Lesson 15 Phasors Complex Numbers and Impedance 1

Lesson 15: Phasors, Complex Numbers and Impedance 1

Learning Objectives • Define a phasor and use phasors to represent sinusoidal voltages and currents. • Determine when a sinusoidal waveform leads or lags another Graph a phasor diagram that illustrates phase relationships. • Define and graph complex numbers in rectangular and polar form. • Perform addition, subtraction, multiplication and division using complex numbers and illustrate them using graphical methods. • Represent a sinusoidal voltage or current as a complex number in polar and rectangular form. • Define time domain and phasor (frequency) domain. • Use the phasor domain to add/subtract AC voltages and currents. 2

Learning Objectives • For purely resistive, inductive and capacitive elements define the voltage and current phase differences. • Define inductive reactance. • Understand the variation of inductive reactance as a function of frequency • Define capacitive reactance. • Understand the variation of capacitive reactance as a function of frequency • Define impedance. • Graph impedances of purely resistive, inductive and capacitive elements as a function of phase. 3

Complex numbers • A complex number (C) is a number of the form: , which is known as the rectangular form. • where a and b are real and • a is the real part of C and jb is the imaginary part. • Complex numbers are merely an invention designed to allow us to talk about the quantity j. • j is used in EE to represent the imaginary component to avoid confusion with CURRENT (i). • Solving AC circuits is simplified (no, really) through the use of phasor transforms, which we will now discuss at length… 4

Geometric Representation • In the rectangular form (C=a+jb), the x-axis is the real axis and the y-axis is the imaginary (j) axis. • The polar form (C=Z Ѳ), where Z is the distance (magnitude) from the origin and Ѳ is the angle measured counterclockwise (CCW) from the positive, x (or real) axis (the y-axis is still the imaginary (j) axis). C = 6 + j 8 (rectangular form) C = 10 53. 13º (polar form) 5

Conversion Between Forms • To convert between forms where: • apply the following relations: 6

Example Problem 1 • Convert (5∠ 60) to rectangular form. • Convert 6 + j 7 to polar form. 7

Properties of j 8

Addition and Subtraction of Complex Numbers • Easiest to perform in rectangular form. Example: Given A =6 +j 12 and B =7 + j 2 ADDITION: • Add the real and imaginary parts separately. SUBTRACTION: • Subtract the real and imaginary parts separately. 9

Multiplication and Division of Complex Numbers • Multiplication and Division is easiest to perform in polar form: • Multiplication: multiply magnitudes and add the angles: • Division: Divide the magnitudes and subtract the angles: • Also: the reciprocal of • The conjugate of C is C* and has the same real value but the OPPOSITE imaginary part: 10

Example Problem 2 Given A =1 +j 1 and B =2 – j 3 • Determine A+B: • and A-B: Given A =1. 41 45° and B =3. 61 -56° • Determine A/B: • and A*B: 11

Example Problem 3 Now, make sure you know how to use your calculator: 1. (3 -i 4) + (10∠ 44) then convert to rectangular: ANS: 10. 6∠ 16. 1 (polar) ANS: 10. 2 + j 2. 9 (rectangular) 2. (22000+i 13)/(3∠-17) then convert to rectangular: ANS: 7. 3*103∠ 17. 0 (polar) ANS: 7. 01*103 + j 2. 15 (rectangular) 3. Convert 95 -12 j to polar: 12 ANS: 95. 8∠-7. 2 (polar)

Phasor Transform • To solve problems that involve sinusoids (such as AC voltages and currents) we use the phasor transform: 1. We transform sinusoids into complex numbers in polar form… 2. solve the problem using complex arithmetic (as described previously)… 3. Then transform the result back to a sinusoid. 13

Phasors • A phasor is a rotating vector whose projection on the vertical axis can be used to represent a sinusoid. • The length of the phasor is the amplitude of the sinusoid (Vm) • The angular velocity of the phasor is . 14

Representing AC Signals with Complex Numbers • By replacing e(t) with it’s phasor equivalent E, we have transformed the source from the time domain to the phasor domain. • Phasors allow us to convert from differential equations to simple algebra. • KVL and KCL still work in the phasor domain. 15

Formulas from Trigonometry • Sometimes signals are expressed in cosines instead of sines. • Below are some formulas to refresh your memory of some of the trig functions: 16

Using Phasors to Represent AC Voltage and Current • Looking at the sinusoid equation, determine Vm and the phase offset Ѳ: • Using Vm, determine VRMS using the formula: • The voltage phasor is then: • The same holds true for current: 17

