Natural and Step Response of Series & Parallel RLC ... · PDF fileNatural and Step Response of...
Transcript of Natural and Step Response of Series & Parallel RLC ... · PDF fileNatural and Step Response of...
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Natural and Step Response of Series & Parallel RLC Circuits (Second-order Circuits)
Objectives: Determine the response form of the circuit Natural response parallel RLC circuits Natural response series RLC circuits Step response of parallel and series RLC circuits
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Natural Response of Parallel RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.
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It is convenient to calculate v(t) for this circuit because
A. The voltage must be continuous for all time
B. The voltage is the same for all three components
C. Once we have the voltage, it is pretty easy to calculate the branch current
D. All of the above
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Natural Response of Parallel RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.
0)(1
)(1)(
0)(1
)(1)(
0)(
)(1)(
2
2
2
2
0
0
dt
tdv
RCtv
LCdt
tvdC
dt
tdv
Rtv
Ldt
tvdC
R
tvIdxxv
Ldt
tdvC
t
:form standard in place to by sides both Divide
:integral the remove to sides both ateDifferenti
:KCL
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Natural Response of Parallel RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.
0)(1
)(1)(
2
2
dt
tdv
RCtv
LCdt
tvd:equation Describing
This equation is Second order Homogeneous Ordinary differential equation With constant coefficients
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Once again we want to pick a possible solution to this differential equation. This must be a function whose first AND second derivatives have the same form as the original function, so a possible candidate is
A. Ksin t
B. Keat
C. Kt2
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Natural Response of Parallel RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.
0)(1
)(1)(
2
2
dt
tdv
RCtv
LCdt
tvd:equation Describing
The circuit has two initial conditions that must be satisfied, so the solution for v(t) must have two constants. Use
0)]1()1([)]1()1([
0)(1
)(1
)(
)(
21
212121
21
22
2
211
2
1
2122112
2
21
2
1
21
tsts
tstststststs
tsts
eALCsRCseALCsRCs
eAeALC
eAseAsRC
eAseAs
eAeAtv :SubstituteV;
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Natural Response of Parallel RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.
0)1()1(
)(
)(1)(
2
21
21
2
2
21
LCsRCs
ss
eAeAtv
tvLCdt
tvd
tsts
:EQUATION STICCHARACTERI the for solutions are and Where
:Solution
0dt
dv(t)
RC
1:equation Describing
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is called the “characteristic equation” because it characterizes the circuit.
A. True
B. False
0)1()1(2 LCsRCs
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Natural Response of Parallel RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.
rad/s) in frequency radian resonant (the and
rad/s) in frequency neper (the where
0LC
RC
LCRCRCs
LCRCRCsLCsRCs
1
2
1
)1()21()21(
2
)1(4)1()1(;0)1()1(
2
0
22
2,1
2
2,1
2
The two solutions to the characteristic equation can be calculated using the quadratic formula:
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So far, we know that the parallel RLC natural response is given by
A. The value of
B. The value of 0
C. The value of (2 - 02)
and where 0LCRC
s
eAeAtv tsts
1
2
1
)(
2
0
2
2,1
2121
There are three different forms for s1 and s2. For a parallel RLC circuit with specific values of R, L and C, the form for s1 and s2 depends on
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Natural Response – Overdamped Example
Given V0 = 12 V and I0 = 30 mA, find v(t) for t ≥ 0.
rad/s rad/s,
case! overdamped the is this so
rad/s
rad/s
2
0
000,2050007500000,12
)000,10()500,12(500,12
000,10)2.0)(05.0(
11
500,12)2.0)(200(2
1
2
1
21
222
0
2
2,1
2
ss
s
LC
RC
o
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Natural Response – Overdamped Example
Given V0 = 12 V and I0 = 30 mA, find v(t) for t ≥ 0.
circuit the in equation the in
circuit the in equation the in
:circuit the in conditions initial the
satisfy to equation the in tscoefficien the use must we Now
for V,
00
00
000,20
2
5000
1
)()(
)()(
0)(
tt
tt
tt
dt
tdv
dt
tdv
tvtv
teAeAtv
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Natural Response – Overdamped Example
Given V0 = 12 V and I0 = 30 mA, find v(t) for t ≥ 0.
21
)0(000,20
2
)0(5000
1
21
0
21
)0(000,20
2
)0(5000
1
000,205000
)000,20()5000()0(
12
12)0(
)0(
AA
eAeAdt
dv
AA
Vv
AAeAeAv
:Equation
V :Circuit
:Equation
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Now we need the initial value of the first derivative of the voltage from the circuit. The describing equation of which circuit component involves dv(t)/dt?
A. The resistor
B. The inductor
C. The capacitor
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Natural Response – Overdamped Example
Given V0 = 12 V and I0 = 30 mA, find v(t) for t ≥ 0.
000,450000,205000
000,450200
1203.0
2.0
1)0()0(
1)0(
))0()0((1
)0(1)0()(
)(
000,205000
)000,20()5000()0(
21
21
)0(000,20
2
)0(5000
1
AA
R
vi
Cdt
dv
iiC
iCdt
dv
dt
tdvCti
AA
eAeAdt
dv
L
RLCC
V/s
:Circuit
:Equation
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Natural Response – Overdamped Example
Given V0 = 12 V and I0 = 30 mA, find v(t) for t ≥ 0.
