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First-Order Circuit

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Chapter 7 First-Order Circuit 1. RC and RL Circuits FORMULATING RC AND RL CIRCUIT EQUATIONS 2. First-order Circuit Complete Response 3. Initial and Final Conditions 4. – PowerPoint PPT presentation

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Title: First-Order Circuit


1
Chapter 7
  • First-Order Circuit

2
Items
  • RC and RL Circuits
  • First-order Circuit Complete Response
  • Initial and Final Conditions
  • First-order Circuit Sinusoidal Response

3
1. RC and RL Circuits
Two major steps in the analysis of a dynamic
circuit
  • use device and connection equations to formulate
    a differential equation.
  • solve the differential equation to find the
    circuit response.

4
FORMULATING RC AND RL CIRCUIT EQUATIONS
5
Eq.(7-1)
Eq.(7-2)
RC
Eq.(7-3)
Eq.(7-4)
Eq.(7-5)
RL
Eq.(7-6)
6
ZERO-INPUT RESPONSE OF FIRST-ORDER CIRCUITS
RC Circuit
makes VT0 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
7
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8
Eq.(7-10)
time constant TCRTC
Fig. 7-3 First-order RC circuit zero-input
response
9
Graphical determination of the time constant T
from the response curve
10
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)    
11
EXAMPLE 7-1 The switch in Figure 7- 4 is closed
at t0, connecting a capacitor with an initial
voltage of 30V to the resistances shown. Find the
responses vC(t), i(t), i1(t) and i2(t) for t  0.
 
Fig. 7-4
12
SOLUTION
13
EXAMPLE 7-2 Find the response of the state
variable of the RL circuit in Figure 7-5 using
L110mH, L230mH, R12k ohm, R26k ohm, and
iL(0)100mA
Fig. 7-5
14
SOLUTION
15
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)  
16
divide solution v(t) into two components
The natural response is the general solution of
Eq.(7-16) when the input is set to zero.
17
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 VFVA into Eq.(7-19)
reduces it to the identity VAVA.
Now combining the forced and natural responses,
we obtain
18
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
t?0
Fig. 7-12 Step response of first-order RC
circuit
19
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 iFIA
20
The constant K is now evaluated from the initial
condition                                     
                    
The initial condition requires that KIO-IA, so
the complete response of the RL circuit is
Eq.(7-22)  
The zero-state response of the RC circuit
t?0
21
The complete response of a first-order circuits
depends on three quantities
  • The amplitude of the step input (VA or IA)
  • The circuit time constant(RTC or GNL) ?
  • The value of the state variable at t0 (VO or IO)

22
EXAMPLE 7-4 Find the response of the RC circuit
in Figure 7-13
SOLUTION
23
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24
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
25
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26
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?
27
SOLUTION (a) The natural response of a
first-order circuit is of the form        .
Therefore, the time constant of the given
responses is Tc1/2005ms (b) The initial (t0)
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                         
28
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?
29
EXAMPLE 7-7 Find the zero-state response of the
RC circuit of Figure 7-15(a) for an input
                                               
Fig. 7-15
30
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 tgt5RC.
However, at tTlt5TC the second step function
initiates an equal and opposite response v2(t).
For tgt T5RC the second response reaches its
final state and cancels the first response, so
that total pulse response returns to zero.
31
3. Initial and Final Conditions
Eq.(7-23)
the general form                              
                                                  
                               
32
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 tlt0
STEP 2 Find the final value by applying dc
analysis to the circuit configuration for
tgt0. STEP 3 Find the time constant TC of the
circuit in the configuration for tgt0 STEP 4
Write the step response directly using Eq.(7-23)
without formulating and solving the circuit
differential equation.
33
Example The switch in Figure 7-18(a) has been
closed for a long time and is opened at t0. 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
34
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35
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,
    TCReC or L/ Re

Then,
36
  • How to get initial value f(0)?
  • the capacitor voltage and inductor current are
    always continuous in some condition.
    Vc(0)Vc(0-) IL(0)IL(0-)
  • ---use 0 equivalent circuit C substituted by
    voltage source L substituted by current
    source
  • Find f(0) in the above DC circuit.

How to get final value f(8)? Use 8 equivalent
circuit(stead state) to get f(8). C open
circuit L short circuit
How to get time constant TC? The key point is to
get the equivalent resistance Re.
37
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38
EXAMPLE 7-8 The switch in Figure 7-20(a) has been
open for a long time and is closed at t0. Find
the inductor current for tgt0.
SOLUTION
Fig. 7-20
39
EXAMPLE 7-9 The switch in Figure 7-21(a) has
been closed for a long time and is opened at t0.
Find the voltage vo(t)
Fig. 7-21
40
another way to solve the problem
41
4. First-Order Circuit Sinusoidal Response
If the input to the RC circuit is a casual
sinusoid
Eq.(7-24)
42
where
43
EXAMPLE 7-12 The switch in Figure 7-26 has been
open for a long time and is closed at t0. Find
the voltage v(t) for t  0 when vs(t)20 sin
1000tu(t)V.
SOLUTION
Fig. 7-26
44
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.

45
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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