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Steady-State Sinusoidal Analysis BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 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. 3. Compute power for steady-state ac circuits. 4. Find Thévenin and Norton equivalent circuits. 5. Determine load impedances for maximum power transfer. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering The sinusoidal function v(t) = VM sin t is plotted (a) versus t and (b) versus t. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering The sine wave VM sin ( t + ) leads VM sin t by radian BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 1 2 Frequency f Angular frequency T T 2f o sin z cos(z 90 ) sin t cos(t 90 ) o sin(t 90 ) costo BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering A graphical representation of the two sinusoids v1 and v2. The magnitude of each sine function is represented by the length of the corresponding arrow, and the phase angle by the orientation with respect to the positive x axis. In this diagram, v1 leads v2 by 100o + 30o = 130o, although it could also be argued that v2 leads v1 by 230o. It is customary, however, to express the phase difference by an angle less than or equal to 180o in magnitude. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Euler’s identity t 2 f A cos t A cos 2 f t In Euler expression, A cos t = Real (A e j t ) A sin t = Im( A e j t ) Any sinusoidal function can be expressed as in Euler form. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 4-6 The complex forcing function Vm e j ( t + ) produces the complex response Im e j (t + ). It is a matter of concept to make use of the mathematics of complex number for circuit analysis. (Euler Identity) BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering The sinusoidal forcing function Vm cos ( t + θ) produces the steady-state response Im cos ( t + φ). The imaginary sinusoidal forcing function j Vm sin ( t + θ) produces the imaginary sinusoidal response j Im sin ( t + φ). BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Re(Vm e j ( t + ) ) Re(Im e j (t + )) Im(Vm e j ( t + ) ) Im(Im e j (t + )) BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Phasor Definition Time function : v1 t V1 cosωt θ1 Phasor : V1 V11 j (t 1 ) V1 Re(e ) j (1 ) V1 Re(e ) by dropping t BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering A phasor diagram showing the sum of V1 = 6 + j8 V and V2 = 3 – j4 V, V1 + V2 = 9 + j4 V = Vs Vs = Ae j θ A = [9 2 + 4 2]1/2 θ = tan -1 (4/9) Vs = 9.8524.0o V. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 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. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Using Phasors to Add Sinusoids v1t 20 cos(t 45 ) v2 (t ) 10 cos(t 30 ) V1 20 45 V2 10 30 BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Vs V1 V2 20 45 10 30 14.14 j14.14 8.660 j5 23.06 j19.14 29.97 39.7 Vs Ae j 19 .14 A 23 .06 ( 19 .14 ) 29 .96 , tan 2 2 39 .7 1 23 .06 vs t 29 .97 cost 39 .7 BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 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.) BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 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 isθ = (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). BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering COMPLEX IMPEDANCES VL jL I L Z L jL L90 VL Z L I L BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering (a) (b) In the phasor domain, (a) a resistor R is represented by an impedance of the same value; (b) a capacitor C is represented by an impedance 1/jC; (c) (c) an inductor L is represented by an impedance jL. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering V Ve jt dV d Ve jt IC C jCVe jt dt dt V 1 I j CV Zc I j C Zc is defined as the impedance of a capacitor The impedance of a capacitor is 1/jC. It is simply a mathematical expression. The physical meaning of the j term is that it will introduce a phase shift between the voltage across the capacitor and the current flowing through the capacitor. As I j C V, if v V cos t , then i CV cos ( t 90 ) BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering As I j C V, if v VM cos t, then i CVM cos( t 90 ) I M CVM BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering I Ie jt dI d Ie jt VL L jLIe jt dt dt V V j LI j L Z L I ZL is defined as the impedance of an inductor The impedance of a inductor is jL. It is simply a mathematical expression. The physical meaning of the j term is that it will introduce a phase shift between the voltage across the inductor and the current flowing through the inductor. As V j C I, if i I cos t, then v LI cos( t 90 ) or i I cos( t 90 ), and v LI cos t. