Complex Numbers, Sinusoidal Sources & Phasors ELEC 308 Elements of Electrical Engineering Dr. Ron Hayne Images Courtesy of Allan Hambley and Prentice-Hall

Complex Numbers  Complex numbers involve the imaginary number EE’s use j instead of i because i is used for current  A complex number Z = x+jy Has a real part x Has an imaginary part y Can be represented by a point in the complex plane ELEC 3082

Basic Concepts  Pure imaginary number has real part zero  Pure real number has imaginary part zero  Complex numbers of the form x+jy are in rectangular form  Complex conjugate of a number in rectangular form is obtained by changing the sign of the imaginary part ex. Complex conjugate of z 3 = 3-j4 is z 3 * = 3+j4 ELEC 3083

Example A.1  Complex Arithmetic in Rectangular Form Given that z 1 = 5+j5 and z 2 = 3-j4, reduce the following to rectangular form: z 1 +z 2 z 1 -z 2 z 1 z 2 z 1 /z 2 ELEC 3084

Polar Form  Complex number z can be expressed in polar form Give length of vector that represents z Denoted as |z| Called the magnitude of the complex number z Give angle of vector that represents z angle between vector and positive real axis Usually represented by θ ELEC 3085

Polar-Rectangular Conversion  Use trigonometry and right triangles: ELEC 3086

Example A.2 ELEC 3087

Example A.3 ELEC 3088

Euler’s Identity  What do complex numbers have to do with sinusoids? Euler’s identity: ELEC 3089

Exponential Form ELEC 30810

Example A.4 ELEC 30811

Arithmetic Operations ELEC 30812

Example A.5 ELEC 30813

Sinusoidal Voltage ELEC 30814

Sinusoidal Signals  Same pattern of values repeat over a duration T, called the period Sinusoidal signals complete one cycle when the angle increases by 2π radians, or ωT = 2π Frequency is number of cycles completed in one second, or f = T -1 Units are hertz (Hz) or inverse seconds (sec -1 ) Angular frequency given by ω = 2πf = 2πT -1 Units are radians per second ELEC 30815

Sinusoidal Signals  Argument of cosine or sine is ωt+θ To evaluate cos(ωt+θ) May have to convert degrees to radians, or vice versa  Relationship between cosine and sine ELEC 30816

Root-Mean-Square (RMS) ELEC 30817

Root-Mean-Square (RMS) ELEC 30818

RMS Value of a Sinusoid  Important Note: THIS ONLY APPLIES TO SINUSOIDS!!!  What is the peak voltage for the AC signal distributed in residential wiring in the United States? ELEC 30819

Example 5.1  Suppose that a voltage given by is applied to a 50-Ω resistance. Sketch v(t) to scale versus time. Find the RMS value of the voltage. Find the average power delivered to the resistance. ELEC 30820

Example 5.1 ELEC 30821

Exercise 5.3  Suppose that the AC line voltage powering a computer has an RMS value of 110 V and a frequency of 60 Hz, and the peak voltage is attained at t = 5 ms.  Write an expression for this AC voltage as a function of time. ELEC 30822

Phasors  Sinusoidal steady-state analysis Generally complicated if evaluating as time- domain functions Facilitated if we represent voltages and currents as vectors in the complex-number plane These vectors are also called PHASORS Convenient methods for adding and subtracting sinusoidal waveforms (for KCL and KVL) Standard trig. techniques too tedious ELEC 30823

Voltage Phasors ELEC 30824

Current Phasors ELEC 30825

Adding Sinusoids ELEC 30826

Exercise 5.4 ELEC 30827

Phasors as Rotating Vectors ELEC 30828

Phase Relationships ELEC 30829

Phase Relationships ELEC 30830

Exercise 5.5 ELEC 30831

Summary  Complex Numbers Rectangular Polar Exponential  Sinusoidal Sources Period Frequency Phase Angle RMS Phasors ELEC 30832