Lectures 16 & 17 Sinusoidal Signals, Complex Numbers, Phasors, Impedance & AC Circuits Nov. 7 & 9, 2011 Material from Textbook by Alexander & Sadiku and Electrical Engineering: Principles & Applications, A. R. Hambley is used in lecture slides.
Sinusoids and Phasors Chapter 9 9.1 Motivation 9.2 Sinusoids features 9.3 Phasors 9.4 Phasor relationships for circuit elements 9.5 Impedance and admittance 9.6 Kirchhoff s laws in the frequency domain 9.7 Impedance combinations 3
How to determine v(t) and i(t)? v s (t) = 10V -> AC How can we apply what we have learned before to determine i(t) and v(t)? 4
War of the Currents Edison s DC power was first Good for lights & motors Ease of generation Not easy transmit long distances Westinghouse & Tesla (AC power) Needed math skils Needed generator and other technology Easy for long distance transmission vs See http://en.wikipedia.org/wiki/ War_of_Currents
Sinusoids A sinusoid is a signal that has the form of the sine or cosine function. A general expression for the sinusoid, where V m = the amplitude of the sinusoid ω = the angular frequency in radians/s (ω = 2 π f) Ф = the phase 6
Sinusoids A periodic function is one that satisfies v(t) = v(t + nt), for all t and for all integers n. Only two sinusoidal values with the same frequency can be compared by their amplitude and phase difference. If phase difference = zero, they are in phase; if phase difference zero, they are out of phase. 7
Sinusoid Examples Example Given a sinusoid,, calculate its amplitude, phase, angular frequency, period, and frequency. Solution: Amplitude = 5, phase = 60 o, angular frequency = 4π rad/s, Period = 0.5 s, frequency = 2 Hz. 8
Sinusoid Example Example 2 Find the phase angle between and, does i 1 lead or lag i 2? Solution: Since sin(ωt+90 o ) = cos ωt therefore, i 1 leads i 2 155 o 9
EE 101 Schedule Version 11-8-11 Class Lecture Date Topic Reading Ahead Homework Quiz 1 1 9-23-11 Introduction Review Math 2 2 9-26-11 Fundamentals of Electrical Engineering A & S Ch 1 Pre-Req 3 3 9-28-11 Circuit Laws, Voltage & Current Dividers A & S Ch 2 4 4 9-30-11 Node/Loop Analysis A& S Ch 3 Hmwk 1 Due 5 5 10-3-11 Node/Loop Analysis 1 6 6 10-5-11 Thévenin Equivalent Circuits A & S Ch 4 7 7 10-7-11 Norton Equivalent Circuits Hmwk 2 Due 8 8 10-10-11 Amplifiers A & S Ch 5 9 10-12-11 Review for Midterm 1 10 10-14-11 Midterm 1 11 9 10-17-11 Op Amps Hmwk 3 Due 2 12 10 10-19-11 Op-Amp Circuits 13 11 10-21-11 Op-Amp Circuits 14 12 10-24-11 Inductance and Capacitance A & S Ch 6 Hmwk 4 Due 3 15 13 10-26-11 First Order Transient Response A & S Ch 7 16 14 10-28-11 RC/RL Circuits, Time Dependent Op Amp Circuits 17 15 10-31-11 Second Order Transient Response A & S Ch 8 Hmwk 5 Due 18 11-2-11 Review for Midterm 2 4 19 11-4-11 Midterm 2 20 16 11-7-11 Sinusoidal Signals, Complex Numbers, Phasors A & S Ch 9 21 17 11-9-11 Phasor Circuits Hmwk 6 Due 11-11-11 Veteran s Day 22 18 11-14-11 AC Power, Thevenin A & S Ch 10 Hmwk 7 Due 23 19 11-16-11 Fourier Analysis, Transfer Function, Decibels, Low Pass Filters A & S Ch 18 + Ch 17 24 20 11-18-11 Bode Plot, Series & Parallel Resonance. High Pass Filters A & S Ch 17 Hmwk 8 Due 5 24 21 11-21-11 2nd Order Filters, Active Filters, Resonances 25 22 11-23-11 Magnetic Circuits, Materials A & S Ch 13 Hmwk 9 Due 11-25-11 Thanksgiving Day 26 23 11-28-11 Mutual Inductance & Transformers 27 24 11-30-11 AC Power Engineering Parts of A&S Chs 11&12 Hmwk 10 Due 6 12-2-11 Review for Final at normal class time and place Final Exam Thursday, December 8th (noon to 3 pm)
Resources to Learn Complex Numbers & other Topics for Engineering http://www.khanacademy.org/ The Kahn Academy has lessons and worked examples on a multitude of math topics, including complex numbers Here are some complex number videos at the Kahn Academy http://www.khanacademy.org/video/introduction-to-i-and-imaginarynumbers http://www.khanacademy.org/video/calculating-i-raised-to-arbitraryexponents http://www.khanacademy.org/video/i-as-the-principal-root-of--1--alittle-technical http://www.khanacademy.org/video/complex-numbers--part-1 http://www.khanacademy.org/video/complex-numbers--part-2 There are also exercises for practice at the Kahn Academy site Others sites for useful links for students http://www.engineerguy.com/ http://www.youtube.com/education http://www.apple.com/education/itunes-u
Phasors A phasor is a complex number that represents the amplitude and phase of a sinusoid. It can be represented in one of the following three forms: a. Rectangular b. Polar c. Exponential where 12
