Transcript Chapter 7

Chapter 7
First-Order Circuit
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Items:
1. RC and RL Circuits
2. First-order Circuit Complete Response
3. Initial and Final Conditions
4. First-order Circuit Sinusoidal Response
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1. RC and RL Circuits
Two major steps in the analysis of a
dynamic circuit
1. use device and connection equations
to formulate a differential equation.
2. solve the differential equation to find
the circuit response.
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FORMULATING RC AND RL CIRCUIT
EQUATIONS
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Eq.(7-1)
Eq.(7-2)
RC
Eq.(7-3)
Eq.(7-4)
Eq.(7-5)
RL
Eq.(7-6)
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ZERO-INPUT RESPONSE OF FIRST-ORDER
CIRCUITS
RC Circuit:
makes VT=0 in Eq.(7-3)
we find the zero-input response
Eq.(7-7)
Eq.(7-7) is a homogeneous equation because the right side is zero.
A solution in the form of an exponential
Eq.(7-8)
where K and s are constants to be determined
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Substituting the trial solution into Eq.(7-7) yields
OR
characteristic equation
Eq.(7-9)
a single root of the characteristic
equation
zero -input response of the RC circuit:
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Eq.(7-10)
time constant
TC=RTC
Fig. 7-3: First-order RC circuit zero-input response
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Graphical determination of the time constant
T from the response curve
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RL Circuit:
Eq.(7-11)
Eq.(7-12)
The root of this equation
The final form of the zero-input response of the RL circuit is
Eq.(7-13)
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EXAMPLE 7-1
The switch in Figure 7- 4 is closed at t=0, connecting a capacitor with an initial voltage of 30V to the
resistances shown. Find the responses v C(t), i(t), i1(t) and i2(t) for t
0.
Fig. 7-4
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SOLUTION:
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EXAMPLE 7-2
Find the response of the state variable of the RL circuit in Figure 7-5 using L1=10mH, L2=30mH, R1=2k
ohm, R2=6k ohm, and iL(0)=100mA
Fig. 7-5
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SOLUTION:
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2. First-order Circuit Complete Response
When the input to the RC circuit is a step function**
Eq.(7-15)
The response is a function v(t) that satisfies this differential
equation for t 0 and meets the initial condition v(0).
If v(0)=0, it is Zero-State Response.
Since u(t)=1 for t  0 we can write Eq.(7-15) as
Eq.(7-16)
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divide solution v(t) into two components:
natural
response
forced
response
The natural response is the general solution of Eq.(7-16)
when the input is set to zero.
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The forced response is a particular solution of
Eq.(7-16) when the input is step function.
seek a particular solution of the equation
Eq.(7-19)
The equation requires that a linear combination of VF(t) and its
derivative equal a constant VA for t  0. Setting VF(t)=VA
meets this condition since . Substituting VF=VA into Eq.(7-19)
reduces it to the identity VA=VA.
Now combining the forced and natural responses, we obtain
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using the initial condition:
 K=(VO-VA)
The complete response of the RC circuit:
Eq.(7-20)
The zero-state response of the
RC circuit:
v(t )  VA (1  e
t / RT C
t0
Fig. 7-12: Step response of firstorder RC circuit
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)
the initial and final values of the response are
The RL circuit in Figure 7-2 is the dual of the RC circuit
Eq.(7-21)
Setting iF=IA
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The constant K is now evaluated from the initial condition:
The initial condition requires that K=IO-IA, so the complete
response of the RL circuit is
Eq.(7-22)
The zero-state response of the RC circuit:
i(t )  I A (1  e
t / GN C
)
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The complete response of a firstorder circuits depends on three
quantities:
1. The amplitude of the step input (VA or IA)
2. The circuit time constant(RTC or GNL) 
3. The value of the state variable at t=0 (VO or IO)
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EXAMPLE 7-4
Find the response of the RC circuit in Figure 7-13
SOLUTION:
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EXAMPLE 7-5 Find the complete response of the RL circuit
in Figure 7-14(a). The initial condition is i(0)=IO
Fig. 7-14
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EXAMPLE 7-6
The state variable response of a first-order RC circuit
for a step function input is
(a) What is the circuit time constant?
(b) What is the initial voltage across the capacitor?
(c) What is the amplitude of the forced response?
(d) At what time is VC(t)=0?
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SOLUTION:
(a) The natural response of a first-order circuit is of the
form
. Therefore, the time constant of the given responses
is Tc=1/200=5ms
(b) The initial (t=0) voltage across the capacitor is
(c) The natural response decays to zero, so the forced response
is the final value vC(t).
(d) The capacitor voltage must pass through zero at some
intermediate time, since the initial value is positive and the final
value negative. This time is found by setting the step response
equal to zero:
which yields the condition
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COMPLETE RESPONSE
The first parts of the above equations are Zero-input response
and the second parts are Zero-state response.
What is s step response?
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EXAMPLE 7-7
Find the zero-state response of the RC circuit of Figure 7-15(a)
for an input
Fig. 7-15
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The first input causes a zero-state response of
The second input causes a zero-state response of
The total response is the superposition of these two responses.
Figure 7-15(b) shows how the two responses combine to produce
the overall pulse response of the circuit. The first step function
causes a response v1(t) that begins at zero and would eventually
reach an amplitude of +VA for t>5RC. However, at t=T<5TC the
second step function initiates an equal and opposite response v2(t).
For t> T+5RC the second response reaches its final state and
cancels the first response, so that total pulse response returns to
zero.
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3. Initial and Final Conditions
Eq.(7-23)
the general form :
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The state variable response in switched dynamic
circuits is found using the following steps:
STEP 1: Find the initial value by applying dc analysis to the
circuit configuration for t<0
STEP 2: Find the final value by applying dc analysis to the
circuit configuration for t>0.
STEP 3: Find the time constant TC of the circuit in the
configuration for t>0
STEP 4: Write the step response directly using Eq.(7-23)
without formulating and solving the circuit differential equation.
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Example: The switch in Figure 718(a) has been closed for a long
time and is opened at t=0. We
want to find the capacitor voltage
v(t) for t0
Fig. 7-18: Solving a switched dynamic circuit using
the initial and final conditions
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There is another way to find the nonstate variables.
Generally, method of “three quantities” can be applied in step
response on any branch of First-order circuit.
1. Get f(0) from initial value of state variable
2. Get f()---use equivalent circuit
3. Get TC---calculate the equivalent resistance Re,
TC=ReC or L/ Re
Then,
f (t )  ( f (0)  f ())e
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
t
TC
 f ()
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How to get initial value f(0)?
1. the capacitor voltage and inductor current are always
continuous in some condition. Vc(0+)=Vc(0-);
IL(0+)=IL(0-)
2. ---use 0+ equivalent circuit C: substituted by voltage
source; L: substituted by current source
3. Find f(0) in the above DC circuit.
How to get final value f(∞)?
Use ∞ equivalent circuit(stead state) to get f(∞). C:
open circuit; L: short circuit
How to get time constant TC?
The key point is to get the equivalent resistance Re.
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f (t )  ( f (0)  f ())e
natural
response
f (t )  f (0)e

