1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is...
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Transcript of 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is...
![Page 1: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/1.jpg)
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SECOND ORDERSECOND ORDER
Examples Examples
![Page 2: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/2.jpg)
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SECOND ORDERSECOND ORDERWhat Circuit do we require and why? What Circuit do we require and why? 1. Circuit at t = 01. Circuit at t = 0--
This circuit is required to find the initial values of i(0-) and/or v(0-). These values are used to form the first equation of finding A1 and A2.
3. Circuit for t > 0 3. Circuit for t > 0
This circuit is required to find the initial values of di(0+)/dt and/or dv(0+)/dt. These values are used to form the second equation of finding A1 and A2.
2. Circuit at t = 02. Circuit at t = 0++
a) Source free circuitWe can divide this circuit into two:
This circuit is used to find the natural response of the circuit.
b) Circuit at t = infinityThis circuit is used to find the steady-state response of the circuit.
Superposition theorem allows us to solve separately and then combine the solution in (a) and (b) together and to get the complete solution easily.
![Page 3: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/3.jpg)
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Example 1
(i) Find v(t) for t > 0.(ii) Find i(t) for t >0.
vi
24 V
R = 6 1 H
0.25 F
+
-
1
t = 0
The switch has been closed for a long time and it is open at t = 0.
![Page 4: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/4.jpg)
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v
i
24V
5
1 H
0.25 F
+
-
1
t = 0
Example 2
t = 0
(i) Find i(t) for t > 0.(ii) Find v(t) for t >0.
The switches have been opened for a long time and they are closed at t = 0.
![Page 5: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/5.jpg)
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(i) Find i(t) for t > 0.(ii) Find iR(t) for t >0.
Example 3
vi
24 V
R = 1 1 H
0.25 F
+
-
1
t = 0
iR
The switch has been opened for a long time and it is closed at t = 0.
![Page 6: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/6.jpg)
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Example 1 : Solution
(i) Find v(t) for t > 0.(ii) Find i(t) for t >0.
vi
24 V
R = 6 1 H
0.25 F
+
-
1
t = 0
The switch has been closed for a long time and it is open at t = 0.
![Page 7: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/7.jpg)
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1. Draw the circuit for t = 0-
This statement means that the circuit is in steady state at t = 0-. Therefore, C is opened and L is shorted.
“The switch has been closed for a long time and it is open at t = 0”
24 V
6
1
i(0) = 3.429 A
v (0) = 3.429 V
+
-
Find i(0) and v(0)
Steps to solve this problem
![Page 8: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/8.jpg)
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2. Draw the circuit for t = 0+
This is a starting point for the circuit to experience transient. Therefore, C is not opened and L is not shorted.
We know that i(0-) = i(0+) and v(0-) = v(0+)
24 V
6
i(0+) = 3.429 A0.25 F
+
-
v(0+) = 3.429 V
Find dv(0+)/dt or/and di(0+)/dt
1 H
![Page 9: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/9.jpg)
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24 V
6
i(0+) = 3.429 A0.25 F
+
-
v(0+) = 3.429 V
Find dv(0+)/dt or/and di(0+)/dt
1 H
sVC
idt
dv /714.1325.0
429.3)0()0(
Lv
dtdi L )0()0(
+ -vL(0+)
-24+ 6(3.429)+vL(0+)+3.429 = 0vL(0+) = 0
KVL around the loop
0)0()0(
Lv
dtdi L
![Page 10: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/10.jpg)
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3. Draw the circuit for t = ∞At t = ∞ the circuit has reached the steady state again. Therefore, C is open and L is shorted.
24 V
6
+
-
Find v(∞) or/and i(∞)
v(∞) = 24 V
i(∞) = 0
![Page 11: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/11.jpg)
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4. Draw the source free circuit for t >0
1H 6
+
-
Voltage source is shorted and current source is opened.
iv0.25F
Generally for this step, we have to find the differential equation for the source free circuit, then its characteristic equation. Since the circuit is a series RLC, we can directly write down its characteristic equation, and determine the type of the response.