Mathematic operation of complex number: 1. Addition 2. Subtraction 3. Multiplication 4. Division 5. Reciprocal 6. Square root 7. Complex conjugate 8. Euler s identity Phasor 13
Phasor Example Evaluate the following complex numbers: a. b. Solution: a. 15.5 + j13.67 b. 8.3 + j2.2 (See Appendix A p.10) 14
Phasor Transform a sinusoid to and from the time domain to the phasor domain: (time domain) (phasor domain) Amplitude and phase difference are two principal concerns in the study of voltage and current sinusoids. Phasor will be defined from the cosine function in all our further study. If a voltage or current expression is in the form of a sine, it will be changed to a cosine by subtracting from the phase. Frequency ω = 2 π f is implicit, i.e. understood from other information. 15
Phasor Example Transform the following sinusoids to phasors: i = 6cos(50t 40 o ) A v = 4sin(30t + 50 o ) V Solution: a. I A (frequency is understood to be ω = 50 r/s b. Since sin(a) = cos(a+90 o ); v(t) = 4cos (30t+50 o +90 o ) = 4cos(30t+140 o ) V Transform to phasor => V V 16
Phasor Example Example: Transform the phasors corresponding to sinusoids: a. b. I = j(12 + j5) A Solution: a) v(t) = 10cos(ωt + 210 o ) V (phases are typically > 0) b) Since I =12 + j5 = 12 2 + 5 2 tan 1 ( 5 ) = 13 22.62 12 i(t) = 13cos(ωt + 22.62 o ) A 17
Phasor Notation The differences between v(t) and V: v(t) is instantaneous or time-domain representation V is the frequency or phasor-domain representation. v(t) is time dependent, V is not. v(t) is always real with no complex term, V is generally complex. Note: Phasor analysis applies ; when it is applied to two or more sinusoid signals only if they have the same frequency. 18
Phasor Relationship between differential, integral operation in phasor listed as follow: 19
Phasor Example Use phasor approach, determine the current i(t) in a circuit described by the integro-differential equation. Answer: I got i(t) = 2.24 cos(2t + 208 o ) A See what you get for this. 20
Phasor In-class exercise, we can derive the differential equations for the following circuit in order to solve for v o (t) in phase domain V o. However, the derivation may sometimes be very tedious. Is there any quicker and more systematic methods to do it? 21
Phasors & Circuits The answer is YES! Instead of first deriving the differential equation and then transforming it into phasor to solve for V o, we can transform all the RLC components into phasor first, then apply the KCL laws and other theorems to set up a phasor equation involving V o directly. 22
Phasor Relationships for Circuit Elements Resistor: Inductor: Capacitor: 23
Phasor Relationships for Circuit Elements Summary of voltage-current relationship Element Time domain Frequency domain R L C 24
Phasor Relationships for Circuit Elements Example If voltage v(t) = 6 cos(100t 30 o ) is applied to a 50 µf capacitor, calculate the current, i(t), through the capacitor. Answer: i(t) = 0.03 cos(100t + 60 o ) ma 25
Impedance and Admittance The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in ohms Ω. where R = Re, Z is the resistance and X = Im, Z is the reactance. Positive X is for L and negative X is for C. The admittance Y is the reciprocal of impedance, measured in siemens (S). 26
Impedance and Admittance Impedances and admittances of passive elements Element Impedance Admittance R L C 27
Impedance and Admittance 28
Impedance and Admittance After we know how to convert RLC components from time to phasor domain, we can transform a time domain circuit into a phasor/frequency domain circuit. Hence, we can apply the KCL laws and other theorems to directly set up phasor equations involving our target variable(s) for solving. 29
Impedance and Admittance Example Refer to Figure below, determine v(t) and i(t). Answers: i(t) = 1.118cos(10t 26.56 o ) A; v(t) = 2.236cos(10t + 63.43 o ) V 30
Amplitude & Phase Relationships v s = 5 cos (10 t) = 5 cos (10 t + 360) i(t) = 1.11 cos(10 t 26.6 ) = 1.11 cos (10 t + 333.4) v(t) = 2.236 cost (10 t + 63.4 )
Kirchhoff s Laws in the Frequency Domain Both KVL and KCL are hold in the phasor domain or more commonly called frequency domain. Moreover, the variables to be handled are phasors, which are complex numbers. All the mathematical operations involved are now in complex domain. 32
Impedance Combinations The following principles used for DC circuit analysis all apply to AC circuit. For example: a. voltage division b. current division c. circuit reduction d. impedance equivalence e. Y-Δ transformation 33
Impedance Combinations Example Determine the input impedance of the circuit in figure below at ω =10 rad/s. Answer: Z in = 32.5 j73.7 34