Kevin D. Donohue, University of Kentucky1 AC Steady-State Analysis Sinusoidal Forcing Functions, Phasors, and Impedance

Kevin D. Donohue, University of Kentucky2 The Sinusoidal Function Terms for describing sinusoids : Maximum Value, Amplitude, or Magnitude Radian Frequency in Radian/second Frequency in cycles/second or Hertz (Hz) Phase.2.4 

Kevin D. Donohue, University of Kentucky3 Trigonometric Identities Radian to degree conversion multiply by 180/  Degree to radian conversion multiply by  /180

Kevin D. Donohue, University of Kentucky4 Sinusoidal Forcing Functions Determine the forced response for i o (t) the circuit below with v s (t) = 50cos(1250t): Note: 0.1 mF 10  40  i o (t) v s (t) +vc(t)-+vc(t)- Show:

Kevin D. Donohue, University of Kentucky5 Complex Numbers Each point in the complex number plane can be represented by in a Cartesian or polar format.

Kevin D. Donohue, University of Kentucky6 Complex Arithmetic Addition: Multiplication and Division: Simple conversions:

Kevin D. Donohue, University of Kentucky7 Euler’s Formula Show: A series expansion ….

Kevin D. Donohue, University of Kentucky8 Complex Forcing Function Consider a sinusoidal forcing function given as a complex function:  Based on a signal and system’s concept (orthogonality), it can be shown that for a linear system, the real part of the forcing function only affects the real part of the response and the imaginary part of the forcing function only affect the imaginary part of the response.  For a linear system excited by a sinusoidal function, the steady-state response everywhere in the circuit will have the same frequency. Only the magnitude and phase of the response will vary.  A useful factorization:

Kevin D. Donohue, University of Kentucky9 Complex Forcing Function Example Determine the forced response for i o (t) the circuit below with v s (t) = 50exp(j1250t): Note: 0.1 mF 10  40  i o (t) v s (t) Show:

Kevin D. Donohue, University of Kentucky10 Phasors Notation for sinusoidal functions in a circuit can be more efficient if the exp(-j  t) is dropped and just the magnitude and phase maintained via phasor notation:

Kevin D. Donohue, University of Kentucky11 Impedance The  affects the resistive force that inductors and capacitors have on the currents and voltages in the circuit. This generalized resistance, which affects both amplitude and phase of the sinusoid, will be called impedance. Impedance is a complex function of . For inductor relation : Show For capacitor relation : Show using passive sign convention show: Given:

Kevin D. Donohue, University of Kentucky12 Finding Equivalent Impedance Given a circuit to be analyzed for AC steady-state behavior, all inductors and capacitors can be converted to impedances and combined together as if they were resistors. 0.5F 50  0.1H0.01F 10  55 2H

Kevin D. Donohue, University of Kentucky13 Impedance Circuit Example Find the AC steady-state value for v 1 (t): 1k  1H 1F1F +v1-+v1- 1F1F

Kevin D. Donohue, University of Kentucky14 Impedance Circuit Example Find AC steady-state response for i o (t): 60  80  3H 4H4H 25mF ioio