Natural and Step Response of Series & Parallel RLC Circuits (Second-order Circuits)
Objectives:Determine the response form of the circuitNatural response parallel RLC circuitsNatural response series RLC circuitsStep response of parallel and series RLC circuits
Natural Response of Parallel RLC Circuits
The problem – given initial energy stored in the inductor and/or capacitor, find v(t) for t ≥ 0.
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
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
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 isSecond orderHomogeneousOrdinary differential equationWith constant coefficients
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
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
)(
)(
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2
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2
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2
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tstststststs
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eALCsRCseALCsRCs
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eAseAsRC
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eAeAtv :SubstituteV;
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
is called the “characteristic equation” because it characterizes the circuit.
A. True
B. False
0)1()1(2 LCsRCs
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:
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 s1and s2 depends on
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
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)000,10()500,12(500,12
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11
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1
2
1
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0
2
2,1
2
ss
s
LC
RC
o
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
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satisfy to equation the in tscoefficien the use must we Now
for V,
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00
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2
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)()(
)()(
0)(
tt
tt
tt
dt
tdv
dt
tdv
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teAeAtv
Natural Response – Overdamped Example
Given V0 = 12 V and I0 = 30 mA, find v(t)for t ≥ 0.
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)0(000,20
2
)0(5000
1
21
0
21
)0(000,20
2
)0(5000
1
000,205000
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12
12)0(
)0(
AA
eAeAdt
dv
AA
Vv
AAeAeAv
:Equation
V :Circuit
:Equation
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
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
Natural Response – Overdamped Example
Given V0 = 12 V and I0 = 30 mA, find v(t)for t ≥ 0.
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(OK) V :Checks
for V,
Thus,
usly,simultaneo Solving
for V,
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0)(
000,205000
21
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000,20
2
5000
1
v
v
teetv
AA
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teAeAtv
tt
tt
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.