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Lectures 18 Sinusoidal (AC) Steady State Circuit Analysis Nov. 14, 2011 Material from Textbook by Alexander & Sadiku and Electrical Engineering: Principles & Applications, A. R. Hambley is used in lecture slides. EE 101 Schedule Version 11-14-11 Class 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Lecture 1 2 3 4 5 6 7 8 22 23 24 24 25 18 19 20 21 22 26 27 23 24 9 10 11 12 13 14 15 16 17 Date 9-23-11 9-26-11 9-28-11 9-30-11 10-3-11 10-5-11 10-7-11 10-10-11 10-12-11 10-14-11 10-17-11 10-19-11 10-21-11 10-24-11 10-26-11 10-28-11 10-31-11 11-2-11 11-4-11 11-7-11 11-9-11 11-11-11 11-14-11 11-16-11 11-18-11 11-21-11 11-23-11 11-25-11 11-28-11 11-30-11 Topic Introduction Fundamentals of Electrical Engineering Circuit Laws, Voltage & Current Dividers Node/Loop Analysis Node/Loop Analysis Thévenin Equivalent Circuits Norton Equivalent Circuits Amplifiers Review for Midterm 1 Midterm 1 Op Amps Op-Amp Circuits Op-Amp Circuits Inductance and Capacitance First Order Transient Response RC/RL Circuits, Time Dependent Op Amp Circuits Second Order Transient Response Review for Midterm 2 Midterm 2 Sinusoidal Signals, Complex Numbers, Phasors Phasor Circuits Veteran’s Day AC Power, Thevenin Fourier Analysis, Transfer Function, Decibels, Low Pass Filters Bode Plot, Series & Parallel Resonance. High Pass Filters 2nd Order Filters, Active Filters, Resonances Magnetic Circuits, Materials Thanksgiving Day Mutual Inductance & Transformers AC Power Engineering 12-2-11 Review for Final at normal class time and place Final Exam Thursday, December 8th (noon to 3 pm) Reading Ahead A & S Ch 1 A & S Ch 2 A& S Ch 3 Homework Review Math Quiz Pre-Req Hmwk 1 Due 1 A & S Ch 4 Hmwk 2 Due A & S Ch 5 Hmwk 3 Due 2 A & S Ch 6 A & S Ch 7 Hmwk 4 Due 3 A & S Ch 8 Hmwk 5 Due 4 A & S Ch 9 Hmwk 6 Due A & S Ch 10 A & S Ch. 14 + bit of 18 Hmwk 7 Due Hmwk 8 Due A & S Ch 13 Parts of A&S Chs 11&12 5 Hmwk 9 Due Hmwk 10 Due 6 Sinusoidal Steady-State Analysis Chapter 10 in Alexander & Sadiku 10.1 10.2 10.3 10.4 10.5 10.6 Basic Approach Nodal Analysis Mesh Analysis Superposition Theorem Source Transformation Thevenin and Norton Equivalent Circuits 3 Steps to Analyze AC Circuits: 1. 2. 3. Transform the circuit to the phasor or frequency domain. Solve the problem using circuit techniques (nodal analysis, mesh analysis, superposition, etc.). Transform the resulting phasor to the time domain. Time to Freq. Solve for variables in Freq. Freq. to Time 4 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. KVL The sum of the phasor currents entering a node must equal the sum of the phasor currents leaving. KCL 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 Chapters 1 - 4, performing the calculations with complex arithmetic. Phasor Relationships for Circuit Elements Summary of voltage-current relationship Element Time domain Frequency domain R L C 8 Example of Series RLC Time t Domain Find i(t): Frequency, ω = 500 r/s is understood Frequency ω or Phasor Domain Phasors in RLC Series Example Find the phasor voltage across each element: Nodal Analysis Example Using nodal analysis, find V1 and V2 in the circuit below. After Transformation to Phasor Domain Answer: V1(t) = 38.7 sin(2t + 69.7°) V V2(t) = 6.75 sin(2t + 165.7°) V 11 Mesh Analysis Example Find Io in the following figure using mesh analysis. 6 After Transformation to Phasor Domain 30 Answer: Io = 3.58 at an angle of 65.4° A 12 Superposition Theorem When a circuit has sources operating at different frequencies, • The separate phasor circuit for each frequency must be solved independently, and • The total response is the sum of timedomain responses of all the individual phasor circuits. 13 Superposition Theorem Example Calculate vo in the circuit of figure shown below using the superposition theorem. Vo = 4.631 sin(5t – 81.12°) + 1.051 cos(10t – 86.24°) V 14 Source Transformation 15 Source Transformation Example Find Io in the circuit of figure below using the concept of source transformation. Io = 3.288 at an angle of 99.46° A 16 Thevenin and Norton Equivalent Circuits Thevenin transform Norton transform 17 Step-by-step Thévenin/ Norton-Equivalent-Circuit Analysis 1. Perform two of these: a. Determine the open-circuit voltage Vt = voc. b. Determine the short-circuit current In = isc. c. Zero the sources and find the Thévenin impedance Zt looking back into the terminals. 2. Use the equation Vt = Zt In to compute the remaining value. 3. The Thévenin equivalent consists of a voltage source Vt in series with Zt . 4. The Norton equivalent consists of a current source In in parallel with Zt . Thevenin Equivalent Example Example Find the Thevenin equivalent at terminals a–b of the circuit below. Zth =12.4 – j3.2 Ω VTH = 18.97 at an angle of -51.57° V 20