Frequency Response and Active Filters

This document is an introduction to frequency response, and an introduction to active filters (filters using active amplifiers, like op amps).  You might also want to read a similar document from National Semiconductor, A Basic Introduction to Filters - Active, Passive, and Switched-Capacitor.

Frequency Response -- Background

Up to now we have looked at the time-domain response of circuits. However it is often useful to look at the response of circuits in the frequency domain. In other words, you want to look at how circuits behave in response to sinusoidal inputs. This is important and useful for several reasons: 1) if the input to a linear circuit is a sinusoid, then the output will be a sinusoid at the same frequency, though its amplitude and phase may have changed, 2) any time domain signal can be decomposed via Fourier analysis into a series of sinusoids. Therefore if there is an easy way to analyze circuits with sinusoidal inputs, the results can be generalized to study the response to any input.

To determine the response of a circuit to a sinusoidal signal as a function of frequency it is possible to generalize the concept of impedance to include capacitors and inductors. Consider a sinusoidal signal represented by a complex exponential:

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where j=(-1)1/2 (engineers use j instead of i, because i is used for current), w is frequency and t is time. It is a common shorthand to use "s" instead of "jw".

Now let us look at the voltage-current relationships for resistors capacitors and inductors.

For a resistor ohms law states:

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where we define the impedance, "Z", of a resistor as its resistance "R".

For a capacitor we can also calculate the impedance assuming sinusoidal excitation starting from the current-voltage relationship:

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Note that for a capacitor the magnitude of the impedance, 1/wC, goes down with increasing frequency. This means that at very high frequencies the capacitor acts as an short circuit, and at low frequencies it acts as an open circuit. What is defined as a high, or low, frequency depends on the specific circuit in question.

Likewise, for an inductor you can show that Z=sL.

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For an inductor, impedance goes up with frequency. It behaves as a short circuit at low frequencies, and an open circuit at high frequencies; the opposite of a capacitor. However inductors are not used often in electronic circuits due to their size, their susceptibility to parisitic effects (esp. magnetic fields), and because they do not behave as near to their ideal circuit elements as resistors and capacitors..


A Simple Low-Pass Circuit

To see how complex impedances are used in practice consider the simple case of a voltage divider.

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If Z1 is a resistor and Z2 is a capacitor then

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Generally we will be interested only in the magnitude of the response:

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Recall that the magnitude of a complex number is the square root of the sum of the squares of the real and imaginary parts. There are also phase shifts associated with the transfer function (or gain, Vo/Vi), thought we will generally ignore these.

This is obviously a low pass filter (i.e., low frequency signals are passed and high frequency signals are blocked).. If w<<1/RC then wCR<<1 and the magnitude of the gain is approximately unity, and the output equals the input. If w>>1/RC (wCR>>1 ) then the gain goes to zero, asdoes the output. At w=1/RC, called the break frequency (or cutoff frequency, or 3dB frequency, or half-power frequency, or bandwidth), the magnitude of the gain is 1/sqrt(2)@0.71. In this case (and all first order RC circuits) high frequency is defined as w>>1/RC; the capacitor acts as a short circuit and all the voltage is across the resistance. At low frequencies, w<<1/RC, the capacitor acts as an open circuit and there is no current (so the voltage across the resistor is near zero).

If Z1 is an inductor and Z2 is a resistor another low pass structure results with a break frequency of R/L.


A Simple High-Pass Circuit

If Z1 is a capacitor and Z2 is a resistor we can repeat the calculation:

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At high frequencies, w>>1/RC, the capacitor acts as a short and the gain is 1 (the signal is passed). At low frequencies, w<<1/RC, the capacitor is an open and the output is zero (the signal is blocked). This is obviously a high pass structure and you can show that the break frequency is again 1/RC.

If Z1 is a resistor and Z2 is an inductor the resulting circuit is high pass with a break frequency of R/L.

This concept of a complex impedance is extremely powerful and can be used when analyzing operational amplifier circuits, as you will soon see.


Active Filters

Low-Pass filters - the integrator reconsidered.

In the first lab with op-amps we considered the time response of the integrator circuit, but its frequency response can also be studied.

First Order Low Pass Filter with Op Amp
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If you derive the transfer function for the circuit above you will find that it is of the form:

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which is the general form for first-order (one reactive element) low-pass filters. At high frequencies (w>>wo) the capacitor acts as a short, so the gain of the amplifier goes to zero. At very low frequencies (w<<wo) the capacitor is an open and the gain of the circuit is Ho. But what do we mean by low (or high) frequency?

We can consider the frequency to be high when the large majority of current goes through the capacitor; i.e., when the magnitude of the capacitor impedance is much less than that of R1. In other words, we have high frequency when 1/wC<<R1, or w>>1/R1C=wo. Since R1 now has little effect on the circuit, it should act as an integrator. Likewise low frequency occurs when w<<1/R1C, and the circuit will act as an amplifier with gain -R1/R2= Ho.


High-Pass filters - the differentiator reconsidered.

The circuit below is a modified differentiator, and acts as a high pass filter.

First Order High Pass Filter with Op Amp

Using analysis techniques similar to those used for the low pass filter, it can be shown that

which is the general form for first-order (one reactive element) low-pass filters. At high frequencies (w>>wo) the capacitor acts as a short, so the gain of the amplifier goes to H0= -R1/R2.  At very low frequencies (w<<wo) the capacitor is an open and the gain of the circuit is Ho. For this circuit w0=1/R2C.  Therefore this circuit is a high-pass filter (it passes high frequency signals, and blocks low frequency signals.

Band-Pass circuits

Besides low-pass filters, other common types are high-pass (passes only high frequency signals), band-reject (blocks certain signals) and band-pass (rejects high and low frequencies, passing only signal areound some intermediate frequency).

The simplest band-pass filter can be made by combining the first order low pass and high pass filters that we just looked at.

Simple Band Pass Filter with Op Amp

This circuit will attenuate low frequencies (w<<1/R2C2) and high frequencies (w>>1/R1C1), but will pass intermediate frequencies with a gain of -R1/R2.  However, this circuit cannot be used to make a filter with a very narrow band.  To do that requires a more complex filter as discussed below.


High Q (Low Bandwidth) Bandpass Filters.

For a second-order band-pass filter the transfer function is given by

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where wo is the center frequency, b is the bandwidth and Ho is the maximum amplitude of the filter. These quantities are shown on the diagram below. The quantities in parentheses are in radian frequencies, the other quantities are in Hertz (i.e. fo=wo/2p, B=b/2p). Looking at the equation above, or the figure, you can see that as w->0 and w->infinity that |H(s=jw)|->0. You can also easily show that at w=wo that |H(s=jwo)|=H0. Often you will see the equation above written in terms of the quality factor, Q, which can be defined in terms of the bandwidth, b, and center frequency, wo, as Q=wo/b. Thus the Q, or quality, of a filter goes up as it becomes narrower and its bandwidth decreases.

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If you derive the transfer function of the circuit shown below:

High-Q Bandpass Filter with Op Amp

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you will find that it acts as a band-pass filter with:

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and the center frequency and bandwidth given by:

Radian frequency Hertz
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The notation R1||R2 denotes the parallel combination of R1 and R2,

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Switched Capacitor Filters

There is a special type of active filter, the switched capacitor filter, that takes advantage of integration to achieve very accurate filter characteristics that are electronically tuneable.  The page Switched Capacitor Filters describes these in more detail.