Phasors

Introduction

        By this time you should know a great deal about DC circuits.  In particular, you should be familiar with the following.

• Resistance

• Ohm's Law

• Series & Parallel Combinations of Resistors

• Kirchhoff's Current Law (KCL)

• Kirchhoff's Voltage Law (KVL)

• The Method of Node Volages

• The Method of Mesh Equations

• Thevinin and Norton Equivalent Circuits

        When you learned these topics you might have assumed that what you were learning was something that only applied to DC circuits.  In a way you were right, but in a way you were wrong.  While the methods you learned only apply to DC circuits, the methods you learned can also be generalized to AC circuits.  It will take a little work, but every one of the analysis tools listed above has a related - and very similar - AC version.

        If you want to generalize your tool set (all of the tools listed above) you will first need to learn about phasors.  Here is a short lesson on phasors and an introduction to using phasors.

• A sinusoidal signal - voltage or current or any other physical variable - can be represented by a phasor which encodes the amplitude and phase information for the signal.  Call that phasor V.

• If the sinusoidal signal is given by:

• v(t) = V_max cos(wt + f)

• NOTE:  We could also use:  v(t) = V_max sin(wt + f).  What matters is the relative phase between signals.

• Then, the signal can be represented by a phasor:

• v(t) <=> V_max/f

• The little symbol "<=>" means that the phasor - the expression on the right - represents a voltage function of time.

• The voltage function of time is sinusoidal with an amplitude of V_max, and a phase angle of f.

• and we would write an expression for the phasor V:

• V = V_max/f

Phasors in linear circuits are related if all of the signals in the circuit are at the same frequency.  If you have signals that are different frequencies you can't use phasors to figure out how signals add, etc.

Important Properties of Phasors - Adding Phasors

        In this section we will examine some simple properties of phasors.  We start with one that may or may not be obvious.

• Adding phasors is equivalent to adding the corresponding time function for each phasor.

        To see how this works out, consider the sum of two voltage phasors

This sum corresponds to the sum of the two voltages (as might occur when you write KVL).  The actual sum would be written:

v_sum(t) = V_sum cos(wt + f_sum) = V₁ cos(wt + f1) + V₂ cos(wt + f2)

The problem is to be able to use the phasor method to do the addition, then interpret that in terms of the time functions.  You and interpret the problem in at least two different ways.

• One way to interpret this is in terms of complex numbers.  The sum voltage - represented as a phasor - is equal to the sum of the other two voltages - represented as phasors.

• V_sum = V₁ + V₂ = V₁/f1+ V₂ /f2

• Each phasor can be represented by a complex number.  Break each phasor into real and imaginary parts.

• V₁ = V₁cos(f1) + jV₁sin(f1)

• V₂ = V₂cos(f2) + jV₂sin(f2)

• So, the sum of the two phasors can be computed by adding the real and the imaginary parts separately, giving:

• = V₁cos(f1) + V₂ cos(f2) + j[V₁sin(f1)+ V₂ sin(f2)]

• Then, we can note the the real part and the imaginary part are the real and imaginary parts of the sum.