Low Pass Filter Transfer Functions

June 28, 2019

My dad recently got a new lease for a car which replaces his tried-and-true, 20-year-old truck. As soon as he gets a new toy he always enjoys pressing all the buttons and seeing what kind of weird things he can do. Recently, he’s been playing with the changing the speaker output for any kind of music or sound and messing with his passengers. It reminds me of playing with the equalizer on an old analog stereo system. Those knobs back then were connected to a filter circuit.

Of the various types of filters, low pass filters are very important for producing clean signals in a number of systems. So what is the easiest way to see examine the behavior of a particular low pass filter? This is low pass filter transfer functions come in handy.

Low Pass Filters and their Transfer Functions

As its name implies, a low pass filter is an electronic device that allows low frequency AC signals to pass a current through the filter circuit. The output from the filter circuit will be attenuated, depending on the frequency of the input signal. A number of different active and passive components can be used to construct filter circuits with various characteristics. Some filters include low pass, high pass, bandpass, all-pass elliptical, Chebyeshev, and Butterworth filters.

The easiest way to summarize the behavior of a filter is to define a transfer function. The transfer function tells you how the output signal is related to the input signal at various frequencies. If you are designing a filter circuit, you can easily determine the transfer function from a graph of the output signal at various frequencies. You can also calculate the transfer function using Kirchoff’s laws to derive the differential equation that governs the circuit.

Definition of a transfer function, its magnitude, and phase for a sinusoidal signal

As a signal passes through a filter, the filter will apply some phase shift to the output signal with respect to the input signal. This means that a transfer function for a filter is a complex function of frequency, and the transfer function contains all the information you need to determine the magnitude of the output signal and its phase. The general definition of a transfer function, its magnitude, and its phase are shown in the image above. Once you know the transfer function for a filter, you can convert this to a Bode plot to get a view of the transfer function in dB.

If this idea of a transfer function that defines the behavior of a filter is unfamiliar, it is convenient to think of the transfer function in terms of impedance. Just like impedance is a complex number that defines frequency-dependent resistance, a transfer function defines frequency dependent attenuation or gain. Instead of relating the output current and input voltage as you would with Ohm’s law, you are simply changing the value of the input voltage to some other value. Low pass filter transfer functions essentially have increased attenuation as frequency increases.

Higher Order Low Pass Filter Transfer Functions

In particular applications, you might find that the roll-off in the transfer function of a low pass filter is not steep enough. This means that there is still some high frequency content that is allowed through the filter and appears in the output signal. If you need better filtration of these higher frequencies, you can create a higher order filter by cascading multiple filters together in series. In general, a simple filter constructed from an RLC or similar circuit is called a first order filter. Two first order filters that are connected in series is called a second order filter, and so on...

A higher order filter is like a stack of coffee filters

If you look at higher order low pass filter transfer functions, you’ll find that the transfer function has a steeper roll-off at higher frequencies. This is very useful for suppressing noise with high frequency content. One example involves gathering a signal from a sensor array that may be sensitive to RF frequencies. If you pass the signal through a low pass filter, you can significantly suppress RF signals at higher frequencies. Mid-range and high frequencies will be suppressed to a greater extent when you use a higher order filter.

More General Transfer Functions

Note that a transfer function is often defined in terms of the Laplace transform for the differential equation describing a circuit. However, you can write the transfer function in terms of the frequency of a sinusoidal source using the equation shown above. This is done by substituting s=iω into a typical transfer function. This gives you a convenient view of the transfer function for individual frequencies.

In the case of a harmonically driven circuit that is not driven away from equilibrium, the transfer function can be easily determined by simply taking the Fourier transform of the differential equation describing the circuit. Using the Laplace transform to derive the transfer function is normally preferable in systems that include feedback, thus you would need to determine whether the system is stable. Unless you are designing a low pass filter with active feedback (e.g., a Butterworth filter), there is no element of stability to be considered under sinusoidal driving, thus the Fourier approach is suitable for most applications.

If you use an AC frequency sweep with a SPICE package and measure the output voltage from your low pass filter, you can determine the magnitude and phase at each driving frequency and construct a plot of the transfer function in the frequency domain. Determining the transient response of a circuit requires using looking in the time domain under impulse driving or step function driving.

Operational amplifiers appear in a number of filter circuits

Finally, when working with active components, you can use a time domain analysis for the circuit at different frequencies if you want to examine stability. This is somewhat time consuming because you are examining the effects of two parameters. If you perform a frequency sweep, you may see a peak in the transfer function near the circuit’s natural frequency, corresponding to gain in the system near cutoff.

Determining the transfer function and creating a bode plot for any circuit and with any type of source is much easier when you use a powerful SPICE package that simplifies frequency domain analysis of complex circuits. The OrCAD PSpice Simulator package from Cadence allows you to perform frequency sweeps, transient analysis, and many other tasks that are important for designing and analyzing analog circuits for any application.

This unique package is adapted to complex PCB designs, interfaces directly with your design data and helps you generate transfer functions for your filter circuits.