Example Problem 4 1. Express 100 sin (ωt)as a voltage phasor. 2. Express 50 sin(ωt+45⁰) as a voltage phasor. 3. Express 50 cos(ωt+45⁰) as a voltage phasor. v Remember, we are representing the sinusoid as a sin function, therefore we need to use one of the trig conversions: cos(ωt+45⁰) = sin(ωt+45⁰+90⁰) 18

Representing AC Signals with Complex Numbers • Phasor representations can be viewed as a complex number in polar form: • NOTE: Before writing the sinusoid equation after finding the phasor, you need to convert the phasor magnitude back to the sinusoid representation (i. e. multiplying the phasor magnitude by 19

Phase Difference • Phase difference is angular displacement between waveforms of same frequency. • If angular displacement is 0°then the waveforms are in phase. • If angular displacement is not 0 o, they are out of phase by amount of displacement. Example: If v 1 = 5 sin(100 t) and v 2 = 3 sin(100 t - 30°), then v 1 leads v 2 by 30° 20

Phase Difference w/ Phasors • The waveform generated by the leading phasor leads the waveform generated by the lagging phasor. 21

Example Problem 5 i 1 = 20 sin ( t) m. A. i 2 = 10 sin ( t+90˚) m. A. i 3 = 30 sin ( t - 90˚) m. A. Determine the equation for i. T. 22

R, L and C circuits with Sinusoidal Excitation • R, L, C have very different voltage-current relationships. Recall: • Sinusoidal (AC) sources are a special case! 23

The Impedance Concept • Impedance (Z) is the opposition (i. e. resistance) that a circuit element presents to current in the phasor domain. It is defined as: • Ohm’s law for AC circuits: 24

Impedance • Impedance is a complex quantity that can be made up of Resistance (R) (real part) and Reactance (X) (imaginary part). • Unit of impedance is Ohms ( ). Z X q R 25

Resistance and Sinusoidal AC • For a purely resistive circuit, current and voltage are in phase. 26

Resistors • For resistors, voltage and current are in phase: 27

Example Problem 6 Two resistors R 1=10 kΩ and R 2=12. 5 kΩ are in series. i(t) = 14. 7 sin (ωt + 39˚) m. A a) b) c) d) Compute VR 1 and VR 2 Compute VT = VR 1 + VR 2 Calculate ZT Compare VT to the results of VT = I ZT Note that these are in phase so we simply sum the magnitudes. May not always be this easy… 28

Inductance and Sinusoidal AC • Voltage-Current relationship for an inductor: • It should be noted that for a purely inductive circuit voltage leads current by 90º. 29

Inductive Impedance • Impedance can be written as a complex number (in rectangular or polar form): • Since an ideal inductor has no real resistive component, this means the reactance (X) of an inductor is the pure imaginary part: 30

Inductance and Sinusoidal AC Again, voltage leads current by 90˚. 31

Inductance • For inductors, voltage leads current by 90º. 32

Impedance and AC Circuits Solution technique 1. 2. 3. 4. Transform time domain currents and voltages into phasors. Calculate impedances for circuit elements. Perform all calculations using complex math. Transform resulting phasors back to time domain (if reqd). 34

Example Problem 7 For the inductive circuit: v. L = 40 sin (ωt + 30˚) V f = 26. 53 k. Hz L = 2 m. H Determine VL and IL Graph v. L and i. L Notice 90°phase difference! i. L = 120 sin (ωt - 60˚) m. A 35

Capacitance and Sinusoidal AC • Current-voltage relationship for an capacitor: • It should be noted that, for a purely capacitive circuit current leads voltage by 90º. 36

Capacitive Impedance • Impedance can be written as a complex number (in rectangular or polar form): • Since a capacitor has no real resistive component, this means the reactance of a capacitor is the pure imaginary part: 37

Capacitance and Sinusoidal AC Again, current leads voltage by 90˚. 38

Capacitance • For capacitors, current leads voltage by 90º. 39

ELI the ICE man Pneumonic Ca C pa urr cit en a t Vo nc lta e ge In Vol du ta ct ge Cu anc rre e nt ELI the ICE man I leads E E leads I When voltage is applied to an inductor, it resists the change of current. The current builds up more slowly, lagging in time and phase. 41 Since the voltage on a capacitor is directly proportional to the charge on it, the current must lead the voltage in time and phase to conduct charge to the capacitor plate and raise the voltage

Frequency Dependency • Inductors: • Capacitors: 42

QUESTIONS? 43