Bilinear transformation

The bilinear transformation is another change of variables. We convert from the z variable into the lU variable. The transformation maps the unit circle in the z plane into the left half of the ID plane. This mapping converts the stability region back to the familiar LHP region. The Routh criterion can then be used. Root locus plots can be made in the 11 plane with the system going closedloop unstable when the loci cross over into the RHP. 

The transformation z = maps the left half of the s plane into the unit circle. Actually, a number of horizontal strips in the s plane are each mapped into the unit circle, one on top of the other. The mapping is, therefore, not unique. The primary strip from co = —coJ2 to co = coJ2, sketched in Fig. 19.8a, is the only one that we are interested in since. 4,, is periodic. 

These two equations differ only in the sign of z. Since we are interested in the interior and exterior of the unit circle, switching signs or quadrants in the z plane does not matter. 

To prove that this transformation maps the unit circle in the z plane into the left half of the U) plane, let us express the complex variables ID and z in the following rectangular and polar forms  

When r is less than 1, u is negative. Thus a point inside the unit circle in the z plane maps into a point in the left half of the TD plane. 

---

Tustin s Rule Tustin s rule, also called the bilinear transformation, gives a better approximation to integration since it is based on a trapizoidal rather than a rectangular area. Tustin s rule approximates the Laplace transform to... 

The examples below illustrate the use of the bilinear transformation to analyze the stability of sampled-data systems. We can use all the classical methods that we are used to employing in the s plane. The price that we pay is the additional algebra to convert to ID from z. 

Bilinear transformation to map the interior of the unit circle in the z-plane onto the left half of the complex variable -plane. (Application of the Routh-Hurwitz stability criterion). ... 

Most audio filters are typically HR filters because a). They are directly transformed from their analog counterparts via the bilinear transform b). They are faster to compute... 

After the bilinear transformation s = -—3—/ so the discrete formula of the controller i... 

Since (B.l) constitutes a bilinear transformation, we know from reference 123 that a general circle will be transformed into a general circle. In particular, we note that the real and imaginary axes will be transformed into themselves while the vertical line through Zl in the Z plane will have its infinity point transformed into (0, 0) in the F plane and the point Rl into 1/Rl- By further noting that the bilinear transformation is conformal [123], we see that the angle between the real axis and the locus in the Y plane must remain 90° that is, the center of the locus circle must be located on the real axis. That completely determines the locus circle in the Y plane as shown. 

Finally, the horizontal lines X = jXi and 7X2 will have their infinity point in the Z plane transformed into point (0, 0) in the Y plane while the two points jXi and 7X2 on the imaginary axis in the Z plane will be transformed into I/7X1 and 1/7X2, respectively, on the imaginary axis in the Y plane. Again, noting that the bilinear transformation preserves angles we reason that the two horizontal lines X = 7X1 and 7X2 will be circles in the Y plane with their centers on the imaginary axis as shown. That completely determines the locus in the Y plane when adding a series reactance to an arbitrary impedance Zl. 

In a bilinear transformation, the variable s in Ha (s) is replaced with a bilinear function of z to obtain H (z). Bilinear transformations for the four standard types of filters, namely, low-pass filter (LPF), high-pass filter (HPF), bandpass filter (BPF), and bandstop filter (BSF), are shown in Table 8.8. The second column in the table gives the relations between the variables s and z. The value of T can be chosen arbitrarily without affecting the resulting design. The third column shows the relations between the analog... 

The upper-half-plane of the Z-plane may be mapped into a unit disk shown in Fig. 2.8 by the following bilinear transformation ... 

---