AC SteadyState Analysis Sinusoidal Forcing Functions Phasors and

AC Steady-State Analysis Sinusoidal Forcing Functions, Phasors, and Impedance Kevin D. Donohue, University of Kentucky 1

The Sinusoidal Function Terms for describing sinusoids: Maximum Value, Amplitude, or Magnitude Phase Frequency in cycles/second or Hertz (Hz) Radian Frequency in Radian/second . 4 . 2 Kevin D. Donohue, University of Kentucky 2

Trigonometric Identities Radian to degree conversion multiply by 180/ Degree to radian conversion multiply by /180 Kevin D. Donohue, University of Kentucky 3

Sinusoidal Forcing Functions Determine the forced response for io(t) the circuit below with vs(t) = 50 cos(1250 t): 10 vs(t) 0. 1 m. F io(t) + vc(t) 40 - Note: Show: Kevin D. Donohue, University of Kentucky 4

Complex Numbers Each point in the complex number plane can be represented by in a Cartesian or polar format. Kevin D. Donohue, University of Kentucky 5

Complex Arithmetic Addition: Multiplication and Division: Simple conversions: Kevin D. Donohue, University of Kentucky 6

Euler’s Formula Show: A series expansion …. Kevin D. Donohue, University of Kentucky 7

Complex Forcing Function Consider a sinusoidal forcing function given as a complex function: Ø Ø Ø 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 Kentucky 8

Complex Forcing Function Example Determine the forced response for io(t) the circuit below with vs(t) = 50 exp(j 1250 t): io(t) 10 vs(t) Note: 0. 1 m. F 40 Show: Kevin D. Donohue, University of Kentucky 9

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 Kentucky 10

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 . Given: using passive sign convention show: For inductor relation Show : For capacitor relation : Show Kevin D. Donohue, University of Kentucky 11

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. 2 H 0. 5 F 50 0. 1 H 5 0. 01 F 10 Kevin D. Donohue, University of Kentucky 12

Impedance Circuit Example Find the AC steady-state value for v 1(t): 1 F 1 H + v 1 - 1 F Kevin D. Donohue, University of Kentucky 1 k 13

Impedance Circuit Example Find AC steady-state response for io(t): 3 H io 60 4 H 25 m. F 80 Kevin D. Donohue, University of Kentucky 14