2  e 10t
38
7.5 The Step-Response
of a RL Circuit (6)
Example 20
At t = 0, switch 1 is closed, and switch 2 is
closed 4 s later. Find i(t) for t > 0. Calculate i
for t = 2 s and t = 5 s.
39
7.6 First-Order Op Amp
Circuits (1)
Example 21
For the op amp circuit, find vo for t > 0, given
that that v(0) = 3 V. Let Rf = 80 k, R1 = 20 k,
and C = 5 μF.
40
7.6 First-Order Op Amp
Circuits (2)
Example 22
Determine v(t) and vo(t) in the circuit.
41
7.6 First-Order Op Amp
Circuits (3)
Example 23
Find the step response vo(t) for t > 0 in the op amp. Let
vi = 2u(t) V, R1 = 20 k, Rf = 50 k, R2 = R3 = 10 k,
and C = 2 μF.
42
7.7 Applications (1)
Delay Circuits
Example 24
Consider the delay circuit, and assume that R1 = 1.5 M,
0 < R2 < 2.5 M. (a) Calculate the extreme limits of the
time constant of the circuit. (b) How long does it take
for the lamp to glow for the first time after the switch is
closed? Let R2 assume its largest value.
43
7.7 Applications (2)
Photoflash Unit
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7.7 Applications (3)
Example 25
An electric flashgun has a current-limiting 6-k
resistor and 2000-μF electrolytic capacitor charged to
240 V. If the lamp resistor is 12 , find (a) the peak
charging current, (b) the time required for the
capacitor to fully charge, (c) the peak discharging
current, (d) the total energy stored in the capacitor,
and (e) the average power dissipated by the lamp.
45
7.7 Applications (4)
Relay Circuits
Example 26
The coil of a certain relay is operated by a 12-V battery.
If the coil has a resistance of 150  and an inductance
of 30 mH and the current needed to put in is 50 mA,
calculate the relay delay time.
46
7.7 Applications (5)
Automobile Ignition Circuit
Example 27
A solenoid with resistance 4  and an inductance 6 mH
is used in an automobile ignition circuit. If the battery
supplies 12-V, determine: the final current through the
solenoid when the switch is closed, the energy stored in
the coil, and the voltage across the air gap, assume that
47
the switch takes 1 μs to open.