Natural Response – Overdamped Example Given V 0 = 12 V and I 0 = 30 mA, find v(t) for t ≥ 0. You can solve this problem using the Second-Order Circuits table: 1. Make sure you are on the Natural Response side. 2. Find the parallel RLC column. 3. Use the equations in Row 4 to calculate and 0 . 4. Compare the values of and 0 to determine the response form (given in one of the last 3 rows). 5. Use the equations to solve for the unknown coefficients. 6. Write the equation for v(t), t ≥ 0. 7. Solve for any other quantities requested in the problem.
Transcript
Natural Response – Overdamped Example
Given V0 = 12 V and I0 = 30 mA, find v(t)for t ≥ 0.
You can solve this problem using the Second-Order Circuits table:
1. Make sure you are on the Natural Response side.2. Find the parallel RLC column.3. Use the equations in Row 4 to calculate and 0.4. Compare the values of and 0 to determine the
response form (given in one of the last 3 rows).5. Use the equations to solve for the unknown coefficients.6. Write the equation for v(t), t ≥ 0.7. Solve for any other quantities requested in the problem.
The values of the __________ determine whether the response is overdamped, underdamped, or critically damped
A. Initial conditions
B. R, L, and C components
C. Independent sources
Natural Response of Parallel RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.
Recap:
teBteBtv
jAjAteAAte
tjteAtjteAtv
xjxexjxe
eeAeeAeAeAtv
js
eAeAtv
LCRCsLCsRCs
d
t
d
t
d
t
d
t
dd
t
dd
t
jxjx
tjttjttjtj
dd
tsts
dddd
sincos)(
)(sin)(cos
)sin(cos)sin(cos)(
sincos;sincos
)(
:
)(:
1
2
1;;0)1()1(
21
2121
21
21
)(
2
)(
1
22
02,1
2
0
2
21
2
0
2
2
0
2
2,1
2
21
:identity sEuler' Note
where so d,underdampe
so ,overdamped
; 0
When the response is underdamped, the voltage is given by the equation
In this equation, the coefficients B1 and B2 are
A. Real numbers
B. Imaginary numbers
C. Complex conjugate numbers
teBteBtv d
t
d
t sincos)( 21
Natural Response – Underdamped Example
Given V0 = 0 V and I0 = 12.25 mA, find v(t) for t ≥ 0.
08.979sin8.979cos
sincos)(
8.979)200()1000(
1000)125.0)(8(
11
200)125.0)(000,20(2
1
2
1
200
2
200
1
21
222
0
2
tteBteB
teBteBtv
LC
RC
tt
d
t
d
t
d
o
V,
rad/s
case! dunderdampe the is this so
rad/s
rad/s
2
2
0
Now we evaluate v(0) and dv(0)/dt from the equation for v(t), and set those values equal to v(0) and dv(0)/dt from the circuit, solving for B1
and B2. The values for v(0) and dv(0)/dt from the circuit do not depend on whether the response is overdamped, underdamped, or critically damped.
A. True
B. False
Natural Response – Underdamped Example
0
0)0(
)0(8.979sin)0(8.979cos)0(
1
0
1
)0(200
2
)0(200
1
B
Vv
BeBeBv
V :Circuit
:Equation
Given V0 = 0 V and I0 = 12.25 mA, find v(t) for t ≥ 0.
Natural Response – Underdamped Example
V/s
V/s
:Circuit
:Equation
000,988.979200
000,98000,20
0)01225.0(
125.0
1
1)0(
1)0(
8.979200
)0(8.979cos8.979)0(8.979sin)200(
)0(8.979sin8.979)0(8.979cos)200()0(
21
00
2121
)0(200
2
)0(200
2
)0(200
1
)0(200
1
BB
R
VI
Ci
Cdt
dv
BBBB
eBeB
eBeBdt
dv
CC
d
Given V0 = 0 V and I0 = 12.25 mA, find v(t) for t ≥ 0.
Natural Response – Underdamped Example
08.979sin100)(
100000,981
8.979200)0(
;0)0(
sincos)(
200
200
2121
01
21
ttetv
BRVIC
BBBBdt
dv
VBv
teBteBtv
t
d
d
t
d
t
V,
Given V0 = 0 V and I0 = 12.25 mA, find v(t) for t ≥ 0.
Natural Response of Parallel RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.
Recap:
0:
sincos)(:
)(:
1
2
1;;0)1()1(
2,1
2
0
2
22
0
21
2
0
2
21
2
0
2
2
0
2
2,1
2
21
s
teBteBtv
eAeAtv
LCRCsLCsRCs
d
d
t
d
t
tsts
so damped, Critically
where
so d,underdampe
so ,overdamped
; 0
When the response is critically damped, a reasonable expression for the voltage is
A. True
B. False
0)( 21 teAeAtv tt V,
Natural Response of Parallel RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.
When the circuit’s response is critically damped, the assumed form of the solution we have been using up until now does not provide enough unknown coefficients to satisfy the two initial conditions from the circuit. Therefore, we use a different solution form:
tttsts eDteDeDteDtv
2121
2
0
2
21)(
: so damped Critically
Natural Response – Critically damped Example
Given V0 = 50 V and I0 = 250 mA, find v(t) for t ≥ 0.
0)(
500)10)(4.0(
11
500)10)(100(2
1
2
1
500
2
500
121
2
teDteDeDteDtv
LC
RC
tttt
o
V,
case! damped critically the is this so
rad/s
rad/s
2
0
Natural Response – Critically damped Example
5050)0(
)0()0(
20
2
)0(500
2
)0(500
1
DVv
DeDeDv
V :Circuit
:Equation
Given V0 = 50 V and I0 = 250 mA, find v(t) for t ≥ 0.
Use the initial conditions from the equation and from the circuit to solve for the unknown coefficients.
Natural Response – Critically damped Example
V/s
V/s
:Circuit
:Equation
000,75500
000,75100
5025.0
10
1
1)0(
1)0(
500
)500()0)(500()0(
21
00
21
)0(500
2
)0(500
1
)0(500
1
DD
R
VI
Ci
Cdt
dv
DD
eDeDeDdt
dv
CC
Given V0 = 50 V and I0 = 250 mA, find v(t) for t ≥ 0.
Natural Response – Critically damped Example
050000,50)(
000,50000,781
500)0(
;50)0(
)(
500500
100
2121
02
500
2
500
1
tetetv
DRVIC
DDDDdt
dv
VDv
eDteDtv
tt
tt
V,
Given V0 = 50 V and I0 = 250 mA, find v(t) for t ≥ 0.
Natural Response of Parallel RLC Circuits – Summary
Use the Second-Order Circuits table:1. Make sure you are on the Natural Response side.2. Find the parallel RLC column.3. Use the equations in Row 4 to calculate and 0.4. Compare the values of and 0 to determine the
response form (given in one of the last 3 rows).5. Use the equations to solve for the unknown coefficients.6. Write the equation for v(t), t ≥ 0.7. Solve for any other quantities requested in the problem.
The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.