(OK)
(OK) V :Checks
for V,
Thus,
usly,simultaneo Solving
for V,
0)(
122614)0(
02614)(
26,14
000,450000,205000;12
0)(
000,205000
21
2121
000,20
2
5000
1
v
v
teetv
AA
AAAA
teAeAtv
tt
tt
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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.
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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.
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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
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When the response is critically damped, a reasonable expression for the voltage is
A. True
B. False
0)( 21 teAeAtv tt V,
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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
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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
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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.
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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.
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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.
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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.
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Step Response of Second-order RLC Circuits
The problem – find the response for t ≥ 0. Note that there may or may not be initial energy stored in the inductor and capacitor!
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The circuit for the parallel RLC step response is repeated here. Consider how this circuit behaves as t ∞. Which component’s final value is non-zero?
A. The resistor
B. The inductor
C. The capacitor
D. All of the above
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Step Response of a Parallel RLC Circuits
As t ∞:
response) natural the of form (the FL Iti )(
The only component whose final value is NOT zero is the inductor, whose final current is the current supplied by the source. We now have to construct a response form that can satisfy two initial conditions and one non-zero final value. We can satisfy the final value directly if we specify the inductor current as the response we will solve for:
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Step Response of a Parallel RLC Circuit
The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.
To begin, find the initial conditions and the final value. The initial conditions for this problem are both zero; the final value is found by analyzing the circuit as t ∞.
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Step Response of a Parallel RLC Circuit
The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.
t ∞:
mA 24FI
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Step Response of a Parallel RLC Circuit
The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.
Next, calculate the values of and 0 and determine the form of the response:
rad/s nm
rad/s n
000,40)25)(25(11
000,50)25)(400(2
1
2
1
0
LC
RC
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We just calculated = 50,000 rad/s and 0 = 40,000 rad/s, so the form of the response is
A. Overdamped
B. Underdamped
C. Critically damped
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Once we know the response form is overdamped, we know we have to calculate
A. d
B. s1 and s2
C. Nothing additional
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Step Response of a Parallel RLC Circuit
The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.
Since the response form is overdamped, calculate the values of s1 and s2:
0024.0)(
000,80000,20
000,30000,50
000,40000,50000,50
000,80
2
000,20
1
21
222
0
2
2,1
teAeAti
ss
s
tt
L A,
rad/s and rad/s
rad/s
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Step Response of a Parallel RLC Circuit
The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.
Next, set the values of i(0) and di(0)/dt from the equation equal to the values of i(0) and di(0)/dt from the circuit.
0)0(
024.0)0(
0
21
Ii
AAi
L
L
:circuit the From
:equation the From
0024.0)( 000,80
2
000,20
1 teAeAti tt
L A,
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Step Response of a Parallel RLC Circuit
The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.
0)0()0(
000,80000,20)0(
0
21
L
V
L
v
dt
di
AAdt
di
LL
L
:circuit the From
:equation the From
0024.0)( 000,80
2
000,20
1 teAeAti tt
L A,
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Step Response of a Parallel RLC Circuit
The problem – there is no initial energy stored in this circuit; find i(t) for t ≥ 0.
083224)(
832
0000,80000,20
0024.0
000,80000,20
21
21
21
teeti
AA
AA
AA
tt
L mA,
mA mA;
and
:Solve
0024.0)( 000,80
2
000,20
1 teAeAti tt
L A,
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We can check this result at t = 0 and as t ∞; from the equation we get
A. iL(0)=32 mA, iL(∞)=0
B. iL(0)=0, iL(∞)=0
C. iL(0)=0, iL(∞)=24 mA
083224)( 000,80000,20 teeti tt
L mA,
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If we now want to find v(t) for t ≥ 0, we need to
A. Find the derivative of the current and multiply by L
B. Find the integral of the current and divide by C
C. Multiply the current by R
083224)( 000,80000,20 teeti tt
L mA,
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Step Response of RLC Circuits – Summary
Use the Second-Order Circuits table: 1. Make sure you are on the Step Response side. 2. Find the appropriate column for the RLC circuit topology. 3. Find the values of the two initial conditions and the one
non-zero final value.Make sure the initial conditions and final value are defined exactly as shown in the figure!
4. Use the equations in Row 4 to calculate and 0. 5. Compare the values of and 0 to determine the
response form (given in one of the last 3 rows). 6. Write the equation for vC(t), t ≥ 0 (series) or iL(t), t ≥ 0
(parallel), leaving only the 2 coefficients unspecified. 7. Use the equations provided to solve for the unknown
coefficients. 8. Solve for any other quantities requested in the problem.
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Step Response of a Series RLC Circuit
The problem –find vC(t) for t ≥ 0.
Find the initial conditions by analyzing the circuit for t < 0:
A
V
V k k
k
0
50
)80(159
15
0
0
I
V
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Step Response of a Series RLC Circuit
The problem –find vC(t) for t ≥ 0.
