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Complex Numbers for AC Circuits Analysis
Complex numbers are very useful in circuit analysis. One of the most
important uses is the use of complex numbers in phasor analysis. A complex
number consists of a real and an imaginary part. Both the real and imaginary
parts are real numbers, but the imaginary part is multiplied with the square
root of -1. Complex numbers can be expressed in numerous forms.
Rectangular Form
A complex number in rectangular form looks like this
where a is the real part and b is the imaginary part. j is sqrt(-1)
Adding and subtracting complex numbers in rectangular form is carried out by
adding or subtracting the real parts and then adding and subtracting the
imaginary parts.
- (5 + j2) + (2 - j7) = (5 + 2) + j(2 - 7) = 7 - j5
- (2 + j4) - (5 + j2) = (2 - 5) + j(4 - 2) = -3 + j2
Multiplying is slightly harder than addition or subtraction. It must be
carried out like the multiplication of two binomials, multiplying both parts of
one by both parts of the other. (Remember that when you multiply the
imaginary parts, j*j = sqrt(-1) * sqrt(-1) = -1, so that part becomes real).
- (2 + j2) (8 - j3) = (2 * 8) + j(2 * 8) + j(2 * -3) + j*j (2 * -3) = 16 + j16
- j6 + 6 = 22 + j10
Division requires a new idea to be introduced. The complex conjugate of
a number is the number that has the same real part as the original number but an
imaginary part that differs only in its sign. The complex conjugate is
denoted by an asterisk immediately following the number or variable.
- A = (2 + j2)
- A* = (2 + j2)* = 2 - j2
When dividing two complex numbers, you must first multiply both the numerator
and denominator by the complex conjugate of the denominator. This
multiplication results in a denominator that has only a real part.
- (4 + j3) / (2 + j2) = ((4 + j3) (2 - j2)) / ((2 + j2) (2 - j2)) = ((8 + 6) +
j(8 - 6)) / ((4 + 4) + j(4 - 4)) = (14 + j2) / 8 = 7/4 + j/4
Polar Form
Another way to represent complex numbers is in polar form. If you look
at the real and imaginary parts of a complex number as coordinates in a plane,
then the real part would be the x coordinate and the imaginary part the y coordinate.
In rectangular form, the x and y coordinate are specified in that way. In
polar form, the point in the plane is instead defined by a magnitude and an
angle. Polar form relates to rectangular form in the following way.
- (magnitude) r = sqrt(a² + b²)
- (angle) theta = tan-1 b/a
and
- a = r cos (theta)
- b = r sin (theta)
A complex number is then represented as
where r = magnitude = |A| and theta = angle = ang A
To find the conjugate of a complex number in polar form, simply reverse the
sign of the angle.
- A = 5 | 25°
- A* = 5 | -25°
Multiplication and division are much simpler for numbers in polar form.
With multiplication you multiply the magnitudes and add the angles, and with
division you divide the second magnitude from the first and subtract the second
angle from the first.
- (5 | 45° ) (2 | 20°) = (5)(2) | (45 + 20)° = 10 | 65°
- (4 | 90°) / (2 | 45°) = 4/2 | (90 - 45)° = 2 | 45°
To add or subtract we basically need to convert the numbers back into
rectangular form
- (2 | 45°) + (8 | 30°) = (2 cos 45 + 8 cos 30) + j(2 sin 45 + 8 sin 30) =
8.342 + j2.414
The answer can then be converted back to polar form if desired
- r = sqrt(8.342² + 2.414²) = 8.684
- theta = tan-1 (2.414/8.342) = 16.1°
- (2 | 45°) + (8 | 30°) = 8.684 | 16.1°
If possible is it best to add and subtract in rectangular form and multiply
and divide in rectangular form.
Complex Exponential Form
The complex exponential form is another way of representing a complex
number. The following formula shows how it relates to rectangular and
polar form
- ej*(theta) = cos (theta) + j sin (theta) = 1 | theta
Basically, the complex exponential works in the same way as polar form,
multiplication and division are carried out simply by multiplying (for
multiplication) or dividing (for division) the coefficients, and then adding
(for multiplication) or subtracting (for division) the angle. The complex
exponential is important for deriving formulas and is the basis for some of the
methods of circuit analysis, but from an algebraic standpoint it behaves in a
similar way as polar form.
