An Introduction to Notch Filter Transfer Function

An Introduction to Notch Filter Transfer Function

By analyzing the passband and stopband of such filters, passband and stopband can be determined. Depending on the topology of the filter, the transfer function will vary for notch filters. In many RF and precision measurement applications, the notch filter transfer function can be difficult to determine by hand for very complex higher order notch filters. Here we will discuss how you can quickly assess the notch filter transfer function for your system through analysis of notch filters.

Topologies of Notch Filters

It is possible to build notch filter circuits with a variety of bandwidths, rolls off, and output voltages using a variety of topologies. Passive components are commonly used in these, but active devices (op-amps) can also be used to provide additional gain.

Circuits of the RLC Series

A RLC series circuits are commonly used as simple bandpass filters. By taking the output and crossing it over the resistor, the circuit will act like a bandpass filter. However, if you add the output voltages over L and C together, the result is a notch filter. There is a relationship between notch filter behavior and resonance in that when resonance is reached, the C and L elements produce equal and opposite voltage drops which result in a 0 V output.
Using an RLC circuit as a notch filter is demonstrated in the following circuit. Through the capacitor and inductor in series, the circuit acts as a notch filter. Kirchhoff’s Laws can be used to calculate the impedance of a circuit and the voltage drop across its elements. In order for the output voltage at the resonant frequency to be 0 V, a voltage drop across the L and C elements must be equal to a voltage drop across their components.

Filters with Higher Order Notches

Higher order filters can be formed by daisy chaining filters. In this arrangement, the gain or attenuation at specific frequencies is obtained by multiplying the transfer functions.
Notch Filters - ULTRA Series Optical Filters - Alluxa

The filters are normally used to provide high rolloff and high loss in the stopband of the transfer function. Filters of 6th order and higher are widely used in RF systems, either as part of integrated circuits or as separate components.

Activated Notch Filters

In order to build active notch filters, linear operational amplifiers are normally used. Filters of this type consist of a feedback loop and an input terminal that feature a reactance for a simple method of adjusting gain or loss over a range of frequencies. The filtration function becomes more complex when the gain spectrum is transformed into a frequency function by adding L or C elements to the input and feedback loop. Parallel resonant notch circuits in the feedback loop can set the gain in any other frequency to zero while leaving the resonant frequency alone.

An Analysis Of The Transfer Function Of A Notch Filter

In each of the notch filter styles discussed above, frequencies within a limited bandwidth are removed by inducing a current in the filter that is inherently inductive, which sets the output voltage to zero. Calculating a notch filter’s transfer function with a SPICE simulation using a real-circuit model is the fastest method. A SPICE solver is easiest to use when you have a collection of real components included in your simulation program.

You should use PCB design and analysis software that integrates SPICE simulators for the calculation of notch filter transfer functions. With Cadence, you can automate many tasks associated with systems analysis, including notch filter behavior analysis based on frequency sweeps and transfer function analysis.