Chapter 3 Phasors & Complex Numbers

Review of Complex Numbers

• A complex number z is written in the rectangular form as z = x + jy with (x, y) belong to RxR where x is the real (Re) part of z : x = Re(z) y is the imaginary (Im) part of z : y = Im(z) j is defined by j2 = -1

• z may be written in polar form as Imaginary axis

θ z = |z| e j = |z| ∠θ z |z|

θ where |z| is the magnitude of z θ is its phase angle Real axis

θ Applying Euler's identity, e j = cosθ + j sinθ , we can convert z from polar form into rectangular form.

• The complex conjugate of z is called “z star” and obtained by replacing j by (-j):

θ z* = (x + jy)* = x – jy = |z| e -j

2 From this definition we have |z| = zz*

www.mywbut.com 3-1 Review of Phasors

Phasor analysis is a useful mathematical tool for solving problems where you have periodic time functions.

It allows us to convert an integro-differential equation into a linear equation with no sinusoidal function.

After solving for the desired variable, conversion from the phasor domain back to the time domain provides the result.

Example for a RC circuit:

The voltage source is a sinusoidally time-varying function,

VS (t) = VO sin (ω t + φΟ )

Application of Kirchhoff's voltage law:

R i(t) + (1/C) i(t) dt = VS(t) (time domain)

Objective: obtain an expression of the current i(t).

www.mywbut.com 3-2 • Use of the phasor analysis to solve this equation:

Step 1: Adopt a cosine reference

We express the forcing function VS(t) as a cosine, knowing that sinx = cos( π/2 – x) and cos(-x) = cosx

then, VS(t) = VO cos ( π/2 – ω t – φΟ ) = VO cos (ω t + φΟ – π/2 )

Step 2: Express time-dependent variables as phasors

Any cosinusoidally time-varying function, z(t), can be written as the real part of a complex number

ω z(t) = Re [ Z ej t ]

where Z is a time-independent, usually complex, function called the phasor of the instantaneous function z(t).

VS(t) = ?

Next we define the unknown variable i(t) in terms of I,

ω i(t) = Re[ I ej t ]

di/dt = ?

i dt = ?

www.mywbut.com 3-3 Step 3: Write the differential equation in phasor form.

jωt jωt jωt R Re[ Ie ] + (1/C) Re[ ( I/jω ) e ] = Re[VS e ]

I ( R + 1/ jCω ) = VS (phasor domain)

Step 4: Solve the phasor-domain equation.

jθ I = VS / ( R + 1/ jCω ) = Io e

To solve we need to convert the right-hand side into the jθ form Ioe , with Io being a real quantity.

I = ?

step 5: Find the instantaneous value

ω We have i(t) = Re[ I ej t ]