Post on 18-Jan-2018
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SECOND ORDERSECOND ORDER
To obtain Coefficient ATo obtain Coefficient A11 and A and A22
General solutionsGeneral solutions
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02 22 oss
222,1 os
General Characteristic Equation:
LR
2
RC21
LCo1
22 od
for series and for parallel
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Three types of response:Three types of response:
the underdamped response , If 3. , If 2.
the overdamped response , If 1.
0
0
0
the critically damped response
Once we know the type of the response, we Once we know the type of the response, we can write its general solution.can write its general solution.
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General Solution for Overdamped Response:tsts eAeAVtv 21
21)()(
tetAAVtv )()()( 21
)sin()cos()()( 21 tAtAeVtv ddt
General Solution for Critically Damped Response:
General solution for Underdamped Response:
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To solve for ATo solve for A11 and A and A22
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Overdamped Response:
tsts eAeAVtv 2121)()(
21)()0( AAVv
tsts eAseAsdt
tdv21
2211)(
2
1
2211)0( AsAs
dtdv
To solve for ATo solve for A11 and A and A22
Expression for v(t) and dv(t)/dtExpression for v(t) and dv(t)/dt
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Critically Damped Response:
tetAAVtv )()()( 21
2)()0( AVv
21)0( AA
dtdv
1
2
)())(()(221
tt eAetAAdt
tdv
Expression for v(t) and dv(t)/dtExpression for v(t) and dv(t)/dt
To solve for ATo solve for A11 and A and A22
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Underdamped Response:
)sin()cos()()( 21 tAtAeVtv ddt
1)()0( AVv
12)0( AA
dtdv
d
1
2
To solve for ATo solve for A11 and A and A22
)sin()cos(
)cos()sin()(
21
21
tAtAe
tAtAedt
tdv
ddt
ddddt
Expression for v(t) and dv(t)/dtExpression for v(t) and dv(t)/dt
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Problem in mathematics
10)(15)(10)( tx
dttdx
dttdx
A Comparison
2)0(10)0( dt
dxx
Find x(t) for t>0.
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Problem in circuits : Problem 1
A Comparison
Find v(t) for t>0.
24 V
R = 5 1 H
v+
-
i
v (0) = 10Vi (0) = 2A
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What are the differences?
A Comparison
1. The differential equation is not given.
2. v(0+) is given but dv(0+)/dt is not.We have to find dv(0+)/dt from the circuit by usingdv(0+)/dt = i(0+)/C.
We have to find from the circuit, in particular from the source free circuit. For a series circuit, it always has the same form. Memorize it.
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Problem in circuits: Problem 2
A Comparison
The switch has been closed for a long time and it is open at t = 0. Find i(t) for t > 0.
vi
24 V
R = 1 1 H
0.25 F
+
-
1
t = 0
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What are the differences?
A Comparison
1. The differential equation is not given.
2. v(0+) and dv(0+)/dt are not given.We have to find v(0+) and dv(0+)/dt from the circuit. From the given information, the circuit is in steady state at t = 0.
We have to find from the source free circuit.Since this circuit is a series RLC circuit, we can write it directly.
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Problem in circuits: Problem 3
A Comparison
The switch has been opened for a long time and it is closed at t = 0. Find i(t) for t > 0.
vi
24 V
R = 1 1 H
0.25 F
+
-
1
t = 0
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What are the differences?
A Comparison
1. The differential equation is not given.
2. v(0+) and dv(0+)/dt are not given.We have to find v(0+) and dv(0+)/dt from the circuit. From the given information, the circuit is in steady state at t = 0.
We have to find from the source free circuit.Since this circuit is a general second order; like it or not, we have to derive the differential equation from the source free circuit. Then, obtain the characteristic equation.
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(Problem 2)Steps to solve this problem
The switch has been closed for a long time and it is open at t = 0. Find i(t) for t > 0.
vi
24 V
R = 1 1 H
0.25 F
+
-
1
t = 0
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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 open and L is shorted.
“The switch has been closed for a long time and it is open at t = 0”
24 V
1
1
i
v
+
-
Find i(0) and v(0)
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2. Draw the circuit for t = 0+
This is a starting point 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 V
1
i(0+) =0.25 F
+
-
v(0+) =
Find dv(0+)/dt or/and di(0+)/dt
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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.
24 V
1
+
-
Find v(∞) or/and i(∞)
i
v
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4. Draw the source free circuit for t >0
1H 1
+
-
Voltage source is shorted and current source is opened.
i
v0.25F
Find the differential equation for the source free circuit. Then its characteristic equation. Since the circuit is RLC series, we can directly write its characteristic equation. Determine the type of the response.
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5. Write the general solution for the circuit for t > 0.
24 V
1
i0.25 F
+
-
v
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6. Find A1 and A2
24 V
1
i0.25 F
+
-
v
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7. Find other circuit quantities for t > 0.
24 V
1
i0.25 F
+
-
v
+ -vL
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Example 1
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 2
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.
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Example 4The 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 3
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Example 5
Find i(t) in the circuit of Fig. 8.10. Assume that the circuit has reached steady state at t = 0-.Fig 8.10
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Example 8.9
• Find v(t) for t > 0 in Fig. 8.29.
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Example 8.9
• Find v(t) for t > 0 in Fig. 8.29.
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Problem 8.56
• Find i(t) for t > 0 in Fig. 8.102.