Phasors A phasor is a complex number that represents the magnitude and phase of a sinusoid:

A = z  q = z ejq = z cos q + j z sin q Complex Exponentials A complex exponential is the mathematical tool needed to obtain phasor of a sinusoidal function. A complex exponential is ejwt = cos wt + j sin wt A complex number A = z  q can be represented A = z  q = z ejq = z cos q + j z sin q We represent a real-valued sinusoid as the real part of a complex exponential. Complex exponentials provide the link between time functions and phasors. Complex exponentials make solving for AC steady state an algebraic problem

Complex Exponentials (cont’d) What do you get when you multiple A with ejwt for the real part? Aejwt = z ejq ejwt = z ej(wt+q) z ej(wt+q) = z cos (wt+q) + j z sin (wt+q) Re[Aejwt] = z cos (wt+q)

Sinusoids, Complex Exponentials, and Phasors z cos (wt+q)= Re[Aejwt] Complex exponential: Aejwt = z ej(wt+q), A= z ejq, Phasor for the above sinusoid: V = z  q

Phasor Relationships for Circuit Elements Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor.

I-V Relationship for a Capacitor v(t) + - i(t) Suppose that v(t) is a sinusoid: v(t) = VM ej(wt+q) Find i(t).

Computing the Current

Phasor Relationship Represent v(t) and i(t) as phasors: V = VM  q I = jwC V The derivative in the relationship between v(t) and i(t) becomes a multiplication by jw in the relationship between V and I.

I-V Relationship for an Inductor + i(t) v(t) L - V = jwL I

Kirchhoff’s Laws KCL and KVL hold as well in phasor domain.

Impedance AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohm’s law: V = I Z Z is called impedance.

Impedance (cont’d) Impedance depends on the frequency w. Impedance is (often) a complex number. Impedance is not a phasor (why?). Impedance allows us to use the same solution techniques for AC steady state as we use for DC steady state. Impedance in series/parallel can be combined as resistors

Impedance Example: Single Loop Circuit 20kW + + VC 10V  0 1mF - - w = 377 Find VC

Impedance Example (cont’d) How do we find VC? First compute impedances for resistor and capacitor: ZR = 20kW= 20kW  0 ZC = 1/j (377 1mF) = 2.65kW  -90

Impedance Example (cont’d) 20kW  0 + + VC 2.65kW  -90 10V  0 - -

Impedance Example (cont’d) Now use the voltage divider to find VC: ÷ ø ö ç è æ ° Ð W + = k 20 90 - 65 . 2 10V C V

Analysis Techniques All the analysis techniques we have learned for the linear circuits are applicable to compute phasors KCL&KVL node analysis/loop analysis superposition Thevenin equivalents/Notron equivalents source exchange The only difference is that now complex numbers are used. Phasors can then converted to corresponding sinusoidal functions to get the time-varying function.

Impedance V = I Z, Z is impedance, measured in ohms () Resistor: The impedance is R Inductor: The impedance is jwL Capacitor: The impedance is 1/jwC

Analysis Techniques All the analysis techniques we have learned for the linear circuits are applicable to compute phasors KCL&KVL node analysis/loop analysis superposition Thevenin equivalents/Notron equivalents source exchange The only difference is that now complex numbers are used. Phasors can then converted to corresponding sinusoidal functions to get the time-varying function.

Admittance I = YV, Y is called admittance, the reciprocal of impedance, measured in siemens (S) Resistor: The admittance is 1/R Inductor: The admittance is 1/jwL Capacitor: The admittance is jwC

Phasor Diagrams A phasor diagram is just a graph of several phasors on the complex plane (using real and imaginary axes). A phasor diagram helps to visualize the relationships between currents and voltages.

An Example I = 2mA  40 VR = 2V  40 VC = 5.31V  -50 – 1mF VC + 1kW VR V I = 2mA  40 VR = 2V  40 VC = 5.31V  -50 V = 5.67V  -29.37

Phasor Diagram Imaginary Axis Real Axis V VC VR

Homework #9 How to determine initial conditions for a transient circuit. When a sudden change occurs, only two types quantities will definitely remain the same before and after the change. IL(t), inductor current Vc(t), capacitor voltage Find these two types of the values before the change and use them as initial conditions of the circuit after change