Crash Course in Complex Numbers

(for Electronics Students and Others)

These rules will help keep all your operations straight when you work with complex numbers.  In electronics we use the imaginary operator "j" instead of "i" (the mathematician's operator) because "I" and "i" represent electric current.  The effect is the same.  Remember:
 

        j2 = -1

All complex numbers can be represented in two forms, the rectangular form and the polar form.  The rectangular form is the one that uses the real and imaginary values - (R + jX) where R is the real (resistive) component and X is the imaginary (reactive) component.  Polar form means that the complex number is expressed as a magnitude and an angle - (Z / q) where Z is the magnitude (impedance) and q is the angle (phase).
 

Converting Complex Numbers

Complex numbers can be converted from one form to the other using the Rule of Pythagoras and by remembering that the two forms of the complex numbers make a right triangle.  Thus, we can use the following equations to convert from polar to rectangular form:
 

        Z2 = R2 + X2

 

        q = tan-1( X / R )

We can also make the conversion the other way.  These equations will allow you to convert from rectangular to polar form:
 

        R = Z cos ( q )

 

        X = Z sin ( q )

where cos and sin are the familiar cosine and sine functions on your calculators.   Please, note:  YOUR CALCULATOR PROBABLY HAS THESE FUNCTIONS BUILT INTO IT.  Check the manual to find out how to get it to give you a polar to rectangular conversion and a rectangular to polar conversion.  Most scientific calculators manufactured today will do this.
 

Adding Complex Numbers

When you want to add two complex numbers, what you have to do is make sure they are both in the rectangular form.  That way, you can add the real parts and the imaginary parts separately.  The sum of the two complex numbers is basically the sum of the reals and the sum of the imaginaries.  Subtraction is the same - subtract the reals and subtract the imaginaries.  Use these equations to add and subtract complex numbers:
 

        Z1 + Z2 = ( R1 + R2 ) + j( X1 + X2 )

 

        Z1 - Z2 = ( R1 - R2 ) + j( X1 - X2 )

As you can see, this would be rather difficult to do if we had not expressed both of our complex numbers in rectangular form.  In electronics we find ourselves using this technique a fair amount to solve AC circuits with both resistors and reactive elements such as capacitors and inductors.  In writing our loop equations, we add resistors and reactive elements separately because of the way they work in the circuits.
 

Multiplying Complex Numbers

When you want to multiply complex numbers, they need to be in the same form - either both in rectangular form or both in polar form. If they are in rectangular form, the answer will be in rectangular form.  You need to be careful to get the terms straight and to remember the distributive property.  Use this equation to multiply complex numbers in rectangular form:
 

        Z1 x Z2 = ( R1 R2 + j2 X1 X2 ) + j ( R1 X2 + R2 X1 )

Expressed another way:
 

        Z1 x Z2 = ( R1 R2 - X1 X2 ) + j ( R1 X2 + R2 X1 )

since j2 = -1.   This method will give you an answer and the answer will be correct.  If you enjoy keeping up with details, this method is for you and you are finished.  There is a method that I find easier since there are fewer operations to do.  If that appeals to you, then use this equation to multiply complex numbers in polar form:
 

        Z1 x Z2 = Z1 Z2 / ( q1 + q2 )

Now, isn't that easy?  Well, easier.  OK, less painful.  There are fewer operations to worry about and if your calculator gave you the magnitudes and angles, it really is simpler.  This only leaves the division operation and we are experts in the use of complex numbers.
 

Dividing Complex Numbers

 

 

To understand division, we need to recall a couple of things.  The first is that mathematicians have a hissy fit if you leave a complex number (or anything with an imaginary operator) in the denominator of a fraction.  Since we want to turn a complex number into a real number we need to remember the other thing.  The other thing is that in algebra we learned that:
 

        ( x + a ) ( x - a ) = x2 - a2

This gives us a way to turn a complex number into a real number.  If we multiply the denominator by the same complex number with the opposite sign, we get:
 

        ( R + jX ) ( R - jX ) = R2 - j2 X2 = R2 + X2


 Notice that the denominator is now made up of the sum of two positive real numbers, so it is positive and real.  The magic number we used to get rid of the imaginary part is called the COMPLEX CONJUGATE.  It is the same complex number with the opposite sign.  This is regardless of the sign of both R and X, though we usually only work with positive resistors.  The main point is that it gives us no imaginary numbers in the denominator.  Well, that was what we wanted, but there is another of those rules in math that tells us if we do something to the denominator we have to do the same thing to the numerator.  There is always a catch.  Multiply the top and bottom of the complex fraction by the complex conjugate of the denominator.  This will give you a complex result.  Use this equation to divide complex numbers in rectangular form:
 

        Z1 / Z2 = (R1 + jX1)(R2 - jX2) / [ (R2 + jX2)(R2 - jX2) ]

 

      Z1 / Z2 = ( R1 R2 + X1 X2 ) / (R22 + X22) + j (R2 X1 - R1 X2) / (R22 + X22)

As with multiplication, we have to keep up with all the terms and especially the signs.  This is a BE CAREFUL zone if there ever was one in math.  Notice there is a real part of the answer and an imaginary part, so the answer is complex as we would expect.  Do I hear you asking if there might be an easier way?

Dividing complex numbers is somewhat simpler to do in polar form.  Just as with multiplying complex numbers, this operation is best done in polar form.  To divide two complex numbers in polar form all you have to do is divide the magnitudes and subtract the angles.  This even makes sense because we learned in math that division means "invert and multiply."  Use this equation to divide complex numbers in polar form:
 

        Z1 / Z2 = Z1 / Z2 / (q1 - q2)

I consider this easier to do than dividing in rectangular form.

So, let's see what we can do now.  We can convert complex numbers from rectangular to polar and back.  We can add and subtract complex numbers, and we can multiply and divide complex numbers in both rectangular and polar form.  With all these operations, we can solve circuits using resistors and reactive elements like capacitors and inductors.
 

Coming soon!  Examples.