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Chapter 5 Steady-State Sinusoidal Analysis 1. Identify the frequency, angular frequency, peak value, rms value, and phase of a sinusoidal signal. 2. Solve steady-state ac circuits using phasors and complex impedances. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis 3. Compute power for steady-state ac circuits. 4. Find Thévenin and Norton equivalent circuits. Lightly. 5. Determine load impedances for maximum power transfer. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis SINUSOIDAL CURRENTS AND VOLTAGES Vt = Vm cos(ωt +θ) Vm is the peak value ω is the angular frequency in radians per second θ is the phase angle ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis 1 f T Frequency Angular frequency 2 T 2f sin z cosz 90 ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Root-Mean-Square Values Vrms 1 T Pavg T v t dt 2 0 2 rms V R T I rms 1 2 i t dt T 0 Pavg I ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis 2 rms R RMS Value of a Sinusoid Vrms Vm 2 The rms value for a sinusoid is the peak value divided by the square root of two. This is not true for other periodic waveforms such as square waves or triangular waves. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Phasor Definition Time function : v1 t V1 cosωt θ1 Phasor : V1 V1θ1 ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Adding Sinusoids Using Phasors Step 1: Determine the phasor for each term. Step 2: Add the phasors using complex arithmetic. Step 3: Convert the sum to polar form. Step 4: Write the result as a time function. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Using Phasors to Add Sinusoids v1 t 20 cost 45 v2 t 10 cost 60 V1 20 45 ELECTRICA L ENGINEERING Principles and Applications V2 10 30 Chapter 5 Steady-State Sinusoidal Analysis Vs V1 V2 20 45 10 30 14.14 j14.14 8.660 j5 23.06 j19.14 29.97 39.7 v s t 29.97 cost 39.7 ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Sinusoids can be visualized as the realaxis projection of vectors rotating in the complex plane. The phasor for a sinusoid is a snapshot of the corresponding rotating vector at t = 0. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Phase Relationships To determine phase relationships from a phasor diagram, consider the phasors to rotate counterclockwise. Then when standing at a fixed point, if V1 arrives first followed by V2 after a rotation of θ , we say that V1 leads V2 by θ . Alternatively, we could say that V2 lags V1 by θ . (Usually, we take θ as the smaller angle between the two phasors.) ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis To determine phase relationships between sinusoids from their plots versus time, find the shortest time interval tp between positive peaks of the two waveforms. Then, the phase angle is θ = (tp/T ) × 360°. If the peak of v1(t) occurs first, we say that v1(t) leads v2(t) or that v2(t) lags v1(t). ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis COMPLEX IMPEDANCES VL jL I L Z L jL L90 VL Z L I L ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis VC Z C I C 1 1 1 ZC j 90 C jC C VR RI R ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Kirchhoff’s Laws in Phasor Form We can apply KVL directly to phasors. The sum of the phasor voltages equals zero for any closed path. The sum of the phasor currents entering a node must equal the sum of the phasor currents leaving. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Circuit Analysis Using Phasors and Impedances 1. Replace the time descriptions of the voltage and current sources with the corresponding phasors. (All of the sources must have the same frequency.) ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis 2. Replace inductances by their complex impedances ZL = jωL. Replace capacitances by their complex impedances ZC = 1/(jωC). Resistances have impedances equal to their resistances. 3. Analyze the circuit using any of the techniques studied earlier in Chapter 2, performing the calculations with complex arithmetic. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis “SPOOKY” And “SPECTRE” ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis AC-130 GUNSHIP ELECTRICA L ENGINEERING Principles and Applications AC Power Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis AC Power Calculations P Vrms I rms cos PF cos v i Q Vrms I rms sin ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis apparent power Vrms I rms P Q Vrms I rms 2 PI QI 2 rms 2 rms R X 2 2 P Q ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis V 2 Rrms R V 2 Xrms X ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis THÉVENIN EQUIVALENT CIRCUITS ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis The Thévenin voltage is equal to the open-circuit phasor voltage of the original circuit. Vt Voc We can find the Thévenin impedance by zeroing the independent sources and determining the impedance looking into the circuit terminals. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis The Thévenin impedance equals the open-circuit voltage divided by the short-circuit current. Voc Vt Z t I sc I sc I n I sc ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Maximum Power Transfer If the load can take on any complex value, maximum power transfer is attained for a load impedance equal to the complex conjugate of the Thévenin impedance. If the load is required to be a pure resistance, maximum power transfer is attained for a load resistance equal to the magnitude of the Thévenin impedance. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis BALANCED THREE-PHASE CIRCUITS Much of the power used by business and industry is supplied by three-phase distribution systems. Plant engineers need to be familiar with three-phase power. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Phase Sequence Three-phase sources can have either a positive or negative phase sequence. The direction of rotation of certain three-phase motors can be reversed by changing the phase sequence. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Wye–Wye Connection Three-phase sources and loads can be connected either in a wye configuration or in a delta configuration. The key to understanding the various threephase configurations is a careful examination of the wye–wye circuit. ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Pavg p t 3VYrms I Lrms cos VY I L Q3 sin 3VYrms I Lrms sin 2 ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis Z 3ZY ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis ELECTRICA L ENGINEERING Principles and Applications Chapter 5 Steady-State Sinusoidal Analysis
