Understanding the Transfer Function of a Band-Reject Filter

Key Takeaways

• All analog systems requiring precision signal measurement, reproduction, or manipulation need filter circuits.

• Band-reject filters are the inverse of a bandpass filter: they filter out unwanted power within a specific frequency range.

• The transfer function of a band-reject filter can be calculated by hand or using a SPICE simulator.

Anyone who has played an analog synthesizer probably understands filters

Analog filter circuits are essential tools in a huge range of applications, and they need to be carefully designed to have the desired attenuation and gain behavior at the desired frequencies. The common topologies for most analog filter circuits involve RLC circuits with various arrangements of resistors, capacitors, and inductors. These circuits can be easily analyzed by hand, and the filter behavior can be determined.

Among the various types of filter circuits, there are various passband behaviors that can be intentionally designed, depending on the arrangement of elements in the RLC circuit. In many analog systems, a simple band-reject filter is useful in filtering specific analog signals that might fall within a certain frequency range.

If you’re planning to design a band-reject filter, then you’ll need to get the transfer function of a band-reject filter and understand how the filter should interface with your load component. To get the transfer function of a band-reject filter and create a connection to a load component, we’ll look at a simple example and discuss the process for more complex filter circuits.

Transfer Function of a Band-Reject Filter

The transfer function of a band-reject filter, and the connection you make between the filter and a load, rely on some basic tasks in circuit theory. A band-reject filter has the same characteristics and functions as other filters built from RLC circuits:

• Attenuation: Band-reject filters with a single pole only provide attenuation; they do not provide gain. 

• Resonance: All band-reject filters have a resonant frequency, which corresponds to a zero measured at the circuit’s output.

• Phase shift: The reactive elements in a band-reject filter will create a phase shift. Phase reversal can be seen at the resonant frequency. 

• Bandwidth: The bandwidth is determined primarily by the resistance in the circuit, and exactly how the resistance affects bandwidth depends on the topology of the filter (series, parallel, mixed, etc.).

Band-reject filters find uses in multiple filtering applications, particularly when specific noise sources need to be suppressed in an interconnect. Due to the high Q-value (low bandwidth) of these filters, they can be used to address noise sources that might emit at a single signal or to remove analog crosstalk between interconnects operating at different frequencies.

The impact of different circuit elements and the function of a particular band-reject filter can be seen directly in the filter’s transfer function. Let’s look at a common example to see how these filters work.

Example: Resistor With Parallel LC Tank Circuit

A basic example of a band-reject filter is a resistor in series with a parallel LC tank circuit. In general, any bandpass filter can be configured as a band-reject filter; this simply requires taking the output from the circuit as the voltage dropped across a different circuit element at resonance. Our example circuit is shown below with the output voltage labeled.

RLC band-reject filter

With the output voltage taken across the resistor, we can now calculate the transfer function using Ohm’s law and Kirchhoff’s laws. This involves calculating the total impedance, total current, and, finally, the voltage drop across the resistor. We see very high-Q attenuation within the stopband with the output voltage dropping to 0 V at resonance. We also see clear phase reversal as resonance is approached.

Transfer function of a band-reject filter (magnitude)

Transfer function of a band-reject filter (phase)

In this circuit, we’ve only considered the series resistor, R, and we can see how it determines the bandwidth, similar to a series RLC circuit used as a bandpass filter. Here, we’ve made an un-mentioned assumption about the load impedance, which needs to be addressed.

How the Load Impedance Affects the Transfer Function

In these types of filter circuit calculations, we are assuming the load component has high impedance, i.e., much higher than the value of the resistor. This is quite important when calculating the transfer function of a band-reject filter, or any other filter circuit for that matter.

Equivalent Output Impedance

There are two reasons for this high impedance load assumption. First, when the load impedance is similar to R, the two act in parallel, and the equivalent resistance of the output resistor is quite different from the design value. When the load impedance is very high, the equivalent impedance is very close to the output resistance, and the transfer function you calculate with no load component will be very similar to the transfer function with the laid connected.

CMOS Input Impedance

It just so happens that buffer stages in CMOS integrated circuits have very high impedance. Therefore, you can get an accurate calculation of the input voltage given to a CMOS buffer by simply looking at the transfer function in isolation. At very high frequencies, we need to consider the load capacitance on the input of CMOS circuits, which acts like a shunt capacitor to ground. More complicated circuits that use op-amps, transistors, or other ICs should consider high reactive impedance for real components when calculating transfer functions.

Use a SPICE Package to Calculate Transfer Functions

For linear band-reject filters, a SPICE package can be used to calculate the transfer function, either by comparing time-domain results at specific frequencies or by running frequency sweeps to calculate the transfer function directly. The results can be used to easily construct a Bode plot that will summarize the magnitude and phase information, as shown above.

More sophisticated types of band-reject filters require more specialized techniques to calculate the transfer function of the filter circuit:

• Active linear filters: The transfer function of this type of filter can be calculated using a frequency sweep as long as simulation models can be assigned to any active components.

• Nonlinear filters: These filters could be active or inactive, but the transfer function can’t be calculated using frequency sweeps. Instead, these filters must be examined using a small-signal linear approximation, using X-parameters, or by looking at specific frequencies in the time domain.

When you need to calculate the transfer function of a band-reject filter, 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 band-reject filter behavior in your system. 

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