Review of Second Order Electrical Circuits: Series RLC Circuit, Parallel RLC Circuit

1. Lecture 21 Review: Second order electrical circuits Series RLC circuit Parallel RLC circuit Second order circuit natural response Sinusoidal signals and complex exponentials Related educational materials: Chapter 8.2, 8.3

2. Summary: Series & parallel RLC circuits • Series RLC circuit: • Parallel RLC circuit

3. Second order input-output equations • In general, the governing equation for a second order system can be written in the form: • Where •  is the damping ratio (  0) • n is the natural frequency (n  0)

4. Solution of second order differential equations • The solution of the input-output equation is (still) the sum of the homogeneous and particular solutions: • We will consider the homogeneous solution first:

5. Homogeneous solution (Natural response) • Assume form of solution: • Substituting into homogeneous differential equation: • We obtain two solutions:

6. Homogeneous solution – continued • Natural response is a combination of the solutions: • So that: • We need two initial conditions to determine the two unknown constants: • ,

7. Natural response – discussion •  and n are both non-negative numbers •   1  solution composed of decaying exponentials •  < 1  solution contains complex exponentials

8. Sinusoidal functions • General form of sinusoidal function: • Where: • VP = zero-to-peak value (amplitude) •  = angular (or radian) frequency (radians/second) •  = phase angle (degrees or radians)

9. Sinusoidal functions – graphical representation • T = period • f = frequency • cycles/sec (Hertz, Hz) •  = phase • Negative phase shifts sinusoid right

10. Complex numbers • Complex numbers have real and imaginary parts: • Where:

11. Complex numbers – Polar coordinates • Our previous plot was in rectangular coordinates • In polar coordinates: • Where:

12. Complex exponentials • Polar coordinates are often expressed as complex exponentials • Where

13. Sinusoids and complex exponentials • Euler’s Identity:

14. Sinusoids and complex exponentials – continued • Unit vector rotating in complex plane: • So

15. Complex exponentials – summary • Complex exponentials can be used to represent sinusoidal signals • Analysis is (nearly always) simpler with complex exponentials than with sines, cosines • Alternate form of Euler’s identity: • Cosines, sines can be represented by complex exponentials

16. Second order system natural response • Now we can interpret our previous result

17. Classifying second order system responses • Second order systems are classified by their damping ratio: •  > 1  System is overdamped (the response consists of decaying exponentials, may decay slowly if  is large) •  < 1  System is underdamped (the response will oscillate) •  = 1  System is critically damped (the response consists of decaying exponentials, but is “faster” than any overdamped response)

18. Note on underdamped system response • The frequency of the oscillations is set by the damped natural frequency, d