# Understanding a Notch Filter Transfer Function

### Key Takeaways

• Notch filters are used in many applications where a specific frequency needs to be removed from a circuit.

• These filters can be designed with multiple topologies and can be used to address specific noise sources in a system.

• Just like other filters, a notch filter can be analyzed using a transfer function.

These coils can be used as notch filters in power systems

Filters are central components in any analog system and there are four basic types of filter circuits:

These filters can be built from basic components and analyzed to determine the passband/stopband. For notch filters, the transfer function will be quite different depending on the topology of the filter. For very complex higher order notch filters, the notch filter transfer function can be difficult to determine by hand, which is quite important in many RF and precision measurement applications. In this article, we will discuss how you can analyze notch filters and quickly determine a notch filter transfer function for your system.

## Notch Filter Topologies

There are a number of topologies that can be used to build notch filter circuits with various bandwidths, rolloffs, and output voltages. These are commonly built from passive components, but they can also be built from active components (op-amps) to provide additional gain.

### Series RLC Circuits

A series RLC circuit is often used as a simple type of bandpass filter. When the output is taken across the resistor, the circuit will function as a bandpass filter. In contrast, taking the output voltage across the L and C elements together gives notch filter behavior. Notch filter behavior is related to resonance in that, at resonance, the L and C elements give equal and opposite voltage drops, leading to 0 V output from the filter circuit.

The circuit below shows how an RLC circuit can also be used as a notch filter. By taking the output voltage across the series capacitor and inductor, the circuit will act like a notch filter. This can be seen when we use Kirchhoff’s Laws to calculate the circuit impedance and the voltage drop across each element. When we look at the voltage drop across the L and C elements together, the output voltage will be 0 V at the resonant frequency.

Series RLC circuit used as a notch filter

In this circuit, the resistor R determines the bandwidth, just like in a series RLC circuit used as a bandpass filter. Larger values of R will give wider bandwidth (Q-value). When used without the resistor, we have a high-Q bandpass filter with high gain, where the bandwidth is limited by any parasitic resistance on the component leads.

### Comparison With a Bandpass Filter: RLC Circuit With Parallel Resonant Tank

The above example shows a notch filter with a series RLC circuit, but this particular circuit has a low Q-factor. A high-Q circuit uses a parallel LC resonant tank, as shown below. In this type of circuit, the output voltage is taken across the resistor R and given to a high impedance load.

Parallel RLC circuit used as a notch filter

The image below shows a comparison of the series filter transfer function for the series RLC circuit and the parallel resonant tank RLC circuit. Note that only the magnitude is shown here, as a simple phase reversal occurs at resonance. Here, we have taken L = C = 1e-6, R = 2 Ohms, and a high impedance load. Here, we see that the tank circuit has a much higher Q-value (lower bandwidth) than the series RLC circuit. The tank circuit is most commonly used when we need to pass power to a load at single frequency, while the series circuit can be used to pass power at a range of frequencies or when working with broadband signals. The Q-factor can be tuned by adjusting the value of the series resistor.

A comparison of the series filter transfer function for the series RLC circuit and the parallel resonant tank RLC circuit

### Higher Order Notch Filters

Filters can be daisy chained to form higher order filters. In this arrangement, filter transfer functions multiply together to give the total gain or attenuation at specific frequencies. These filters are normally used to give a transfer function with high rolloff and high loss in the stopband. It is common to see 6th order and higher filters used in RF systems, either built into RF integrated circuits or built from discrete components.

### Active Notch Filters

Active notch filters are normally built with operational amplifiers working in the linear range. These filters are constructed by adding some reactance on the input terminal and the feedback loop, giving a simple method for applying gain or loss over a specific frequency range. By adding L or C elements to the input and feedback loop, the gain spectrum becomes a function frequency, giving a more complex filtration function. Placing a parallel resonant notch circuit into the feedback loop sets the gain to 0 at the resonant frequency while providing gain at all other frequencies.

## Calculating a Notch Filter Transfer Function

All of the notch filter styles discussed above provide the same basic function: they remove frequencies within a limited bandwidth by providing infinite impedance (for series topologies) or zero impedance (for parallel topologies) for a current in the filter, which sets the output voltage to 0 V. The fastest way to calculate a notch filter transfer function is to use a SPICE simulation with real subcircuit models. This is easiest when your simulation program includes a SPICE solver and a complete library of real components.

When you need to calculate a notch filter transfer function, use PCB design and analysis software with an integrated SPICE simulator. Cadence provides a powerful set of software tools that help automate many important tasks in systems analysis, including frequency sweeps and transfer function analysis to determine notch filter behavior in your system.