Pi Filter Circuit Design Formulas and Calculator

I would venture to guess that the most popular type of filter circuit I encounter in PCB designs and schematics is a pi filter, specifically a doubly terminated passive Butterworth pi filter. A pi filter circuit made from L and C elements can be easily scaled up to higher-order filtering. These filter circuits are also very easy to simulate, and the results are straightforward to interpret. Whether for a simple EMI solution or a filter on an analog circuit, LC filters are often the first choice to consider and evaluate in circuit design.

In this article, I'll demonstrate how to design, simulate, and analyze a pi filter circuit. There is also a calculator included to scale your pi filter circuit up to higher-order filtering. The analysis procedure in SPICE will also be presented in terms of transfer functions, although you could also use S-parameters.

LC Pi Filter Circuit Design

LC pi filters come in two varieties: high-pass or low-pass filters. In my opinion, the LC low-pass filter is the most commonly used type of pi filter. The two circuit topologies for high-pass and low-pass functionality are shown in the image below.

As a fully passive linear time-invariant (LTI) system, these circuits support transfer function analysis methods, even when scaled to an infinite number of elements. Most often, you will use a three-element LC pi filter circuit in either a high-pass or low-pass configuration.

An alternative topology that produces the same -3 dB frequency, characteristic impedance, and roll-off is the T filter, which also uses three elements. There are also active low-pass or high-pass filters that use op-amps and can achieve higher-order filtering without ripple in the pass band. These filters will be discussed in a future article.

Pi LC Filter Topologies

Pi LC filters can be designed in two main configurations: Butterworth and Chebyshev topologies. Butterworth pi filter circuits are very easy to design and involve two primary design formulas for determining the values of L and C. Chebyshev pi filters require an additional parameter called passband ripple, which will be discussed below.

Butterworth Passive Pi Filter Design Formulas

All pi filters have a characteristic impedance value and a cutoff frequency value. When the shunt elements are equal to each other, the input and output impedances of the circuit match the selected impedance (doubly terminated). The frequency value is a resonant frequency, but the presence of a high-Q resonance in the transfer function depends on the value of the load, as we will see later.

The design approach involves selecting a frequency (f) and impedance value (Z) and using these to determine the L and C values.

For a low-pass pi filter circuit, the inductance and total capacitance are determined by:

Low-pass pi filter inductance and total capacitance values

The individual values of C1​ and C2 are:

Low-pass pi filter capacitance values

For a high-pass filter circuit, slightly different relations for inductance and capacitance are used:

High-pass pi filter total inductance and capacitance values

The total inductance (L) is related to the individual inductances as follows:

High-pass pi filter inductance values

This straightforward procedure completes the design. To verify functionality in terms of the transfer function, we need a source and load impedance and simulate the transfer function to determine the voltage that reaches the load.

Butterworth LC Pi Filter Simulation

To simulate a Butterworth LC pi filter, refer to the schematic shown below. This schematic includes a simple voltage source connected to the pi filter circuit and a 50-ohm load. Plugging the L and C values into the above equations reveals that this LC low-pass filter circuit has a characteristic impedance of 50 ohms and targets a roll-off frequency of 100 MHz.

The magnitude of the transfer function is calculated by dividing the voltage across the load by the input voltage at the node of C1​. Note that if there is a source impedance, the input voltage will differ at the C1 node as the source impedance forms a voltage divider with the loaded circuit, thus it has been omitted in the schematic for simplicity.

To observe the impact of load impedance on the transfer function, the load is varied across five different values. The results demonstrate that a higher load impedance results in a higher-Q resonance and higher voltage gain. This is evident from the 1 kOhm curve.

We can easily see that sufficiently loading the pi filter is sufficient to dampen the observed LC resonance. One may correctly note that parasitics in the components will also affect the resonance, but that is an exercise which we will leave for readers.

Butterworth Pi Filter Calculators

For higher-order filters, an iterative procedure can be used to calculate the L and C elements. While the pi filter can theoretically scale up to any number of elements, most practical designs stop at five or seven elements. Although manual calculations are possible, a pi filter calculator simplifies the process.

Below we have embedded the calculator apps from CalculatorEdge.com. We highly encourage anyone looking for easy-to-use online engineering calculators to visit the CalculatorEdge.com website.

The high-pass filter calculator operates similarly, taking target impedance, cutoff frequency, and the number of components to calculate the required L and C values.

Butterworth vs. Chebyshev Filters

Simulation results demonstrate that the Butterworth pi LC filter exhibits a flat response in the pass band, maintained even in higher-order designs. This is a major advantage over other filter topologies, whether active or passive.

Chebyshev filters, particularly for higher-order filtering, exhibit a stronger roll-off beyond the cutoff frequency but introduce ripple in the pass band. The passband ripple limit is specified as a decibel value and is a critical design parameter. Bandpass configurations for both filter types will also be discussed in a future article.

Butterworth and Chebyshev low-pass pi filter comparison. Image source: The Open University.

Stay tuned for more detailed discussions on Chebyshev filter designs in upcoming articles.

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