1. 2nd Order Circuits
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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.
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3. Important Concepts The differential equation
Forced and homogeneous solutions
The natural frequency and the damping ratio
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4. A 2nd Order RLC Circuit R + - C
vs(t)
i (t) L
The source and resistor may be equivalent to a circuit with many resistors and sources.
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5. Applications Modeled by a 2nd Order RLC Circuit
Filters
A bandpass filter such as the IF amp for the AM radio.
A lowpass filter with a sharper cutoff than can be obtained with an RC circuit.
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6. The Differential Equation
vr(t)
i (t) + - R + +
vc(t)
vs(t) C - -
vl(t) + - L
KCL around the loop:
vr(t) + vc(t) + vl(t) = vs(t)
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7. Differential Equation
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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:
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9. Important Concepts The differential equation
Forced and homogeneous solutions
The natural frequency and the damping ratio
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10. The Particular Solution
The particular 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.
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11. The Complementary Solution
The complementary 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.
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12. Complementary Solution
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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.
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14. Complementary Solution
Each root (s1 and s2) contribute a term to the complementary solution.
The complementary solution is (usually)
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15. Important Concepts The differential equation
Forced and homogeneous solutions
The natural frequency and the damping ratio
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16. Damping Ratio and Natural Frequency
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.
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17. Roots of the Characteristic Equation
The roots of the characteristic equation determine whether the complementary solution wiggles.
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18. Real Unequal Roots If  > 1, s1 and s2 are real and not equal.
This solution is over damped.
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19. Over Damped
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20. Complex Roots If  < 1, s1 and s2 are complex.
Define the following constants:
This solution is under damped.
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21. Under Damped
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22. Real Equal Roots If  = 1, s1 and s2 are real and equal.
This solution is critically damped.
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23. Example
i (t)
10W +
vs(t)
769pF -
159mH
This is one possible implementation of the filter portion of the IF amplifier.
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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.
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27. Slightly Different Example
i (t)
1kW +
vs(t)
769pF -
159mH
Increase the resistor to 1kW
What are z and w0?
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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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