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Bilinear Transform

Module by: Bill Wilson. E-mail the author

Summary: Introduction of bilinear transform.

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There is a way that we can make things a good bit easier for ourselves however. The only drawback is that we have to do some complex analysis first, and look at a bilinear transform! Let's do one more substitution, and define another complex vector, which we can call rs r s :

rs| Γ v |ei( θ r 2βs) r s Γ v θ r 2 β s

The vector rs r s is just the rotating part of the crank diagram which we have been looking at. It has a magnitude equal to that of the reflection coefficient, and it rotates around at a rate 2βs 2 β s as we move down the line. For every rs r s there is a corresponding Zs Z s which is given by:

Zs= Z 0 1+rs1rs Z s Z 0 1 r s 1 r s

Figure 1: The vector rs r s .
Figure 1 (717.png)

Now, it turns out to be easier if we talk about a normalized impedance, which we get by dividing Zs Zs by Z 0 Z 0 .

Zs Z 0 =1+rs1rs Z s Z 0 1 r s 1 r s
which we can solve for rs r s
rs=Zs Z 0 1Zs Z 0 +1 r s Z s Z 0 1 Z s Z 0 1
This relationship is called a bilinear transform. For every rs r s that we can imagine, there is one and only one Zs Z 0 Z s Z 0 and for every Zs Z 0 Z s Z 0 there is one and only one rs r s . What we would like to be able to do, is find Zs Z 0 Z s Z 0 , given an rs r s . The reason for this should be readily apparent. Whereas, as we move along in s, Zs Z 0 Z s Z 0 behaves in a most difficult manner (dividing one phasor by another), rs r s simply rotates around on the complex plane. Given one r s 0 r s 0 it is easy to find another rs r s . We just rotate around!

We shall find the required relationship in a graphical manner. Suppose I have a complex plane, representing Zs Z 0 Z s Z 0 . And then suppose I have some point "A" on that plane and I want to know what impedance it represents. I just read along the two axes, and find that, for the example in Figure 2, "A" represents an impedance of Zs Z 0 =4+2i Z s Z 0 4 2 . What I would like to do would be to get a grid similar to that on the Zs Z 0 Z s Z 0 plane, but on the rs r s plane instead. That way, if I knew one impedance (say Z0 Z 0 = Z L Z 0 Z 0 Z 0 Z L Z 0 then I could find any other impedance, at any other s, by simply rotating rs r s around by 2βs 2 β s , and then reading of the new Zs Z 0 Z s Z 0 from the grid I had developed. This is what we shall attempt to do.

Figure 2: The complex impedance plane
Figure 2 (718.png)

Let's start with the above equation and re-write it as:

rs=Zs Z 0 +12Zs Z 0 +1=1+-2Zs Z 0 +1 r s Z s Z 0 1 2 Z s Z 0 1 1 -2 Z s Z 0 1

In order to use Equation 5, we are going to have to interpret it in a way which might seem a little odd to you. The way we will read the equation is to say: "Take Zs Z 0 Z s Z 0 and add 1 to it. Invert what you get, and multiply by -2. Then add 1 to the result." Simple isn't it? The only hard part we have in doing this is inverting Zs Z 0 +1 Z s Z 0 1 . This, it turns out, is pretty easy once we learn one very important fact.

The one fact about algebra on the complex plane that we need is as follows. Consider a vertical line, s, on the complex plane, located a distance d away from the imaginary axis. There are a lot of ways we could express the line s, but we will choose one which will turn out to be convenient for us. Let's let:

φ,π2φπ2:s=d(1itanφ) φ 2 φ π 2 s d 1 φ

Figure 3: A Vertical line, s, a distance, d, away from the imaginary axis
Figure 3 (719.png)

Now we ask ourselves the question: what is the inverse of s?

1s=1d11jtanφ 1 s 1 d 1 1 j φ
We can substitute for tanφ φ :
1s=1d11isinφcosφ=1dcosφcosφisinφ 1 s 1 d 1 1 φ φ 1 d φ φ φ
And then, since cosφisinφ=e(iφ) φ φ φ
1s=1dcosφe(iφ)=1dcosφeiφ 1 s 1 d φ φ 1 d φ φ

Figure 4: A plot of 1/s
Figure 4 (720.png)

A careful look at Figure 4 should allow you to convince yourself that Equation 9 is an equation for a circle on the complex plane, with a diameter = 1/d. If s is not parallel to the imaginary axis, but rather has its perpendicular to the origin at some angle ΦΦ, to make a line s s . Since s=seiΦ s s Φ , taking 1/s simply will give us a circle with a diameter of 1/d, which has been rotated by an angle ΦΦ from the real axis. And so we come to the one fact we have to keep in mind: "The inverse of a straight line on the complex plane is a circle, whose diameter is the inverse of the distance between the line and the origin."

Figure 5: The line s multiplied by eiΦ Φ
Figure 5 (721.png)
Figure 6: Inverse of rotated line
Figure 6 (722.png)

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