Understanding Transfer Function Notation in Circuit Engineering

[FrogPad]

Just a quick question about some notation used in my book.

The proper form of the transfer function used in my book is as follows:

$$\bar H(j\omega) = \frac{K_0(j\omega)^{\pm N} (1+j\omega\tau_1)(1+2\zeta_3(j\omega\tau_3)+(j\omega\tau_3)^2)\cdot\cdot\cdot }{(1+j\omega \tau_a)(1+2\zeta_b(j \omega \tau_b)+(j \omega \tau_b)^2 )\cdot \cdot \cdot}$$

I'm kinda just being picky here, but I would like to understand the convention that they used.

Why the jump from $\tau_1$ to $\tau_3$, the choice of starting with $\zeta_3$ in the numerator. Just curious if someone could shed some light on this.

Thank you

 

[FredGarvin]

Is this in circuits, controls, vibrations...? What is the topic?

By the looks of it, it's a formulation from an infinite series.

 

[FrogPad]

Woops I forgot about this post

Sorry, I should have specified where this came from. This is from a basic circuit engineering course, specfically from the book: "Basic Engineering Circuit Analsysis" 8th Edition, Irwin/Nelms.

It looks like an expansion of some sort. The lecture notes have been very good, so I haven't been reading chapters like I typically do, just skimming them...it looks like I missed a page or two.

$$\bar H(s) = \frac{N(s)}{D(s)}=\frac{K_0(s-z_1)(s-z_2)\cdot \cdot \cdot(s-z_m)}{(s-p_1)(s-p_2)\cdot\cdot\cdot(s-p_n)}$$ (1)

where:
$$s= j\omega$$
$$N(s) =$$ a polynomial of degree m
$$D(s) =$$ a polynomial of degree n

Also, it says that in general (1) can be expressed in the form that I gave in the OP. Hope that helps clear things up.