SECOND ORDER CIRCUIT

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Things You Should Know
SECOND ORDER CIRCUIT
Source Free Circuits
2
Series RLC Circuit

-
-

vR
vL

vC
We want to solve for i(t).
-
Applying KVL,
Differentiate once,
Characteristic Equation
3
Series RLC Circuit
We want to solve for v(t).

-
-

vR
vL

Applying KVL,
v
-
and
Characteristic Equation
Divided by 1/ LC
4
Parallel RLC Circuit
We want to solve for v(t).
Applying KCL,
Differentiate once,
Divided by C
Characteristic Equation
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Parallel RLC Circuit
We want to solve for i(t).
Applying KVL,
and
Characteristic Equation
Divided by 1/ LC
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General RLC Circuit Type-1
1 W
a
1H

i(t)
v(t)
0.25 F
1 W
-
Applying KCL at node a
Applying KVL to the left mesh
Characteristic Equation
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SECOND ORDER Complete Solution
What Circuit do you require and why?
1. 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.
2. Circuit at 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.
3. Circuit for t 0
We can divide this circuit into two
a) Source free circuit
This circuit is used to find the natural response
of the circuit.
b) Circuit at t infinity
This 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.
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SECOND ORDER
General solutions
To obtain Coefficient A1 and A2
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General Characteristic Equation
for series and
for parallel
10
Three types of response
Once we know the type of the response, we can
write its general solution.
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General Solution for Overdamped Response
General Solution for Critically Damped Response
General solution for Underdamped Response
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To solve for A1 and A2
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Overdamped Response
Expression for v(t) and dv(t)/dt
To solve for A1 and A2
1
2
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Critically Damped Response
Expression for v(t) and dv(t)/dt
To solve for A1 and A2
1
2
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Underdamped Response
Expression for v(t) and dv(t)/dt
To solve for A1 and A2
1
2
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Things You Should Know
RESONANT CIRCUITS
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SERIES RESONANT CIRCUIT
You have to focus on (but not limited to)
What happens to the Z at resonance ?
What happens to the I at resonance ?
What happens to the VR, VL, VC at resonance ?

• Know how to calculate their values
• Know how to draw their Phasor diagram

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SERIES RESONANT CIRCUIT
You have to focus on (but not limited to)
What happens to the Z at w1 ?
What happens to the I at w1?
What happens to the VR, VL, VC at w1 ?

• Know how to calculate their values
• Know how to draw their Phasor diagram

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SERIES RESONANT CIRCUIT
You have to focus on (but not limited to)
What happens to the Z at w2 ?
What happens to the I at w2?
What happens to the VR, VL, VC at w2 ?

• Know how to calculate their values
• Know how to draw their Phasor diagram

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And know how to solve this problem
Vs 10 Vrms, R 10 W, L 100 mH, C 10 mF
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(No Transcript)
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• (xi) The impedance of the circuit at w2 in polar
  form
  

(xii) The current at w2 in polar form
(xiii) The real power P at w2
(xiv) The expression for i(t) at w2
(xv) The expressions for vL(t), vC(t) and
vL(t)vC(t) at w2
(xvi) Draw the voltage phasor diagram at wo
(xvii) Draw the voltage phasor diagram at w1
(xviii) Draw the voltage phasor diagram at w2
(ixx) Draw the waveforms of vC(t), vL(t) and
vC(t)vL(t) at wo (xx) Draw the waveforms of vC(t
), vL(t) and vC(t)vL(t) at w1
(xxi) Draw the waveforms of vL(t), vC(t) and
vL(t)vC(t) at w2
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(xxii) The resonant frequency, fo
(xxiii) The lower cut-off frequency, f1
(xxiv) The upper cut-off frequency, f2
(xxv) The bandwidth, BW in Hertz
(xxvi) The Quality factor, Q
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PARALLEL RESONANT CIRCUIT
You have to focus on (but not limited to)
What happens to the Y at resonance ?
What happens to the V at resonance ?
What happens to the IR, IL, IC at resonance ?

• Know how to calculate their values
• Know how to draw their Phasor diagram

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PARALLEL RESONANT CIRCUIT
You have to focus on (but not limited to)
What happens to the Y at w1 ?
What happens to the V at w1?
What happens to the IR, IL, IC at w1 ?

• Know how to calculate their values
• Know how to draw their Phasor diagram

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PARALLEL RESONANT CIRCUIT
You have to focus on (but not limited to)
What happens to the Y at w2 ?
What happens to the V at w2?
What happens to the IR, IL, IC at w2 ?

• Know how to calculate their values
• Know how to draw their Phasor diagram

27
And know how to solve this problem
Is 0.1 Arms, R 1k W, L 100 mH, C 10 mF
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(No Transcript)
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• (xi) The admittance of the circuit at w2 in polar
  form
  

(xii) The voltage at w2 in polar form
(xiii) The real power P at w2
(xiv) The expression for v(t) at w2
(xv) The expressions for iL(t), iC(t) and
iL(t)iC(t) at w2
(xvi) Draw the current phasor diagram at wo
(xvii) Draw the current phasor diagram at w1
(xviii) Draw the current phasor diagram at w2
(ixx) Draw the waveforms of iC(t), iL(t) and
iC(t)iL(t) at wo (xx) Draw the waveforms of iC(t
), iL(t) and iC(t)iL(t) at w1
(xxi) Draw the waveforms of iL(t), iC(t) and
iL(t)iC(t) at w2
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(xxii) The resonant frequency, fo
(xxiii) The lower cut-off frequency, f1
(xxiv) The upper cut-off frequency, f2
(xxv) The bandwidth, BW in Hertz
(xxvi) The Quality factor, Q