Analyzing a Band-Stop Filter Transfer Function

Key Takeaways

• Band-stop filters can be built as active filters or as RLC circuits.

• RLC circuits are normally analyzed as filters, and there are two RLC circuits that can be specifically designed to have a band-stop filter transfer function.

• These circuits are simple to design and analyze with Ohm’s law and Kirchhoff’s laws.

Band-stop filters work just like their optical analogues

RLC circuits are so ubiquitous in analog circuits that many modern applications can’t operate without them. These circuits provide essential functions like filtering with simple component arrangements, and they make possible a range of applications as diverse as antenna impedance matching and PDN decoupling. Because the function of an RLC filter can get a bit complex, it’s not always obvious how they enact filtering action at the output port. In addition, there can be multiple circuits that produce the same type of filtering behavior, but with different bandwidth and rolloff.

One example of such a filter is a band-stop filter, which can come in two possible RLC circuits. The two possible band-stop filter arrangements provide the same basic function, namely to prevent a narrow band of frequencies from reaching the output port. However, they are different in terms of their transfer functions. In this article, we’ll look at these band-stop filter transfer function possibilities and examine the transfer functions directly.

Types of RLC Band-Stop Filters and Their Transfer Functions

Band-stop filters can be built from passive linear components or from an active filter circuit like an op-amp (see below for an example). Building these circuits using RLC circuits is simplest and the transfer functions for these filters are easy to analyze. RLC band-stop filters come in two forms:

• As a parallel LC tank circuit in series with a resistor

• As a series RLC circuit

The filtering action in these circuits arises due to where the output voltage is measured in the circuit. The image below shows how the output voltage is measured for a given input voltage source. The output voltage (magnitude and phase) is then compared with the input source voltage to determine the transfer function for the circuit.

Band-stop filter transfer functions for the two common RLC circuit topologies

The functions in the above image are the transfer functions for each circuit. Simply plot these as a function of angular frequency to get the transfer function. In the above example, we can see that there is a resonant frequency, as is generally the case in RLC circuits. At resonance, the behavior in each circuit is different and depends on the reactive nature of the LC section. To better understand how these measurement positions give band-stop filter transfer function behavior, let’s analyze each of these circuits in terms of behavior at resonance.

Series RLC Circuit

In the series RLC circuit, the impedance of the series LC section drops to zero at resonance. In this case, the current in the circuit is maximized, but the output voltage is zero because the output impedance is zero, according to Ohm’s law. This is found by taking the limit at the resonant frequency in the transfer function:

Series RLC band-stop filter transfer function at resonance

In this way, the resonant frequency is a zero of a band-pass filter transfer function. Pole-zero analysis in a SPICE simulation can be used to identify zeros in the transfer function when an equation for the system is not known. The same applies in a parallel LC version of a band-stop filter, as shown below.

Parallel LC Tank With Series Resistor

In the parallel LC tank circuit with the series resistor, we can take the same limit to see what happens at resonance. In this case, there is an infinity in the denominator and the transfer function converges to zero at resonance. We have the same effect; the impedance in the circuit is infinity at resonance, thus the current in the resistor will be zero and the output voltage will be zero.

Parallel LC tank circuit band-stop filter transfer function at resonance

In both versions of a band-stop filter transfer function, the resistor R will determine the bandwidth of the transfer function, but they don’t have the exact same dependence. This dependence can be determined by plotting the transfer functions for various values of R.

Active Band-Stop Filters With Op-Amps

In our opinion, active filters don’t get the attention they deserve. However, if you understand something about op-amps, you can design a band-stop filter circuit with a precise stop band by engineering the feedback loop to have zero gain at a specific frequency. The closed-loop gain in an op-amp is normally designed with resistors to provide a flat frequency response throughout the amplifier’s bandwidth. However, filtering can be applied to the gain curve with L and C elements.

The diagram below shows one common way to implement band-stop filtering in an op-amp. This is done on the output, where the LC string shorts the output voltage to ground at resonance. Another possibility is to place the series LC circuit inside the feedback loop. At resonance, the feedback loop will have zero impedance, which will set the gain to zero.

Simple active band-stop filter with an op-amp

This is only an active filter in the sense that there is a band-stop filter attached to the output that will shunt current to ground within a specific bandwidth. Here, the value of R2 will determine the bandwidth of the series LC filtering section. Note that the resonant frequency of the LC section will need to fall inside the bandwidth of the op-amp to observe any useful effects. You can evaluate the transfer function of the R2-L-C voltage loop using a standard SPICE simulation with a frequency sweep.

At high frequencies, there is a stability problem with this circuit due to the input capacitance at the load component and the output inductance from the op-amp. The load’s input capacitance can be anything from ~100 fF to ~10 pF, depending on package size and how circuitry is constructed on the die, but this will usually only be noticeable at very high frequencies. The output inductance and load capacitance both add a new pole to the R2-L-C transfer function, so the total transfer function can be a bit complicated with limit cycle transient behavior. A transfer function can be used to spot these additional poles (or zeroes) in the transfer function.

When you need to simulate your band-stop filter transfer function, use PCB design and analysis tools with integrated circuit simulation and analysis features. Cadence offers a range of applications that automate many important tasks in systems analysis, including signal and power integrity analysis through an integrated set of field solvers. 

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