Chapter 9Complex Numbers and Phasors

Huseyin BilgekulEENG224 Circuit Theory IIDepartment of Electrical and Electronic EngineeringEastern Mediterranean UniversityChapter Objectives: Understand the concepts of sinusoids and phasors. Apply phasors to circuit elements. Introduce the concepts of impedance and admittance. Learn about impedance combinations. Apply what is learnt to phase-shifters and AC bridges.

Complex Numbers A complex number may be written in RECTANGULAR FORM as: x is the REAL part. y is the IMAGINARY part. r is the MAGNITUDE. is the ANGLE. A second way of representing the complex number is by specifying the MAGNITUDE and r and the ANGLE in POLAR form. The third way of representing the complex number is the EXPONENTIAL form.

Complex Numbers A complex number may be written in RECTANGULAR FORM as: forms.

Complex Number Conversions We need to convert COMPLEX numbers from one form to the other form.

Mathematical Operations of Complex Numbers Mathematical operations on complex numbers may require conversions from one form to other form.

Phasors A phasor is a complex number that represents the amplitude and phase of a sinusoid. Phasor is the mathematical equivalent of a sinusoid with time variable dropped. Phasor representation is based on Eulers identity.

Given a sinusoid v(t)=Vmcos(t+).

Phasors Given the sinusoids i(t)=Imcos(t+I) and v(t)=Vmcos(t+ V) we can obtain the phasor forms as:

ExampleTransform the following sinusoids to phasors:i = 6cos(50t 40o) Av = 4sin(30t + 50o) VPhasorsAmplitude 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.Solution:a. I Ab. Since sin(A) = cos(A+90o); v(t) = 4cos (30t+50o+90o) = 4cos(30t+140o) V Transform to phasor => V V

Example 5: Transform the sinusoids corresponding to phasors:a)b) Phasors

Phasor as Rotating Vectors

Phasor Diagrams The SINOR Rotates on a circle of radius Vm at an angular velocity of in the counterclockwise direction

Phasor Diagrams

Time Domain Versus Phasor Domain

Differentiation and Integration in Phasor Domain Differentiating a sinusoid is equivalent to multiplying its corresponding phasor by j. Integrating a sinusoid is equivalent to dividing its corresponding phasor by j.

Adding Phasors Graphically Adding sinusoids of the same frequency is equivalent to adding their corresponding phasors. V=V1+V2

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? 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.Solving AC Circuits