# Passive filter design from transfer function

I would like to implement a passive RIAA filter circuit. This is the bode plot of a RIAA equalizer.

The transfer function of this:

H(jw) = (jwT1 + 1)/[(jwT2 + 1)(jwT3 + 1)] where time constants are defined: T1=318us, T2=3180us, T3=75us

So there is one zero and two poles. The circuit can be implemented in several ways, but I would like to implement it in the circuit below:

simulate this circuit – Schematic created using CircuitLab

I determined the transfer function of the circuit, which is as follows:

H(jw) = jw[(R2(C1+C2)]+1/[jw^2(R1R2C1C2)+jw(R2C1+R2C2+R1C2)+1]

The zero can be easily determined 1/[R2(C1+C2)], so T1=R2(C1+C2) The poles are given by the solution of the quadratic equation.

My question is how can I determine the component values based on the transfer function? Many people know how to determine poles and zeros by seeing the circuit, how?

• Why not take a rough guess at approximate values and simulate it? Commented Apr 21 at 13:09
• I assume you're doing this for your own personal education, because surely there must be thousands of articles describing every detail of every type of RIAA equalization circuit.
– pipe
Commented Apr 21 at 13:13
• pipe: Yes, this is for my own personal education to understand filter design. Commented Apr 21 at 13:40
• @slimcolt Also, how do you expect to get $+20\:\text{dB}$ of gain from a passive network? (I would like to know. I think I could make bank on the idea if you share it with me.) Commented Apr 21 at 13:45
• periblepsis: The equation has two real roots. So I think you're wrong about that. The circuit was the example, anything else could have been. I have studied many books, etc., but in most cases the derivation of the equations is missing. To fully understand, I need to know everything step by step. +20dB amplification is impossible with a passive circuit, but you know what to imagine for the output of the circuit, an operational amplifier with 10x amplification! :) If you don't know the answer to my question either, I think we'll leave the word battle behind. Commented Apr 21 at 13:51

Many people know how to determine poles and zeros by seeing the circuit, how?

Yes, and this would be my preferred method. So, it's easy to get stuff on RIAA equalization circuits on-line and here's one that's nearly what you want (ignoring the op-amps): -

It has, as expected, the standard response: -

So, use a simulator and make the above response a template for comparison with your circuit's spectrum. Like this (about 5 minutes messing around): -

The red line is the one that's from your circuit and, it's pretty close to the blue line (the reference). Probably within about 0.5 dB. But you could tweak it a tad more and get it exact if you were an audiophile.

The transfer function solution for your circuit is the following KCL result:

$$\frac{s\,R_2\left(C_1 + C_2\right) + 1}{s^2\,C_1\,C_2\,R_1\,R_2 + s\left(C_1\,R_2 + C_2\,R_1 + C_2\,R_2\right) + 1}$$

It's obvious from the above that $$\T_1= R_2\left(C_1 + C_2\right)\$$. Find the roots for the denominator using the usual, well-known quadratic equation can then get $$\T_2\$$ and $$\T_3\$$. I'm sure you can handle the algebra.

That's all there is to it. Three equations, three unknowns, one specified value. So given any one value, the other three can be found. There are a few more interesting ratios to find. But they aren't necessarily important. Just interesting.