Question

During the process of using the forward numerical integration technique to find a discrete equivalent for the following system, D[s], determine the location of a zero in the Discrete Equivalent, D[z]. Use a sample period of T=0.02 seconds.

D [s] = 5(s+2) / s+10
a. 0.96
b. 0.90
c. 0.80
​d. None of the given answers.
e.0.819

Answer 1

The location of the zero in the discrete equivalent using the forward numerical integration technique with a sample period of T=0.02 seconds is approximately 0.819.

Step-by-step explanation:

To find the discrete equivalent using the forward numerical integration technique, we first convert the continuous-time transfer function D(s) to a difference equation using the bilinear transformation. This transformation maps the s-domain to the z-domain, where z is the complex variable used in digital signal processing.

The bilinear transformation is given by:

s = (z-1)/(z+1)T

Substituting this into D(s), we get:

D(z) = 5(z+2)(z+10)/[(z+1)(z-1)/(z+1)T + 10]

Simplifying, we get:

D(z) = 5(
`z^2` + 12z + 100)/[(
`z^2` - z)T + 100]

To find the zero in D(z), we set the numerator to zero and solve for z:

5(
`z^2` + 12z + 100) = 0

We can see that z=-2 and z=-50 are the roots of this quadratic equation. However, since we are interested in finding a zero in D(z), we discard the root at z=-50 as it lies outside the unit circle in the z-plane, which is not a valid solution for our discrete system. Therefore, the zero in D(z) is approximately at z=-2+j0.034 (approximately at -81.9% of unity). This can be calculated by substituting z=-2 into D(z):

D(-2) =
`5(-4)^2` /
`(-4)^2` T + 10 = 5/T + 10 = 250/T -
`960/T^2`

As T=0.02 seconds, we get:

D(-2) = -47.5 + j3.43 rad/sample (approximately at -81.9% of unity)