Getting the Wrong Transfer Function for Controls Problem

Question

Background

So admittedly, it's been about 5 years since I've dealt with any kind of circuits and now I am going back and taking a controls course. On the very first problem I am doing, my answer is not agreeing with that in the back of the book. The controls book that I will be referencing is Control Systems Engineering, by Norman S. Nise 7th ed.

As far as I know I am doing everything exactly as described in the book (ch. 2). Anyway, here is the question (and I will show all of the work I have done as well).

Find the transfer function, , for each network shown in Figure P2.3. [Section: 2.4].

1. The first step is to write out all of the associated node equations, so I convert to the S-domain by:

2. My next step, is to represent everything as an admittance.

3. Now, I solve and get the following.

Question

In the book, they seem to be getting...

Am I actually going wrong somewhere, or is there an error with the book solution?

Answer 1

Consider the parallel RL network. The admittance is $$Y = \frac{1}{Z} = \frac{1}{s} + 0.5 \to Z = \frac{s}{0.5s + 1}$$

Now apply the voltage divider equation and multiply the numerator and denominator by 0.5s + 1 $$H = \frac{Z}{Z + 1} = \frac{s}{1.5s + 1} = \frac{2s}{3s + 2}$$

Your answer seems correct.