# Designing a circuit from a transfer function

I have this transfer function of a bandpass filter with 2 poles and 2 zeros:

$$H(s)= 10^{-0.8} \frac{\left( \dfrac{s}{10} +1 \right) \left( \dfrac{s}{100.000} +1 \right)} {\left( \dfrac{s}{100} +1 \right) \left( \dfrac{s}{10.000} +1 \right)}$$

I have a gain, 2 poles, 2 zeros and no idea how to design a circuit that has this transfer function.

Maybe it could be an active filter but it only can have 1 opamp (instructions) It could also be a passive filter. R, C and L can have any value.

How do I design a circuit from transfer function?

Your transfer function can be obtained in different ways and one of them could be by cascading two stages with a buffer in-between. Your original expression follows the form

$H(s)=H_0\frac{1+\frac{s}{\omega_{z1}}}{1+\frac{s}{\omega_{p1}}}\frac{1+\frac{s}{\omega_{z2}}}{1+\frac{s}{\omega_{p2}}}$

If you design a first stage that introduces a pole-zero, you can already calculate the component values to get a first zero at 10 and a pole at 100. Then you buffer and drive another similar passive network this time loaded by resistor $R_5$. It will give you the required attenuation.

simulate this circuit – Schematic created using CircuitLab

If you apply the Fast Analytical Circuits Techniques (FACTs), you can show that poles and zeros are located as follows:

$\omega_{p1}=\frac{1}{(R_1+R_2)C_1}$ $\omega_{z1}=\frac{1}{R_2C_1}$ $\omega_{p2}=\frac{1}{(R_4+R_5||R_3)C_2}$ $\omega_{z2}=\frac{1}{R_4C_2}$ $H_0=\frac{R_5}{R_3+R_5}$

The component values for the 1st stage are easy to determine. For the second stage, you need to account for $R_5$ which also introduces attenuation but nothing insurmountable.

The FACTs are truly unbeatable in terms of execution speed to determine transfer functions of any kind (passive or active). Check out the following links to know more about them:

and

Step 1) Find a topology that works, if you can only use 1 op amp then you'll need a two pole op amp topology, I'll pick sallen-key.

Step 2) Determine what the transfer function response is, it might be a low pass, high pass or band pass. This will help you determine what kind of resistors or capacitors you will use for the next step.

Step 3) Match up your transfer function with the topolgies transfer function:

Capacitors are: $Z = \frac{1}{C*s}$, you probably don't want to use inductors but they are $Z = {L*s}$ and resistors are $Z = R$

The sallen-key transfer function is:

$$\frac{vout}{vin}= \frac{Z_3 Z_4}{Z_1Z_2+Z_3(Z_1+Z_2)+Z_3Z_4}$$

You can also use calculators if you know what the poles of your transfer function is.

If the transfer function doesn't match up then find a different topology.

Hint, you'll probably use two capacitors and two resistors.

• Yep, it is but I wanted to answer the question generally anyway. – Voltage Spike Mar 30 '17 at 3:52