![Page 12: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/12.jpg)
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For second order circuit:
02 22 oss
012 LC
sLRs
For second order circuit, its characteristics equation is
General form
Series RLC circuit
3)1(2
62
L
R 225.01
11
LCo rad/srad/s
o the response is overdamped
the roots are real and distincts1
2o
2 s1 0.764
s2 2 o2 s2 5.236
![Page 13: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/13.jpg)
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5. Write the general solution for the circuit for t > 0.
24 V
6
i0.25 F
+
-
v
1H
tsts eAeAvtv 2121)()( tsts eBeBiti 21
21)()( or
tsts eAeA 212124 tsts eBeB 21
210
![Page 14: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/14.jpg)
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6. Find A1 and A2
24 V
6
i0.25 F
+
-
v
1H
21)(429.3)0( AAvv 2211
)0( AsAsdt
dv
21 236.5764.0714.13 AA 2124429.3 AA
448.02 A019.211 A
1 2
tt eetv 236.5764.0 448.00198.2124)(
![Page 15: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/15.jpg)
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7. Find other circuit quantities for t > 0.
24V
6
i0.25 F
+
-
v
1H
tt eedtd
dttdvCti 236.5764.0 448.00198.212425.0)()(
tt eeti 236.5764.0 )448.0(236.5)764.0(019.2125.0)( tt ee 236.5764.0 586.0015.4
tt eetv 236.5764.0 448.00198.2124)(
![Page 16: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/16.jpg)
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Verification Using PSpice
24V
6
i0.25 F
+
-
v
1H
0
1 2 3
-+
1h 2h
E1h G2h
1 1
Hand Calculation Verification: Example 1**Overdamped response: Case 1.Param R=6 L=1 C=0.25**************************************************************************************E1h 1h 0 value={24-21.019*EXP(-0.764*TIME)+0.448*EXP(-5.236*TIME)}R1h 1h 0 1 ; V(1h) = output voltage****************************************************************************************************************************************************************************G2h 0 2h value={4.015*EXP(-0.764*TIME)-0.586*EXP(-5.236*TIME)}R2h 2h 0 1 ; V(2h) = output current***************************************************************************************V1 1 0 DC 24VR1 1 2 {R}L1 2 3 {L} IC=3.429AC1 3 0 {C} IC=3.429V.Tran 5m 5s 0 5m UIC.Probe V(1h) V(3) .Probe I(R2h) I(L1).End
![Page 17: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/17.jpg)
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(i) Find i(t) for t > 0.(ii) Find v(t) for t >0.
1
Example 2 : Solution
vi
24 V
5
1 H
0.25 F
+
-
t = 0
t = 0
The switches have been opened for a long time and they are closed at t = 0.
![Page 18: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/18.jpg)
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1. Draw the circuit for t = 0-
This statement means that the circuit is in steady state at t = 0-. Therefore, C is opened and L is shorted.
“The switch have been opened for a long time and it is closed at t = 0”
24 V
5
1 v(0-) = 24 V
+
-
Find i(0) and v(0)
Steps to solve this problem
i(0-) = 0 A
![Page 19: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/19.jpg)
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2. Draw the circuit for t = 0+
This is a starting point for the circuit to experience transient. Therefore, C is not open and L is not shorted.
We know that i(0-) = i(0+) and v(0-) = v(0+)
24 Vi(0+) = 0
0.25 F
+
-v(0+) = 24 V
Then we can find dv(0+)/dt or/and di(0+)/dt
1
1H
![Page 20: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/20.jpg)
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i(0+) = 00.25 F
+
-v(0+) = 24 V 1
1H
iR(0+) = 24/1 = 24 AiC(0+)
iC(0+) = -iR(0+) = -24 A
sVC
idt
dv C /9625.024)0()0(
Lv
dtdi L )0()0(
and from the circuit 0)0( Lv
0
![Page 21: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/21.jpg)
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3. Draw the circuit for t = ∞At t = ∞ the circuit reaches steady state again. Therefore, C is open and L is shorted.
