1. Second-Order Circuits (6.3)
Dr. Holbert
November 20, 2001
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.
ECE201 Lect-23

3. Important Concepts The differential equation
Forced and homogeneous solutions
The natural frequency and the damping ratio
ECE201 Lect-23

4. A 2nd Order RLC Circuit
i (t) R + –
vs(t) C L
The source and resistor may be equivalent to a circuit with many resistors and sources.
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.
ECE201 Lect-23

6. The Differential Equation
vr(t)
i (t) + – R + + –
vc(t)
vs(t) C –
vl(t) – + L
KVL around the loop:
vr(t) + vc(t) + vl(t) = vs(t)
ECE201 Lect-23

7. Differential Equation
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:
ECE201 Lect-23

9. Important Concepts The differential equation
Forced and homogeneous solutions
The natural frequency and the damping ratio
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.
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.
ECE201 Lect-23

12. Complementary Solution
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.
ECE201 Lect-23

14. Complementary Solution
Each root (s1 and s2) contributes a term to the complementary solution.
The complementary solution is (usually)
ECE201 Lect-23

15. Important Concepts The differential equation
Forced and homogeneous solutions
The natural frequency and the damping ratio
ECE201 Lect-23

16. Damping Ratio () and Natural 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 w0
It determines how fast sinusoids wiggle.
ECE201 Lect-23

17. Roots of the Characteristic Equation
The roots of the characteristic equation determine whether the complementary solution wiggles.
ECE201 Lect-23

18. Real Unequal Roots If  > 1, s1 and s2 are real and not equal.
This solution is overdamped.
ECE201 Lect-23

19. Overdamped
ECE201 Lect-23

20. Complex Roots If  < 1, s1 and s2 are complex.
Define the following constants:
This solution is underdamped.
ECE201 Lect-23

21. Underdamped
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22. Real Equal Roots If  = 1, s1 and s2 are real and equal.
This solution is critically damped.
ECE201 Lect-23

23. Example
i (t)
10W + –
vs(t)
769pF
159mH
This is one possible implementation of the filter portion of the IF amplifier.
ECE201 Lect-23

24. More of the Example
For the example, what are z and w0?
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25. Even More Example z = 0.011 w0 = 2p455000
Is this system over damped, under damped, or critically damped?
What will the current look like?
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26. Example (cont.)
The shape of the current depends on the initial capacitor voltage and inductor current.
ECE201 Lect-23

27. Slightly Different Example
i (t)
1kW + –
vs(t)
769pF
159mH
Increase the resistor to 1kW
What are z and w0?
ECE201 Lect-23

28. More Different Example
z = 2.2
w0 = 2p455000
Is this system over damped, under damped, or critically damped?
What will the current look like?
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29. Example (cont.)
The shape of the current depends on the initial capacitor voltage and inductor current.
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30. Damping Summary
ECE201 Lect-23

31. Class Examples Learning Extension E6.9 Learning Extension E6.10
ECE201 Lect-23