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ELECTRICALCIRCUIT II
Chapter 1
First-Order Circuits
Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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First-Order Circuits
Chapter 11.1 The Source-Free RC Circuit
1.2 The Source-Free RL Circuit
1.3 Unit-step Function
1.4 Step Response of an RC Circuit
1.5 Step Response of an RL Circuit
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1.1 The Source-FreeRC Circuit (1)
A first-order circuit is characterized by a first-order differential equation.
Apply Kirchhoffs laws to purely resistive circuit results inalgebraic equations.
Apply the laws to RC and RL circuits produces differential
equations.
Ohms law Capacitor law
0dt
dvC
R
v0 CR iiBy KCL
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1.1 The Source-FreeRC 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.
The time constant of a circuit is the time required for theresponse to decay by a factor of1/e or 36.8% of its initial value.
v decays faster for small and slower for large .
CRTime constant
Decays more slowly
Decays faster
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1.1 The Source-FreeRC Circuit (3)
The key to working with a source-free RC circuit is
finding:
1. The initial voltage v(0) = V0 across thecapacitor.
2. The time constant = RC.
/0)(t
eVtv CRwhere
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1.1 The Source-FreeRC Circuit (4)
Example 1
Refer to the circuit below, determine vC, vx, andio for t 0.
Assume that vC(0) = 30 V.
Please refer to lecture or textbook for more detail elaboration.
Answer: vC= 30e0.25tV; vx= 10e
0.25t; io = 2.5e0.25tA
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1.1 The Source-FreeRC Circuit (5)
Example 2
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) = 8e2tV
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1.2 The Source-FreeRL Circuit (1)
A first-order RL circuit consists of a inductorL (or its equivalent) and a resistor (or itsequivalent)
0RL vvBy KVL
0 iRdt
diL
Inductors law Ohms law
dtL
R
i
di
LtReIti
/
0)(
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1.2 The Source-FreeRL Circuit (2)
The time constant of a circuit is the time required for the responseto decay by a factor of1/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.
/
0)(
teIti
R
L
A general form representing a RL
where
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1.2 The Source-FreeRL Circuit (3)
/
0)( teItiR
L
A RL source-free circuit
where /0)( teVtv RC
A RC source-free circuit
where
Comparison between a RL and RC circuit
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1.2 The Source-FreeRL Circuit (4)
The key to working with a source-free RLcircuit is finding:
1. The initial voltage i(0) = I0 through theinductor.
2. The time constant = L/R.
/
0)(t
eIti
R
Lwhere
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1.2 The Source-FreeRL Circuit (5)
Example 3
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) = 5e53t
A
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1.2 The Source-FreeRL Circuit (6)
Example 4
For the circuit, find i(t) for t > 0.
Please refer to lecture or textbook for more detail elaboration.
Answer: i(t) = 2e2t
A
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1.3 Unit-Step Function (1)
The unit step functionu(t) is 0 for negative
values oftand 1 for positive values oft.
0,1
0,0)(
t
t
tu
o
o
o
tt
ttttu
,1
,0)(
o
o
o
tt
ttttu
,1
,0)(
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1.3 Unit-Step Function (2)
1. voltage source.
2. for current source:
Represent an abrupt change for:
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Initial condition:
v(0-) = v(0+) = V0
Applying KCL,
or
Where u(t) is the unit-step function
1.4 The Step-Responseof a RC Circuit (1)
The step response of a circuit is its behavior when theexcitation is the step function, which may be a voltageor a current source.
0)(
R
tuVv
dt
dvc
s
)(tuRC
Vv
dt
dvs
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1.4 The Step-Responseof a RC Circuit (2)
Integrating both sides and considering the initialconditions, the solution of the equation is:
0)(0)(
/
0
0
teVVV
tVtvt
ss
Final valueat t ->
Initial valueat t = 0
Source-freeResponse
Complete Response = Natural response + Forced Response(stored energy) (independent source)
= V0et/ + Vs(1e
t/)
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1.4 The Step-Responseof a RC Circuit (3)
Three steps to find out the step responseof an RC circuit:
1. The initial capacitor voltage v(0).
2. The final capacitor voltage v() DCvoltage across C.
3. The time constant.
/)]()0([)()( tevvvtv
Note: The above method is a short-cut method. You may alsodetermine the solution by setting up the circuit formula directly
using KCL, KVL , ohms law, capacitor and inductor VI laws.
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Example 5
Find v(t) for t > 0 in the circuit in below.Assume the switch has been open for a longtime and is closed at t = 0.
Calculate v(t) at t = 0.5.
1.4 The Step-Responseof a RC Circuit (4)
Please refer to lecture or textbook for more detail elaboration.
Answer: and v(0.5) = 0.5182V515)(
2 tetv
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1.5 The Step-responseof a RL Circuit (1)
The step response of a circuit is its behavior when theexcitation is the step function, which may be a voltage ora current source.
Initial currenti(0-) = i(0+) = Io
Final inductor currenti() = Vs/R
Time constant = L/R
)()()( tueR
VI
R
Vti
t
s
o
s
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1.5 The Step-Responseof 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.
Note: The above method is a short-cut method. You may alsodetermine the solution by setting up the circuit formula directly
using KCL, KVL , ohms law, capacitor and inductor VI laws.
/
)]()0([)()(t
eiiiti
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Example 6
The switch in the circuit shown below has beenclosed for a long time. It opens at t = 0.
Find i(t) for t > 0.
1.5 The Step-Responseof a RL Circuit (4)
Please refer to lecture or textbook for more detail elaboration.
Answer: teti102)(
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