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering As V j C I, if i I M cos t , then v LI M cos( t 90 ) or i I M cos( t 90 ), and v LI M cos t, VM LI M BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Complex Impedance in Phasor Notation VC ZC IC 1 1 1 ZC j 90 jC C C VL Z LI L Z L jL L90 VR RI R BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Vm Im BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 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. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 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 remain the same as their resistances. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 3. Analyze the circuit using any of the techniques studied earlier performing the calculations with complex arithmetic. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Ztotal 100 j (150 50) 100 j100 141.445 V 10030 I 0.70730 45 Ztotal 141.445 0.707 15 VR 100 I 70.7 15 , vR (t ) 70.7 cos(500t 15 ) VL j150 I 15090 0.707 15 106.0590 15 106.0575 , vL (t ) 106.05 cos(500t 75 ) VC j50 I 50 90 0.707 15 35.35 90 15 35.35 105 , vL (t ) 35.35 cos(500t 105 ) BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 1 1 Z RC 1 / R 1 / Zc 1 / 100 1 /( j100) 1 1 j j 0.01 As j 0.01 j100 j100 j j 2 1 10 Z RC 70.71 45 0.01 j 0.01 0.0141445 50 j50 BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Z RC Vc Vs ( voltage division) Z L Z RC 70.71 45 70.71 45 10 90 10 90 j100 50 j50 50 j50 70.71 45 10 45 10 135 70.7145 vc (t ) 10 cos (1000t 135 ) 10 cos 1000t V BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Vs I Z L Z RC 10 90 10 90 j100 50 j50 50 j50 10 90 0.414 135 70.7145 i (t ) 0.414 cos (1000t 135 ) BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering VC IR R 10 135 0.1 135 100 iR (t ) 0.1 cos (1000t 135 ) VC 10 135 10 135 IC 0.1 45 Zc j100 100 90 iR (t ) 0.1 cos (1000t 45 ) A BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Solve by nodal analysis V1 V1 V2 2 90 j 2 eq(1) 10 j5 V2 V2 V1 1.50 1.5 eq(2) j10 j5 BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering V1 V1 V2 V2 V2 V1 2 90 j 2 eq(1) 1.50 1.5 eq( 2) 10 j5 j10 j5 From eq (1) 1 1 j j 1 0.1V1 j 0.2V1 j 0.2V2 j 2 As j, j j j j 1 j (0.1 j 0.2)V1 j 0.2V2 j 2 From eq ( 2) j 0.2V1 j 0.1V2 1.5 Solving V1 by eq(1) 2 eq( 2) (0.1 j 0.2)V1 3 j 2 3 j2 3.6 33.69 V1 16.1 33.69 63.43 0.1 j 0.2 0.2236 63.43 16.129.74 v1 16.1cos( t 29.74 ) V 100 BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Vs= - j10, ZL=jL=j(0.5×500)=j250 Use mesh analysis, BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering Vs VR VZ 0 ( j10) I 250 I ( j 250) 0 j10 10 90 I 0.028 90 45 250 j 250 353.3345 I 0.028 135 i 0.028 cos(500t 135 ) A VL I Z L (0.028 135 ) 25090 7 45 vL (t ) 7 cos(500t 45 ) V VR I R (0.028 135 ) 250 7 135 vR (t ) 7 cos(500t 135 ) BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 5 j200 -j50 BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 5 j200 -j50 BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering j100 -j200 100 j100 BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering j100 -j200 100 j100 BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering AC Power Calculations P Vrm sI rm s cos PF cos v i Q Vrm sI rm s sin BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering apparent power VrmsI rms P Q VrmsI rms 2 2 2 2 PI 2 rm s R P V Rrm s R 2 QI 2 rm s X Q V Xrm s X BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering THÉVENIN EQUIVALENT CIRCUITS BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 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. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 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 BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering 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. BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Department of Electronic Engineering

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posted: | 10/5/2011 |

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