Steady-State Sinusoidal Analysis: Phasors, Complex Impedances, and Power in AC Circuits, Exams of Electrical and Electronics Engineering

Download Steady-State Sinusoidal Analysis: Phasors, Complex Impedances, and Power in AC Circuits and more Exams Electrical and Electronics Engineering in PDF only on Docsity! 1 Chapter 5 Steady-State Sinusoidal Analysis Sinusoidal Currents and Voltages Phasors Complex Impedances Circuit Analysis with Phasors&Complex Impedances Power in AC Circuits Thevenin and Norton Equivalent Circuits Balanced Three-Phase Circuits 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. 4. Find Thévenin and Norton equivalent circuits. 5. Determine load impedances for maximum power transfer. 6. Solve balanced three-phase circuits. 3. Compute power for steady-state ac circuits. SINUSOIDAL CURRENTS AND VOLTAGES Vm is the peak value ω is the angular frequency in radians per second θ is the phase angle T is the period T πω 2= fπω 2= ( ) ( )o90cossin −= zz Frequency T f 1= Angular frequency 2 Root-Mean-Square Values ( )dttv T V T 2 0 rms 1 ∫= R V P 2 rms avg = ( )dtti T I T 2 0 rms 1 ∫= RIP 2rmsavg = RMS Value of a Sinusoid 2 rms mVV = 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. Phasor Definition ( ) ( )111 cos :function Time θtωVtv += 111 :Phasor θV ∠=V 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. Using Phasors to Add Sinusoids( ) ( )o45cos201 −= ttv ω ( ) ( )o60cos102 += ttv ω o45201 −∠=V o30102 −∠=V 5 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. 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.) 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. Figure 5.13, Figure 5.14 Phar diagram for Example S.A ii aeee “=O Fe Figure 5.15 Circuit for Example 5.5 Figure 5.16 Chet and phasor gram for Exrcie $9. 1002 Figure 5.17 Circuit for Exercise 5.10. ont Suk 10 cos1000) Figure 5.18 Circuit for Exercise 5.11 7 AC Power Calculations ( )θcosrmsrms IVP = ( )θcosPF = iv θθθ −= ( )θsinrmsrmsIVQ = rmsrmspower apparent IV= ( )2rmsrms22 IVQP =+ RIP 2rms= XIQ 2rms= R V P R 2 rms= X V Q X 2 rms= 10 Maximum Average 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. 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. 11 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. 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 three- phase configurations is a careful examination of the wye–wye circuit. ( ) ( )θcos3 rmsrmsavg LY IVtpP == ( ) ( )θθ sin3sin 2 3 rmsrms LY LY IV IV Q == Figure 5.39 Phasor diagram showing the elationship between the ine-tosine voltage vin and the ine-to-neutal voltages Vox and Vs rw the aso diagram Figure 5.40 Phasordagram showing tinetotine voltages and tine omneuta wotages Figure 5.42 Delta-connected three-phase source zy a 25 wy () Wye-connected load (6) Detta-connected load Figure 5.43 Loads can be either wye-connected or deta-connected, ty 7 (@) Wye-connected load (b) Delta-connected load 12