Find v(∞) or/and i(∞)
24 Vi(∞) = 24 A +
-
v(∞) = 24 V 1
![Page 22: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/22.jpg)
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4. Draw the source free circuit for t >0Voltage source is shorted and current source is open.
i(t)0.25 F
+
-
v(t) 1
1H
Generally for this step, we have to find the differential equation for the source free circuit, then its characteristic equation. Since the circuit is a parallel RLC, we can directly write down its characteristic equation, and determine the type of the response.
![Page 23: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/23.jpg)
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For second order circuit:
02 22 oss
0112 LC
sRC
s
For second order circuit, its characteristics equation is
General form
Parallel RLC circuit
2)25.0)(1(2
12
1
RC 2
25.0111
LC
o rad/srad/s
o the response is critically damped
21 s
22 sThe roots are real and equal
![Page 24: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/24.jpg)
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5. Write the general solution for the circuit for t > 0.
24 Vi(t)
0.25 F
+
-
v(t) 1
1H
tetAAiti )()()( 21
tetAAti 221 )(24)(
![Page 25: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/25.jpg)
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6. Find A1 and A2
24 Vi(t)
0.25 F
+
-
v(t) 1
1H
1240)0( Ai 241 A
21)0( AA
dtdi
02)0(21 AA
dtdi 48)24)(2(2 12 AA
tt etetti 22 )4824(24)4824(24)(
![Page 26: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/26.jpg)
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7. Find other circuit quantities for t > 0.
24 Vi(t)
0.25 F
+
-
v(t) 1
1H
tL et
dtditv 2)4824(241)(
tetti 2)4824(24)(
)48()4824(21)( 22 ttL eettv
tL tetv 248)(
tL tetvtv 24824)(24)(
![Page 27: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/27.jpg)
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(i) Find i(t) for t > 0.(ii) Find iR(t) for t >0.
vi
24 V
R = 1 1 H
0.25 F
+
-
1
t = 0
Example 3: Solution
iR
The switch has been opened for a long time and it is closed at t = 0.
![Page 28: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/28.jpg)
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1. Draw the circuit for t = 0-
This statement means that the circuit is in steady state at t = 0-. Therefore, C is opened and L is shorted.
“The switch has been opened for a long time and it is closed at t = 0”
24 V
1 i(0-) = 0
v(0-) = 24 V
+
-
Find i(0) and v(0)
Steps to solve this problem
![Page 29: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/29.jpg)
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2. Draw the circuit for t = 0+
This is a starting point for the circuit to experience transient. Therefore, C is not opened and L is not shorted.
We know that i(0-) = i(0+) and v(0-) = v(0+)
24 Vi(0+) = 0
0.25 F
+
-
v(0+) = 24
Then we can find dv(0+)/dt or/and di(0+)/dt
1
1H
1 iR(0+) = 24A
![Page 30: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/30.jpg)
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24 Vi(0+) = 0
0.25 F+
-v(0+) = 24 1
1H
1 iR(0+) = 24A
sVC
idt
dv C /9625.024)0()0(
iC(0+)
iC(0+) = -iR(0+)
![Page 31: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/31.jpg)
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3. Draw the circuit for t = ∞At t = ∞ the circuit reaches steady state again. Therefore, C is open and L is shorted.
Find v(∞) or/and i(∞)
24 Vi(∞) = 12 A +
-
v(∞) = 12V 1
1
![Page 32: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/32.jpg)
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4. Draw the source free circuit for t >0.Voltage source is shorted and current source is opened.
Since this circuit is neither a parallel nor a series RLC. This circuit must be a general second order. So, we have to find the differential equation in terms of i(t) or v(t).