t
TC

t
TC
 f ()
forced
response
 f ()(1  e
Zero-input
response

t
TC
)
Zero-state
response
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EXAMPLE 7-8
The switch in Figure 7-20(a) has
been open for a long time and is
closed at t=0. Find the inductor
current for t>0.
SOLUTION:
Fig. 7-20
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EXAMPLE 7-9 The switch in Figure
7-21(a) has been closed for a long time
and is opened at t=0. Find the voltage
vo(t)
Fig. 7-21
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another way to solve the problem:
R2VA
V0 (0)  
R1  R2
V0 ()  0 TC  R2C
V0 (t )  (V0 (0)  V0 ())e
R2VA

e
R1  R2


t
TC
 V0 ()
t
R2C
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4. First-Order Circuit Sinusoidal Response
If the input to the RC circuit is a casual sinusoid
Eq.(7-24)
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where
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EXAMPLE 7-12
The switch in Figure 7-26 has been open for a long time and is
closed at t=0. Find the voltage v(t) for t 0 when vs(t)=[20 sin
1000t]u(t)V.
SOLUTION:
Fig. 7-26
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Summary
•Circuits containing linear resistors and the equivalent of one
capacitor or one inductor are described by first-order differential
equations in which the unknown is the circuit state variable.
•The zero-input response in a first-order circuit is an exponential
whose time constant depends on circuit parameters. The amplitude
of the exponential is equal to the initial value of the state variable.
•The natural response is the general solution of the homogeneous
differential equation obtained by setting the input to zero. The
forced response is a particular solution of the differential equation
for the given input. For linear circuits the total response is the sum
of the forced and natural responses.
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Summary
•For linear circuits the total response is the sum of the zero-input
and zero-state responses. The zero-input response is caused by the
initial energy stored in capacitors or inductors. The zero-state
response results form the input driving forces.
•The initial and final values of the step response of a first and
second-order circuit can be found by replacing capacitors by open
circuits and inductors by short circuits and then using resistance
circuit analysis methods.
•For a sinusoidal input the forced response is called the sinusoidal
steady-state response, or the ac response. The ac response is a
sinusoid with the same frequency as the input but with a different
amplitude and phase angle. The ac response can be found from the
circuit differential equation using the method of undetermined
coefficients
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