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 ] www.mywbut.com 3-4 .