Is this a series RLC circuit? Is this a parallel RLC circuit?
i(t)0.25 F
+
-
v(t) 1
1H
1
![Page 33: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/33.jpg)
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i(t)0.25 F
+
-
v(t) 1
1H
1
Applying KCL at node a:
a
dtdvvi
41
1
Applying KVL to the left mesh:
0411
41
2
2
vdt
vddtdv
dtdvv
011 vdtdii
0852
2
vdtdv
dtvd
![Page 34: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/34.jpg)
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For second order circuit:
02 22 oss
0852 ss
For second order circuit, its characteristics equation is
General form
The circuit under-study
5.2 828.28 o rad/srad/s
o the response is underdamped
djs 1
djs 2
The roots are complex conjugate
d o2 2 d 1.323
![Page 35: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/35.jpg)
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5. Write the general solution for the circuit for t > 0.
24 Vi(t)
0.25 F
+
-
v(t) 1
1H
1
tdd etAtAvtv )sin()cos()()( 21
tetAtAtv 5.221 )323.1sin()323.1cos(12)(
![Page 36: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/36.jpg)
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6. Find A1 and A2
24 Vi(t)
0.25 F
+
-
v(t) 1
1H1
2412)0( 1 Av
tetAtAtv 5.221 )323.1sin()323.1cos(12)(
121 A
12)0( AA
dtdv
d
12 5.2323.196 AA 89.492 A
tetttv 5.2)323.1sin(89.49)323.1cos(1212)(
![Page 37: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/37.jpg)
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7. Find other circuit quantities for t > 0.
24 Vi(t)
0.25 F+
-v(t) 1
1 H1
tetttv 5.2)323.1sin(89.49)323.1cos(1212)(
iC(t)
tC ett
dtdti 5.2)323.1sin(89.49)323.1cos(121225.0)(
iR(t)
1)()( tvtiR )()()( tititi RC
![Page 38: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/38.jpg)
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Example 4 t = 0
5 H
3
8 A 0.05 F
i(t)
-
+
v(t)1
Find v(t) for t >0
![Page 39: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/39.jpg)
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Example 5
6
4u(t)
0.1 F1 H v(t)
+
-
i(t)
Find i(t) for t >0
![Page 40: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/40.jpg)
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Example 1a
A series RLC circuit has R = 10 k, L = 0.1 mH, and C = 10 F. What type of damping is exhibited by the circuit.
Example 2a
A parallel RLC circuit has R = 10 k, L = 0.1 mH, and C = 10 F. What type of damping is exhibited by the circuit.
![Page 41: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/41.jpg)
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Example 4aThe responses of a series RLC circuit are
ttL
ttc
eeti
eetv1020
1020
3040)(
301030)(
Determine the values of R, L, and C.
V
mA
(a) Overdamped (b) Critically damped(c) Underdamped
If R = 20 , L = 0.6 H, what value of C will make an RLC series circuit:
Example 3a
![Page 42: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/42.jpg)
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Example 5a
Find i(t) in the circuit of Fig. 8.10. Assume that the circuit has reached steady state at t = 0-.Fig 8.10
![Page 43: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/43.jpg)
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Example 8.9
• Find v(t) for t > 0 in Fig. 8.29.
![Page 44: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/44.jpg)
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Example 8.9
• Find v(t) for t > 0 in Fig. 8.29.
![Page 45: 1 SECOND ORDER Examples. 2 What Circuit do we require and why? 1. Circuit at t = 0 - This circuit is required to find the initial values of i(0 - ) and/or.](https://reader036.fdocuments.net/reader036/viewer/2022081605/5a4d1ae87f8b9ab059979cbd/html5/thumbnails/45.jpg)
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Problem 8.56
+ v -
iR
i
0.75
0.25 H
4
0.04 F
20 V
Find iR(t), v(t), and i(t) for t > 0.
Given v(0) = 0 and i(0) = 0