Find the final value of the capacitor voltage by analyzing the circuit as t ∞:
V 100FV
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Step Response of a Series RLC Circuit
The problem –find vC(t) for t ≥ 0.
Use the circuit for t ≥ 0 to find the values of and 0:
rad/s
dunderdampe
rad/s
rad/s
0
6000
8000000,10
000,10
)2)(005.0(11
8000)005.0(2802
2222
0
2
0
2
d
LC
LR
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Step Response of a Series RLC Circuit
The problem –find vC(t) for t ≥ 0.
Write the equation for the response and solve for the unknown coefficients:
06000sin67.666000cos50100)(
67.6650
060008000)0(
50100)0(
06000sin6000cos100)(
80008000
21
210
21
101
8000
2
8000
1
ttetetv
BB
BBC
IBB
dt
dv
BVBVv
tteBteBtv
tt
C
dC
FC
tt
C
V,
V V,
V,
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Natural Response of Series RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find i(t) for t ≥ 0.
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The difference(s) between the analysis of series RLC circuit and the parallel RLC circuit is/are:
A. The variable we calculate.
B. The describing differential equation.
C. The equations for satisfying the initial conditions
D. All of the above
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Natural Response of Series RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find i(t) for t ≥ 0.
0)(1)()(
0)(
)(1)(
0)()(1)(
2
2
2
2
0
0
tiLCdt
tdi
L
R
dt
tidL
dt
tdiRti
Cdt
tidL
tRiVdxxiCdt
tdiL
t
:form standard in place to by sides both Divide
:integral the remove to sides both ateDifferenti
:KVL
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The describing differential equation for the series RLC circuit is Therefore, the characteristic equation is
A. s2 + (1/RC)s + 1/LC = 0
B. s2 + (R/L)s + 1/LC = 0
C. s2 + (1/LC)s + 1/RC = 0
0)(1)()(
2
2
tiLCdt
tdi
L
R
dt
tid
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Natural Response of Series RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find i(t) for t ≥ 0.
rad/s) in frequency radian resonant (the and
rad/s) in frequency neper (the where
0LC
L
R
sLCsLRs
1
2
;0)1()( 2
0
2
2,1
2
The two solutions to the characteristic equation can be calculated using the quadratic formula:
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Natural Response Series RLC Problems
You can solve these problem using the Second-Order Circuits table:
1. Make sure you are on the Natural Response side. 2. Find the series 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 i(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 i(t) for t ≥ 0.
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Natural Response Series RLC Example
The capacitor is charged to 100 V and at t = 0, the switch closes. Find i(t) for t ≥ 0.
09600sin9600cos)(
9600
000,10)1.0)(1.0(
11
2800)1.0(2
560
2
2800
2
2800
1
22
0
2
tteBteBti
LC
L
R
tt
d
o
A,
rad/s
case! dunderdampe the is this so
rad/s
rad/s
2
0
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Natural Response Series RLC Example
The capacitor is charged to 100 V and at t = 0, the switch closes. Find i(t) for t ≥ 0.
circuit the in equation the in
circuit the in equation the in
:circuit the in conditions initial the
satisfy to equation the in tscoefficien the use must we Now
A,
00
00
2800
2
2800
1
)()(
)()(
09600sin9600cos)(
tt
tt
tt
dt
tdi
dt
tdi
titi
tteBteBti
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The following quantities used to calculate the unknown coefficients are defined by different equations in both the series and parallel RLC natural response problems:
A. The initial values of voltage or current from the equation.
B. The initial values of voltage or current from the circuit.
C. The initial values of the derivative of voltage or current from the equation.
D. The initial values of the derivative of voltage or current from the circuit.
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Natural Response Series RLC Example
0
0)0(
)0(
09600sin9600cos)(
1
0
1
2800
2
2800
1
B
Ii
Bi
tteBteBti tt
:Circuit
)case! parallel the as (same :Equation
A,
The capacitor is charged to 100 V and at t = 0, the switch closes. Find i(t) for t ≥ 0.
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Natural Response Series RLC Example
09600sin104.0)(
104.010009600)0(2800
1000)0(560)100(1.0
1
1)0()0(
1)0(
1)0(
)0(
2800
22
00
21
tteti
BB
RIVL
vvL
vLdt
di
BBdt
di
t
RCL
d
A,
A/s
:Circuit
)case! parallel the as (same :Equation
The capacitor is charged to 100 V and at t = 0, the switch closes. Find i(t) for t ≥ 0.
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Natural Response of RLC Circuits – Summary
Use the Second-Order Circuits table: 1. Make sure you are on the Natural Response side. 2. Find the appropriate column for the RLC circuit topology. 3. Make sure the initial conditions are defined exactly as
shown in the figure! 4. Use the equations in Row 4 to calculate and 0. 5. Compare the values of and 0 to determine the
response form (given in one of the last 3 rows). 6. Use the equations to solve for the unknown coefficients. 7. Write the equation for v(t), t ≥ 0 (parallel) or i(t), t ≥ 0
(series). 8. Solve for any other quantities requested in the problem.