Lect5EEE 2021 Loop (Mesh) Analysis Dr. Holbert January 30, 2008.
ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.
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Transcript of ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.
![Page 1: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/1.jpg)
ECE201 Lect-23 1
Second-Order Circuits (6.3)
Dr. Holbert
November 20, 2001
![Page 2: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/2.jpg)
ECE201 Lect-23 2
2nd Order Circuits
• Any circuit with a single capacitor, a single inductor, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 2.
• Any voltage or current in such a circuit is the solution to a 2nd order differential equation.
![Page 3: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/3.jpg)
ECE201 Lect-23 3
Important Concepts
• The differential equation
• Forced and homogeneous solutions
• The natural frequency and the damping ratio
![Page 4: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/4.jpg)
ECE201 Lect-23 4
A 2nd Order RLC Circuit
• The source and resistor may be equivalent to a circuit with many resistors and sources.
R
Cvs(t)
i (t)
L
+–
![Page 5: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/5.jpg)
ECE201 Lect-23 5
Applications Modeled by a 2nd Order RLC Circuit
• Filters
– A lowpass filter with a sharper cutoff than can be obtained with an RC circuit.
![Page 6: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/6.jpg)
ECE201 Lect-23 6
The Differential Equation
KVL around the loop:
vr(t) + vc(t) + vl(t) = vs(t)
R
Cvs(t)
+
–
vc(t)
+ –vr(t)
L
+– vl(t)
i (t)
+–
![Page 7: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/7.jpg)
ECE201 Lect-23 7
Differential Equation
)()(1)(
)( tvdxxiCdt
tdiLtRi s
t
dt
tdv
Lti
LCdt
tdi
L
R
dt
tid s )(1)(
1)()(2
2
![Page 8: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/8.jpg)
ECE201 Lect-23 8
The Differential Equation
Most circuits with one capacitor and inductor are not as easy to analyze as the previous circuit. However, every voltage and current in such a circuit is the solution to a differential equation of the following form:
)()()(
2)( 2
002
2
tftidt
tdi
dt
tid
![Page 9: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/9.jpg)
ECE201 Lect-23 9
Important Concepts
• The differential equation
• Forced and homogeneous solutions
• The natural frequency and the damping ratio
![Page 10: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/10.jpg)
ECE201 Lect-23 10
The Particular Solution
• The particular (or forced) solution ip(t) is usually a weighted sum of f(t) and its first and second derivatives.
• If f(t) is constant, then ip(t) is constant.
• If f(t) is sinusoidal, then ip(t) is sinusoidal.
![Page 11: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/11.jpg)
ECE201 Lect-23 11
The Complementary Solution
The complementary (homogeneous) solution has the following form:
K is a constant determined by initial conditions.
s is a constant determined by the coefficients of the differential equation.
stc Keti )(
![Page 12: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/12.jpg)
ECE201 Lect-23 12
Complementary Solution
02 2002
2
ststst
Kedt
dKe
dt
Ked
02 200
2 ststst KesKeKes
02 200
2 ss
![Page 13: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/13.jpg)
ECE201 Lect-23 13
Characteristic Equation
• To find the complementary solution, we need to solve the characteristic equation:
• The characteristic equation has two roots-call them s1 and s2.
02 200
2 ss
![Page 14: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/14.jpg)
ECE201 Lect-23 14
Complementary Solution
• Each root (s1 and s2) contributes a term to the complementary solution.
• The complementary solution is (usually)
tstsc eKeKti 21
21)(
![Page 15: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/15.jpg)
ECE201 Lect-23 15
Important Concepts
• The differential equation
• Forced and homogeneous solutions
• The natural frequency and the damping ratio
![Page 16: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/16.jpg)
ECE201 Lect-23 16
Damping Ratio () andNatural Frequency (0)
• The damping ratio is .
• The damping ratio determines what type of solution we will get:
– Exponentially decreasing ( >1)
– Exponentially decreasing sinusoid ( < 1)
• The natural frequency is 0
– It determines how fast sinusoids wiggle.
![Page 17: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/17.jpg)
ECE201 Lect-23 17
Roots of the Characteristic Equation
The roots of the characteristic equation determine whether the complementary solution wiggles.
12001 s
12002 s
![Page 18: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/18.jpg)
ECE201 Lect-23 18
Real Unequal Roots
• If > 1, s1 and s2 are real and not equal.
• This solution is overdamped.
tt
c eKeKti
1
2
1
1
200
200
)(
![Page 19: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/19.jpg)
ECE201 Lect-23 19
Overdamped
0
0.2
0.4
0.6
0.8
1
-1.00E-06
t
i(t)
-0.2
0
0.2
0.4
0.6
0.8
-1.00E-06
ti(t)
![Page 20: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/20.jpg)
ECE201 Lect-23 20
Complex Roots
• If < 1, s1 and s2 are complex.
• Define the following constants:
• This solution is underdamped.
tAtAeti ddt
c sincos)( 21
0 2
0 1 d
![Page 21: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/21.jpg)
ECE201 Lect-23 21
Underdamped
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1.00E-05 1.00E-05 3.00E-05
t
i(t)
![Page 22: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/22.jpg)
ECE201 Lect-23 22
Real Equal Roots
• If = 1, s1 and s2 are real and equal.
• This solution is critically damped.
ttc teKeKti 00
21)(
![Page 23: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/23.jpg)
ECE201 Lect-23 23
Example
• This is one possible implementation of the filter portion of the IF amplifier.
10
769pFvs(t)
i (t)
159H
+–
![Page 24: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/24.jpg)
ECE201 Lect-23 24
More of the Example
dt
tdv
Lti
LCdt
tdi
L
R
dt
tid s )(1)(
1)()(2
2
)()()(
2)( 2
002
2
tftidt
tdi
dt
tid
For the example, what are and 0?
![Page 25: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/25.jpg)
ECE201 Lect-23 25
Even More Example
• = 0.011
• 0 = 2455000
• Is this system over damped, under damped, or critically damped?
• What will the current look like?
![Page 26: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/26.jpg)
ECE201 Lect-23 26
Example (cont.)
• The shape of the current depends on the initial capacitor voltage and inductor current.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1.00E-05 1.00E-05 3.00E-05
t
i(t)
![Page 27: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/27.jpg)
ECE201 Lect-23 27
Slightly Different Example
• Increase the resistor to 1k• What are and 0?
1k
769pFvs(t)
i (t)
159H
+–
![Page 28: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/28.jpg)
ECE201 Lect-23 28
More Different Example
• = 2.2
• 0 = 2455000
• Is this system over damped, under damped, or critically damped?
• What will the current look like?
![Page 29: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/29.jpg)
ECE201 Lect-23 29
Example (cont.)
• The shape of the current depends on the initial capacitor voltage and inductor current.
0
0.2
0.4
0.6
0.8
1
-1.00E-06
t
i(t)
![Page 30: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/30.jpg)
ECE201 Lect-23 30
Damping Summary
ζ Roots (s1, s2) Damping
ζ>1 Real and unequal Overdamped
ζ=1 Real and equal Critically damped
0<ζ<1 Complex Underdamped
![Page 31: ECE201 Lect-231 Second-Order Circuits (6.3) Dr. Holbert November 20, 2001.](https://reader034.fdocuments.in/reader034/viewer/2022052618/5516e225550346fe558b4572/html5/thumbnails/31.jpg)
ECE201 Lect-23 31
Class Examples
• Learning Extension E6.9
• Learning Extension E6.10