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Unformatted text preview: 11/29/2010 1 Electrical Engineering Principles & Applications 5 Steady State Steady State Sinusoidal Sinusoidal Analysis Analysis Slide 1 Outline 1. Identify the frequency, angular frequency, peak value RMS value and phase of a peak value, RMS value, and phase of a sinusoidal signal 2. Solve steadystate AC circuits using phasors and complex impedances Slide 2 11/29/2010 2 Importance of Sinusoidal Sources • Appear in many practical applications – Electric power is distributed by sinusoidal currents and voltages – Sinusoidal signals are used widely in radio communications • Any signal can be represented by a sum of sinusoidal components (Fourier Analysis) • Sinusoids have good mathematical properties – Derivative is a sinusoid – Integral is a sinusoid Slide 3 Sinusoidal steadystate • Whenever the forced input to the circuit is sinusoidal the response will be sinusoidal • If the input persists, the response will persist and we call it steadystate response Sinusoidal Currents or Voltages Slide 4 11/29/2010 3 Sinusoidal Currents and Voltages V m is the peak value ) cos( ) ( t V t v m ω is the angular frequency in radians per second θ is the phase angle T is the period , where is the frequency T f 1 Slide 5 T 2 f 2 90 cos sin z z T is the angular frequency , where: RootMeanSquare (RMS) Values of a Sinusoid ) ( 2 2 i R V R I P eff eff For DC circuit the power is For AC circuit the power is T T T T ii dt v RT dt i T R Rdt i T P 2 2 2 1 ) ( 1 1 Slide 6 T rms eff rms eff dt v T V V I dt i T I 2 2 1 & 1 Equate (i) and (ii) 11/29/2010 4 RootMeanSquare (RMS) Values of a Sinusoid T dt t v T V 2 rms 1 T dt t i T I 2 rms 1 R V P 2 rms avg R I P 2 rms avg The RMS value for a sinusoid is the peak value divided by the square root of 2. Slide 7 2 rms m V V This is NOT true This is NOT true for other periodic waveforms such as square waves or triangular waves Voltage applied to a 50 Ω resistance 2 V Voltage Applied to Resistors Voltage Voltage W 100 50 ) 2 / 100 ( 2 R V P rms avg 2 Power Power Slide 8 W ) 100 ( cos 200 ) ( ) ( 2 2 t R t v t p Power Power 11/29/2010 5 Phasor Definition Phasors Phasors are complex numbers that can be used to represent sinusoidal signals The magnitude magnitude of the phasor = Peak Peak value 2 1 1 sin t I t i and 1 1 1 cos t V t v Consider The magnitude magnitude of the phasor = Peak Peak value Angle Angle of the phasor = phase phase of the sinusoid (written as a cosine ) 90 is phaso The Slide 9 1 1 1 V V is phasor The 90 2 1 1 I I is phasor The The steady state analysis of sinusoidal signals can be carried out easily if signals are represented as phasors (vectors) Real and Complex Signals jy x Z Real part...
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 Spring '11
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