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
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:
= R2 + X2
= 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
+ Z2 = ( R1 + R2 ) + j( X1
+ X2 )
- 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
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
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
Z1 / Z2
= Z1 / Z2 / (q1
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.