Designing an RLC circuit from a given transfer function.

[MonsterAar]

Designing a circuit from a given transfer function.

I hope you don't mind that I haven't used the default format. It doesn't lend itself well to my problem.

The original problem is as follows:
We were given a 'black box' that contained a passive circuit and we experimentally determined a transfer function for the circuit by passing through various frequencies and recording the magnitude and phase of the outputs.

This was then used to create a bode plot. This bode plot showed a pole at approximately 56Hz and a zero at approximately 1660Hz. The DC gain of the circuit is 0.064.

From this information I deduced the transfer function of the circuit to be:
$$H(\omega)=0.064\frac{1+j\frac{\omega}{1660}}{1+j\frac{\omega}{56}}$$

The task now asks for us to design a circuit to cancel out the effects of the black box on the input signal. Basically we have to design a circuit so that when we have input-blackbox-our circuit-output, the input=output. So, I need to design a circuit with the inverse of the above transfer function.

IE. I need to design a circuit with the following transfer function:
$$H(\omega)=15.625\frac{1+j\frac{\omega}{56}}{1+j\frac{\omega}{1660}}$$

This is where I get stuck. How do I go about designing a circuit with a DC gain of 15.625, a pole at 1660Hz and a zero at 56Hz?
I understand I'll have to use an op-amp to get the DC gain greater than 1. Other than this I'm stuck.

Thanks,
Luke

 

[CEL]

Consider a voltage divider composed of two resistors and a capacitor in series. The input voltage acts on all three components and the output voltage is taken on the series of the capacitor and one of the resistors.
What is the transfer function of this circuit?

 

[Mene]

I would approach this problem by breaking it down into smaller steps and using my knowledge of circuit design and transfer functions to guide my approach.

Firstly, I would start by identifying the type of circuit that would have a transfer function with a pole at 1660Hz and a zero at 56Hz. In this case, an RLC circuit would be a suitable choice as it can have both a pole and a zero in its transfer function.

Next, I would focus on designing the RLC circuit to have the desired pole and zero frequencies. This would involve selecting appropriate values for the resistor, inductor, and capacitor. I would use the standard equations for calculating the pole and zero frequencies in an RLC circuit and adjust the values until they match with the given frequencies.

Once the RLC circuit is designed, I would then move on to incorporating the desired DC gain of 15.625. This can be achieved by using an op-amp with a suitable gain value, as you have mentioned in your question. The op-amp can be placed in the feedback path of the RLC circuit to amplify the output and achieve the desired DC gain.

Finally, I would test the circuit and make any necessary adjustments to ensure that it meets the required transfer function. This could involve fine-tuning the values of the components or adjusting the gain of the op-amp.

In summary, designing an RLC circuit from a given transfer function requires a good understanding of circuit design principles and transfer functions. By breaking down the problem into smaller steps and using appropriate equations and components, a circuit can be designed to meet the given specifications.