1. lecture 181 Second-Order Circuits (6.3) Prof. Phillips April 7, 2003

2. lecture 182 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.

3. lecture 183 Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio

4. lecture 184 A 2nd Order RLC Circuit The source and resistor may be equivalent to a circuit with many resistors and sources. R Cv s (t) i (t) L +–+–

5. lecture 185 Applications Modeled by a 2nd Order RLC Circuit Filters –A lowpass filter with a sharper cutoff than can be obtained with an RC circuit.

6. lecture 186 The Differential Equation KVL around the loop: v r (t) + v c (t) + v l (t) = v s (t) R Cv s (t) + – v c (t) + – v r (t) L +– v l (t) i (t) +–+–

7. lecture 187 Differential Equation

8. lecture 188 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:

9. lecture 189 Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio

10. lecture 1810 The Particular Solution The particular (or forced) solution i p (t) is usually a weighted sum of f(t) and its first and second derivatives. If f(t) is constant, then i p (t) is constant. If f(t) is sinusoidal, then i p (t) is sinusoidal.

11. lecture 1811 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.

12. lecture 1812 Complementary Solution

13. lecture 1813 Characteristic Equation To find the complementary solution, we need to solve the characteristic equation: The characteristic equation has two roots- call them s 1 and s 2.

14. lecture 1814 Complementary Solution Each root (s 1 and s 2 ) contributes a term to the complementary solution. The complementary solution is (usually)

15. lecture 1815 Important Concepts The differential equation Forced and homogeneous solutions The natural frequency and the damping ratio

16. lecture 1816 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  0 –It determines how fast sinusoids wiggle.

17. lecture 1817 Roots of the Characteristic Equation The roots of the characteristic equation determine whether the complementary solution wiggles.

18. lecture 1818 Real Unequal Roots If  > 1, s 1 and s 2 are real and not equal. This solution is overdamped.

19. lecture 1819 Overdamped

20. lecture 1820 Complex Roots If  < 1, s 1 and s 2 are complex. Define the following constants: This solution is underdamped.

21. lecture 1821 Underdamped

22. lecture 1822 Real Equal Roots If  = 1, s 1 and s 2 are real and equal. This solution is critically damped.

23. lecture 1823 Example This is one possible implementation of the filter portion of the IF amplifier. 10  769pFv s (t) i (t) 159  H +–+–

24. lecture 1824 More of the Example For the example, what are  and  0 ?

25. lecture 1825 Example continued  = 0.011  0 = 2  455000 Is this system over damped, under damped, or critically damped? What will the current look like?

26. lecture 1826 Example (cont.) The shape of the current depends on the initial capacitor voltage and inductor current.

27. lecture 1827 Slightly Different Example Increase the resistor to 1k  What are  and  0 ? 1k  769pFv s (t) i (t) 159  H +–+–

28. lecture 1828 Example cont.  = 2.2  0 = 2  455000 Is this system over damped, under damped, or critically damped? What will the current look like?

29. lecture 1829 Example (cont.) The shape of the current depends on the initial capacitor voltage and inductor current.

30. lecture 1830 Damping Summary

31. lecture 1831 Class Examples Learning Extension E6.9 Learning Extension E6.10