EECE310_F05_Lecture_Notes__wk_of_10_10_05_

EECE310_F05_Lecture_Notes__wk_of_10_10_05_ - College of...

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Unformatted text preview: College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4 Alternating Current (AC) Steady-State Analysis EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis In this chapter, the analysis of the steady-state forced response (ergo the partial solution) of circuits with alternating current (AC) input sources will be studied. In AC circuits, the input function is sinusoidal in nature. Because the transient response rapidly decays to zero, steady-state analysis is more significant. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis i (t ) = I M cos( t + i ) amperes v ( t ) = V M cos( t + v ) volts IM and VM are the amplitude, maximum, or peak values of the current or voltage. is the angular frequency (measured in radians per seconds) I and V are the phase angles for the current or voltage. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP Sinusoidal current and voltage are given by the corresponding equations: College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis i (t ) = I M cos( t + i ) amperes v ( t ) = V M cos( t + v ) volts Sinusoidal waveforms are periodic, that is they repeat the same pattern of values after a certain point in time, t. This specific point in time is called the period, T. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP Sinusoidal current and voltage corresponding equations (continued): College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis Periodicity can be expressed in general for a function p(t) with a period T as p (t + nT ) = p (t ) n = 1, 2,3 K for all values of t Cosine and sine functions complete one cycle when the argument (t) increases by 2 radians. Hence: T = 2 EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis i (t ) = I M cos( t + i ) amperes v ( t ) = V M cos( t + v ) volts The frequency (f) of a sinusoidal, which is measured in Hertz (Hz), is related to the period, T, by the equation: 1 Sinusoidal current and voltage corresponding equations (continued): f = EECE 310 Preston D. Frazier, Ph.D., P.E., PMP T College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis Two sinusoidal cosine voltage functions: v1 (t ) = V M 1 cos( t + v1 ) volts v 2 (t ) = V M 2 cos( t + v 2 ) volts If v1 = v2, then the waveforms are in phase. If v1 v2, then the waveforms are out of phase. If v1 < v2, then the v2(t) leads v1(t) or v1(t) lags v2(t). If v1 > v2, then the v1(t) leads v2(t) or v2(t) lags v2(t). EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis Sine and cosine function relationships: sin( t ) = cos( t - 2 - A sin( t + ) = A sin( t + ) - A cos( t + ) = A cos( t + ) EECE 310 Preston D. Frazier, Ph.D., P.E., PMP cos( t ) = sin( t + 2 ) ) College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis Sinusoidal functions and complex numbers are related through Euler's equation. e j t = cos( t ) + j sin( t ) j t The complex function may be broken into its real and imaginary parts: Re( e ) = cos( t ) j t Im( e ) = sin( t ) EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis Sinusoidal current and voltage corresponding equations: i (t ) = I M cos( t i ) amperes v ( t ) = V M cos( t v ) volts j I The complex representation of the current and voltage sinusoidal may be written as: i (t ) = I M e amperes j V v (t ) = V M e volts EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis i (t ) = I M cos( t i ) = I M i v ( t ) = V M cos( t v ) = V M v The phasor transformation of sinusoidal sine current and voltage functions: The phasor transformation of sinusoidal cosine current and voltage functions: i ( t ) = I M sin( t i ) = I M ( i - EECE 310 Preston D. Frazier, Ph.D., P.E., PMP 2 v ( t ) = V M sin( t v ) = V M ( v - ) 2 ) College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 4-AC Steady-State Analysis The phasor identities for a voltage function: r V =rV M i volts Re( V ) = V M cos( v ) r Im( V ) = jV M sin( v ) VM r 2 r 2 = (Re( V )) + (Im( V )) r -1 Im( V ) r ) v = tan ( Re( V ) EECE 310 Preston D. Frazier, Ph.D., P.E., PMP College of Engineering, Architecture, and Computer Sciences Department of Electrical and Computer Engineering Chapter 5 Steady-State Sinusoidal Analysis Important Note about Phasors: If the number falls to the left of the imaginary axis (i.e., if the real part is negative), then add 180 degrees to the arctangent (imag/real) to obtain the correct angle. EECE 310 Preston D. Frazier, Ph.D., P.E., PMP ...
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