# FILTER CIRCUITS IN ELECTRONICS !

**RC element as a frequency-dependent voltage divider:**

The diagram below shows the previously considered RC element once again in a slightly different configuration. Here it is supplied with an alternating voltage *U*_{0} (input voltage to the element) and itself supplies an alternating voltage *U*_{C} to the capacitor (output voltage to the element). This configuration makes it clear that an RC element acts as a *frequency-dependent *voltage divider. Whereas the active resistance *R* is frequency-independent, the capacitive reactance *X*_{C} is inversely proportional to the frequency *f*. Consequently, the element's output voltage *U*_{C} drops as the frequency rises. Whereas an RC element permits low frequencies to pass, it progressively blocks higher frequencies. This type of RC element is accordingly termed a *low-pass filter*.

As in the case of the purely ohmic voltage dividers used in DC circuits, the output voltage *U*_{C} can be calculated as follows. The following equation applies.

The ratio between the output and input voltages is

*Z* is the element's apparent resistance (impedance).

If this ratio is plotted graphically for specific values of *R* and *C* as a function of the frequency (usually represented on a logarithmic axis), the RC element's *amplitude response* is obtained, as shown in the example below. At low frequencies, the high capacitive reactance causes practically the entire supply voltage to drop across the capacitor. At very high frequencies, the capacitor's impedance drops to nearly zero, thus causing the output voltage to approach zero too.

The phase displacement j between the voltages also depends on the frequency in accordance with the following equation:

The negative sign here indicates that the output voltage lags behind the input voltage. Displayed graphically, this relationship is termed the RC element's *phase response*. At low frequencies, the two voltages are nearly in phase whereas at high frequencies, the phase shift approaches -90°.

The RC element's amplitude and phase responses together are termed its *frequency response*.

**Cut-off frequency of an RC element:**

A RC element's frequency response depends on the ohmic resistance *R* and the capacitance *C*. However, the basic amplitude and phase characteristics (curve shapes) are always the same. A change in resistance merely results in a corresponding *phase shift* of the curves on the frequency axis. An RC element's characteristic parameter is its *cut-off frequency* *f*_{G}, given by the following equation:

Substituting this parameter in the amplitude response gives the following result:

At the cut-off frequency, the amplitude or rms value of the output variable has dropped to 0.707 of the peak value. The phase response is given by:

The phase shift between the input and output voltages at the cut-off frequency is precisely -45°. The following diagrams illustrate these relationships.

The cut-off frequency depends less on the individual values of *R* and *C* and rather on their product *R*·*C* - termed the RC element's *time constant T*. The larger this time constant, the lower the RC element's cut-off frequency, i.e. the greater its attenuation of the low-frequency range. For instance, increasing an RC element's ohmic resistance by a factor of ten while simultaneously decreasing its capacitance by the same factor does not change the element's frequency response.

When displaying amplitude response in communications engineering, not only the frequency axes but also amplitude axes often have a logarithmic scale. The RC element's resulting characteristic falls *linearly* in the high frequency range. In the diagram below, the same amplitude response depicted earlier is plotted along a logarithmic amplitude axis.

In this type of graph, the amplitude ratio is often specified in *decibels* (dB). An amplitude ratio of 1 corresponds to 0 dB, 0.1 (i.e. 10^{-1}) to -20 dB, 0.01 (i.e. 10^{-2}) to -40 dB and so on. The conversion formula is:

An amplitude ratio of 1/Ö2 = 0.707 at the cut-off frequency thus corresponds to -3 dB.

The following interactive animation shows an RC element's amplitude response at different values of *R* and *C* , as well as the position of the cut-off frequency (red vertical line). On the right-hand edge of the display, you can select a variety of component parameters to observe their effect on the amplitude response.

**Low-pass filtered signals:**

All periodic signals can be represented through *Fourier analysis* as a sum of sinusoidal oscillations whose frequencies are integral multiples of the fundamental signal frequency. For example, a *square-wave signal **s*(*t*) of a frequency *f*_{0} is represented as follows:

The first sine term with a frequency *f*_{0} is called the *fundamental oscillation*; the subsequent terms involving progressively higher frequencies are called *harmonics*. In the case of a square-wave signal, Fourier analysis yields an infinite series of harmonics each of which are odd multiples of the fundamental signal frequency and the amplitude of which decreases as the frequency factor rises.

The following interactive animation demonstrates a square-wave signal of a frequency *f*_{0} (red curve) and amplitude 1. Using Fourier analysis as described above, this signal is resolved into a series of sinusoidal oscillations of different frequencies. The tool on the right-hand side allows you to activate the fundamental and its harmonics up to a frequency of 35·*f*_{0}. Activate the individual frequency components successively and observe how the resulting series of sine terms approximates to a square-wave oscillation more and more closely.

If such a square-wave signal (or other periodic, non-sinusoidal signal) is passed through a low-pass filter, the individual harmonics are attenuated very strongly by the filter's frequency response. The signal output by the low-pass filter is correspondingly distorted.

**RL high-pass filters:**

Like an RC element, an RL element is also a frequency-dependent voltage divider. In this case, however, the inductive reactance *X*_{L} increases with the frequency *f*. Consequently, the element's output voltage *U*_{L} also increases with the frequency. In other words, an RL element allows high frequencies to pass while progressively blocking low frequencies. An RL element is accordingly termed a *high-pass filter*. The diagram below illustrates the relationships between the various voltages.

The output voltage *U*_{L} can be calculated as in the case of the purely ohmic voltage dividers employed in DC circuits according to the following formula:

The ratio between the output and input voltages is defined as follows:

*Z* is the total impedance of the element.

Plotting the ratio for specific values of *R* and *L* as a function of the frequency provides, as in the case of the RC element, the RL element's *amplitude response*, as exemplified in the illustration below. At low frequencies, the low inductive reactance results in practically no voltage drop across the coil, whereas at high frequencies, the coil has a high impedance leading to a correspondingly large drop in supply voltage across the coil. The amplitude response here is therefore entirely the inverse of that for an RC element, which as we have seen acts as a low-pass filter.

Also when represented in dB (see following illustration), RL amplitude response looks like a mirror image of RC amplitude response.

RL high-pass filters have a cut-off frequency too, defined in this case by the following formula:

The phase displacement j between the voltages of an RL element is described by the following equation:

The diagram below shows the associated phase response. At high frequencies, both voltages are nearly in phase; at low frequencies, the phase shift on RL elements approaches +90°. The phase shift at the cut-off frequency is +45°.

The following interactive animation shows the amplitude response of an RL element at different values of *R* and *L*, as well as the position of the cut-off frequency (red vertical line). On the right-hand edge of the display you can select a variety of component parameters to observe their effect on the amplitude response.

**Band-pass filters:**

Combining a low-pass element and a high-pass element by connecting them in series results in a *band-pass* filter circuit. As its name suggests, this type of circuit allows passage of a particular band of frequencies while attenuating frequencies outside this band. The band-pass circuit shown in the example below comprises a series-connected LR low-pass element (green background) and CR high-pass element (blue background).

A band-pass circuit's frequency response is produced through superimposition of the low-pass and high-pass frequency responses. The diagram below illustrates this in terms of the amplitude response.

If the two circuit sections are decoupled by means of an operational-amplifier circuit, the band-pass frequency response corresponds exactly to the superimposition of both partial frequency responses (ideal case). However, in the case of passive circuit sections such as those illustrated above, the second filter section (i.e. high-pass) acts as a load on the preceding voltage divider (i.e. the low-pass filter), so that the overall frequency response deviates somewhat from the ideal model.