Chapter 9 A Sinusoids Complex Numbers SteadyState Analysis

Chapter 9 A Sinusoids, Complex Numbers Steady-State Analysis 1

Sinusoids and Phasor Chapter 9 A 9. 1 9. 2 9. 3 9. 4 9. 5 9. 6 9. 7 Motivation Sinusoids’ features Phasor relationships for circuit elements Impedance and admittance Kirchhoff’s laws in the frequency domain Impedance combinations 2

9. 1 Motivation (1) How to determine v(t) and i(t)? vs(t) = 10 V How can we apply what we have learned before to determine i(t) and v(t)? 3

9. 2 Sinusoids (1) • A sinusoid is a signal that has the form of the sine or cosine function. • A general expression for the sinusoid, where Vm = the amplitude of the sinusoid ω = the angular frequency in radians/s Ф = the phase 4

9. 2 Sinusoids (2) A periodic function is one that satisfies v(t) = v(t + n. T), 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 is zero, they are in phase; if phase difference is not zero, they are out of phase. 5

9. 2 Sinusoids (3) Example 1 Given a sinusoid, , calculate its amplitude, phase, angular frequency, period, and frequency. Solution: Amplitude = 5, phase = – 60 o, angular frequency = 4 p rad/s, Period = 0. 5 s, frequency = 2 Hz. 6

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9. 2 Sinusoids (4) 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. 8

Complex Numbers 9

Complex Numbers • A complex number uses j= to represent a number z = x + jy which has two parts: Real and Imaginary, OR magnitude and phase • It can be represented in one of the following three forms: a. Rectangul ar b. Polar c. Exponenti al where 10

Three forms of a Complex # • Express a = 2 – 3 j and b = -4 + 5 j – in polar form: – in exponential form 11 http: //www. kusashi. com/cartesian-polar. php

Three forms of a Complex # • Express c = 2<30 o and d = 4<200 o – in exponential form: – in Rectangular form: 12

Mathematic operation of complex number: http: //www. 1728. com/compnumb. htm 1. Addition 2. Subtraction 3. Multiplication 4. Division 5. Reciprocal 6. Square root 7. Complex conjugate 8. Euler’s identity 13

Complex Arithmetic Given: a = 3 + 4 i b= -5 + 6 i c = 8<45 d = 12<250 Express (rectangular) a+b a/b c c+d 14

Complex Arithmetic Given: a = 3 + 4 i b= -5 + 6 i c = 8<45 d = 12<250 Express (polar) a c/d ab a+c 15

More complex math to try later Example 3 • Evaluate the following complex numbers: a. b. Solution: a. – 15. 5 + j 13. 67 b. 8. 293 + j 2. 2 16

Complex number Tutorials (about 10 minutes each) R-click then "open hyperlink" Problem 1 This is a tutorial introducing the idea of complex numbers and how they're represented graphically. Problem 2 This is a tutorial that shows how to perform arithmetic operations on complex numbers in the rectangular form. Problem 1 This is a tutorial introducing the concept of polar coordinates in reference to complex numbers. Problem 2 This is a tutorial about how to perform mathematical operations on complex numbers in polar form 17

9. 3 Phasor (1) • A phasor is a complex number that represents the amplitude and initial phase (at t=0) of a sinusoid. Acos( t + ) • It allows us to perform math on sinusoids of the same frequency by using a complex number representation – much easier than trig identities!! 18

http: //ptolemy. eecs. berkeley. edu/eecs 20/berkeley/phasors/demo/phasors. html 9. 3 Phasor (4) • 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 proceeding 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. • Note, is the phase angle of the sinusoid at t=0 19

9. 3 Phasor (5) Example 4 Transform the following sinusoids to phasors: i = 6 cos(50 t – 40 o) A v = – 4 sin(30 t + 50 o) V Solution: a. I A b. Since –sin(A) = cos(A+90 o); v(t) = 4 cos (30 t+50 o+90 o) = 4 cos(30 t+140 o) V Transform to phasor => V V 20

9. 3 Phasor (6) Example 5: Transform to sinusoids the phasors: a. b. Solution: a) v(t) = 10 cos(wt + 210 o) V b) Since i(t) = 13 cos(wt + 22. 62 o) A 21

9. 3 Phasor (7) 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 only when frequency is constant; when it is applied to two or more sinusoid signals only if they have the same frequency. 22

9. 3 Phasor (10) • we can derive the differential equations for the following circuit in order to solve for vo(t) in phase domain Vo. • However, the derivation may sometimes be very tedious. Is there any quicker and more systematic methods to do it? 24

9. 3 Phasor (11) The answer is YES! Instead of first deriving the differential equation and then transforming it into phasor to solve for Vo, 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 Vo directly. 25

Resistors in the Frequency (Phasor) Domain • Ohm's Law, Time Domain – v = i*R – suppose i(t) = Im cos (wt + ) – then v(t) = R*Im cos (wt + ) • Ohm's Law, Frequency Domain – I = Im < – V = R*Im < 26

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9. 4 Phasor Relationships for Circuit Elements (1) Resistor: Inductor: Capacitor: 29

9. 4 Phasor Relationships for Circuit Elements (2) Summary of voltage-current relationship Element Time domain Frequency domain R L C 30

9. 4 Phasor Relationships for Circuit Elements (3) Example 7 If voltage v(t) = 6 cos(100 t – 30 o) is applied to a 50 μF capacitor, calculate the current, i(t), through the capacitor. Answer: i(t) = 30 cos(100 t + 60 o) m. A 31

9. 5 Impedance and Admittance (1) • 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). 32

9. 5 Impedance and Admittance (2) Impedances and admittances of passive elements Element Impedance Admittance R L C 33

9. 5 Impedance and Admittance (3) 34

9. 5 Impedance and Admittance (4) 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. 35

9. 5 Impedance and Admittance (5) Example 8 determine v(t) and i(t). Answers: i(t) = 1. 118 cos(10 t – 26. 56 o) A; v(t) = 2. 236 cos(10 t + 63. 43 o) V

9. 6 Kirchhoff’s Laws in the Frequency Domain (1) • Both KVL and KCL 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. 37

9. 7 Impedance Combinations (1) • 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 38

9. 7 Impedance Combinations (2) Determine the input impedance of the circuit in figure below at ω =10 rad/s. Example 9 39 Answer: Zin = 32. 38 – j 73. 76

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1) Problem 1 Determine the following properties for each of the given sinusoidal voltages: amplitude, peak-to-peak value, cyclic frequency (in Hz), angular frequency (in rad/s), period, and phase. v 1(t): amplitude=52 volts, frequency=90. 7 Hz (570 rad/s), period=11. 0 ms, p-p value = 104 volts, and phase = 30° v 2(t): amplitude = 35 volts, frequency=1000 Hz (6280 rad/s), period=1. 00 ms, p-p value =70 volts, and phase=-70° v 3(t): amplitude=20 volts, frequency=159 Hz (1000 rad/s), period=6. 28 ms p-p value =40 volts, and phase=0° 41

2)Prob 1 Use KVL to find the unknown phasor voltage V. 42 9. 39∠ 242° V

Problem 1 Determine the equivalent impedance at the terminals A-B when the circuit operates at a frequency of 50 rad/s 3) 43 252 ∠ -10. 7Ŷ Ω = 248 - j 47 Ω

4 44 vo(t) = -32 sin(8000 t) (or 32 cos(8000 t + 90) )