# Electronics: Problem with textbook

1. Dec 27, 2008

### D.R.

1. The problem statement, all variables and given/known data
Not really a homework problem. The problem is that while I was busy deriving the equations stated in my textbook about electronics, I get an inconsistency with the transfer equations relating to bandpass filters. The paragraph in my book (Electronic Instrumentation, P.P.L. Regtien) talks about bandpass filters and gives several examples with their transfer functions given without derivation. The electronic network is a lowpass filter connected to a highpass filter with R1,C1 referring to the lowpass filter (R:resistance, C:capacitance) and R2,C2 referring to the highpass filter.

Low-pass network:
High-pass network:

2. Relevant equations
The transfer function is defined as:
$$H = \frac{V_O}{V_I}$$
, where the voltages refer to the input and output ports.

The voltages are written in complex notation for one angular frequency. All impedances are written in the complex notation as well. In addition, the symbol j stands for the complex number i, but since i also refers to current, the book uses j for the complex number i instead.

The correct solution according to the book is:
$$H = \frac{j\omega\tau_2}{1 + j\omega(\tau_1 + \tau_2 + a\tau_2) - \omega^2\tau_1\tau_2}$$

With:
$$\tau_1 = R_1C_1$$
$$\tau_2 = R_2C_2$$
$$a = R_1/R_2$$

3. The attempt at a solution
The transfer function:
$$H = \frac{V_O}{V_I}$$
, can be written as:
$$H = \frac{V_O}{V_I} = \frac{iZ_O}{iZ_I} = \frac{Z_O}{Z_I}$$
, where Z denotes impedance. I believe that I made a mistake here in assuming that the current in the entire network is the same. . But I might as well present the solution I found:

For output impedance:
$$\frac{1}{Z_O} = \frac{1}{R_2} + \frac{1}{\frac{1}{j\omega C_1}+\frac{1}{j\omega C_2}}$$

For input impedance:
$$Z_I = R_1 + \frac{1}{j\omega C_1 + \frac{1}{\frac{1}{j\omega C_2} + R_2}}$$

They can be written respectively as:
$$Z_O = \frac{j\omega (C_1R_2 + \tau_2)}{j\omega (C_1 + C_2) - \omega^2 C_1\tau_2}$$
$$Z_I = \frac{1 + j\omega (\tau_1 + \tau_2 + a\tau_2) - \omega^2\tau_1\tau_2}{j\omega (C_1 + C_2) - \omega^2 C_1\tau_2}$$

The transfer function becomes:
$$H = \frac{j\omega (C_1R_2 + \tau_2)}{1 + j\omega (\tau_1 + \tau_2 + a\tau_2) - \omega^2\tau_1\tau_2}$$

This looks almost exactly like the transfer function given in the book, but I got an extra term of $$j\omega C_1R_2$$, and that's where I get stuck. I think I made an error in the beginning, and that's due to that I have problems understanding the physics of band filters.

Last edited by a moderator: Apr 24, 2017