I am going to do a series of posts and try explaining what impedance is and how to analyse a passive Low and High Pass filters. But for that we need to know what a complex number is and how to handle it. A complex number is in the format of A + Bj, where B is an imaginary number and A was real. That was the entire story I knew. Until recently, I didn’t know why it is necessary or what it represented. I will be going to great depth on this subject and begin with the absolute basics.
Graphical representation of a Complex Number
Let’s say we have a graph as shown in figure 1.
We have three points, A = (0, 15), B = (10, 0) and another point Z. We defined Z as the sum of A and B. So what is the value of Z.? What point does it represent? That is simple:
Z = A + B = (0 + 10, 15 + 0) = (10, 15) .We just add the X values and Y values together and that’s the answer.
A complex number can also be thought of just a point in a graph. The Y axis is represents the imaginary numbers and X real.
For example the graph in figure 1, the point A is represents the imaginary part and point B the real part of a complex number. Therefore, that complex number must be B + Aj, or, 10 + 15j. And this complex number is what Z represents.
Another way of thinking is that all points in a graph represent some complex number. A point that lies on the Y axis is purely imaginary, and, that which lies on the X axis is purely real.
Thus A is a complex number, with no real part and B with no imaginary part. To find A + B we simply do a complex number addition.
Z = B + A = ( 10 + 0j ) + ( 0 + 15j ) = 10 + 15j.
Polar and Cartesian form of a complex number
We can represent a point on a graph either in polar or Cartesian (or rectangular) format. And because a complex number is thought as nothing more than just a point on a graph, we can represent a complex number in either of these forms.
In Cartesian format, we represent a complex number by the point on the graph it represents. For example in the figure 1, the point Z represents the point (10, 15). So its Cartesian form is Z = 10 + 15j.
In the polar format, we represent a complex by its distance from the centre and an angle. The distance from the origin, is, the magnitude of the complex number. The complex number can also be thought as a vector; with a magnitude and a direction.
And the angle
In the polar format we can demote Z with its magnitude and it’s angle, so
- Say we want to represent Z = 10 + 15j in polar format. This is in Cartesian format, so we know, that it denotes a point of (10,15) on the graph.
Take another example. Convert P = 15.5 – 10.5j in polar format.
Again, we know that this complex number denotes the point (15.5 , 10.5) in a graph. So
We can convert a complex number in polar format to Cartesian format in the following way.
We know that the Cartesian format of a complex number is in the form
Z = x + yj.
From the basic trigonometry, and
thus the Cartesian form of the complex number is
For example: A the Cartesian form of the complex number
Multiplication, Division and Reciprocal of a complex number
Multiplication and division is easier when the complex number is represented in the polar format. For example:
We just multiply the scalar and add up the angles.
We divide the magnitude as usual, then to find the angle, we subtract the denominator from the numerator.
Reciprocal of a complex number is also easy to do, when the complex number is in the polar format. Just follow the division procedure, except the angle in numerator is zero.
In the next post, I will design a passive high pass and a low pass filter, and then you will see how important complex numbers are. It makes our lives a lot easier.
PS: I am not an expert in this, so please inform me, if I have done anything wrong, or something that you didn’t understand.