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Filter Basics 

While an ac voltage at a fixed frequency is useful for explaining how electronic components work, it's not a good representation of a realworld signal. Realworld signals cover ranges of frequencies. For example, the sounds we hear throughout our lives exist over a range of 20 Hz to 20,000 Hz. Actual sounds go beyond this range, in fact, but most of us simply don't hear those sounds.
Because electronic signals can also exist over broad ranges of frequencies, we need to know how various combinations of electronic components will affect the different frequencies that make up a signal.
Sometimes we find that our circuit designs have odd, unintended effects on the signal passing through them. The signal might be distorted in various ways. Other times, we want to be able to allow some frequencies to pass easily, while blocking other frequencies. Circuits designed to do this are called filters, and they are used throughout the field of electronics.
Filters are described according to their behavior. For example, a lowpass filter will pass all signals below a specific frequency, but will attenuate or block signals of a higher frequency. Highpass filters do just the opposite. And we can combine these effects to produce bandpass filters and bandstop filters, according to what we need.
A filter consists of a circuit with at least one reactive component, typically used in conjunction with a resistor to form a voltage divider. A filter with a single reactive component, either a capacitor or an inductor, is called a first order filter, and its behavior is very specific. A filter with two reactances is a second order filter, and its behavior, while also quite specific, is different from that of the first order filter.
Consider the circuit shown schematically to the right. It is essentially a voltage divider in that is has two components, arranged so that the input voltage is applied to the series combination of the two, while the output voltage is taken from across one of the two. The difference from the basic resistive voltage divider, of course, is that here one of the components is a capacitor.
As we already know, if the input voltage, V_{IN}, is a fixed DC voltage, capacitor C will charge through resistor R, after which there will be no charging current and therefore no output voltage, V_{OUT}. If V_{IN} changes very slowly, C will charge or discharge with the changes, and again V_{OUT} will be vanishingly small or zero. However, if V_{IN} changes very rapidly, C won't have time to charge or discharge through R, so V_{OUT} is the same as V_{IN}. In this way, this particular circuit will block low frequencies while passing high frequencies without affecting them. Accordingly, this circuit is known as a firstorder highpass filter.
Two basic questions need to be asked about this circuit:
The boundary line in question is somewhat arbitrary, because there is no clear frequency such that all signals above it are passed intact, while all signals below that frequency are entirely blocked. Rather, there will be a "transition zone," or range of frequencies, over which the input signal will be partially transmitted to the output.
Nevertheless, we must select some specific frequency such that we can say (for the highpass filter) that all signals above this frequency will be passed without appreciable loss, while all signals below this frequency will be blocked to a significant extent. This frequency will be designated as the cutoff frequency (often designated f_{CO}) for our filter.
We have already noted that for very high frequencies there will be no appreciable voltage drop across the capacitor, while for very low frequencies there will be no appreciable voltage drop across the resistor. The transition zone, then, must be in between these extremes, where some of the signal voltage will be dropped across each of the components. And the logical place to start is to set the cutoff frequency at the point where the voltage drops across the two components are the same. As you should recall from previous pages in this group, this is also the frequency at which X_{C} = R.
The cutoff frequency is also the frequency at which half of the power in the input signal is absorbed by the filter, and only the other half makes it to the output. Therefore, it is sometimes known as the half power frequency, although that designation is no longer used as much as it was in the past.
We have already seen that the ac voltage and current are 90° out of phase with each other in an inductance or capacitance, and that the phase relationship between voltage and current varies between zero and 90° in an RL or RC circuit. In the case of our RC circuit above, current will lead voltage by some phase angle that will be determined by the values of the components and the frequency of the signal. Therefore the output voltage, which will be in phase with the current through R, will lead the input voltage by that phase angle.
Consider the graph shown to the right. The X axis in this graph shows frequency relative to the cutoff frequency of the filter. At very high frequencies, X_{C} will be negligibly small, and all of the input signal will appear across R. Therefore, the output signal will be the same amplitude as the input signal, and output signal voltage will be in phase with signal current. At very low frequencies, X_{C} will be very much larger than R, so all of the input signal voltage will be dropped across C. This causes the output signal voltage to lead the input voltage by 90°, and to have zero amplitude. Thus, at extremes of frequency, the signal is either passed unchanged, or else it is totally blocked. We need to look at this circuit over a range of frequencies in between those limits, to see what happens there.
Mathematically, the voltage phase shift from input to output will be arctan(X_{C}/R). If X_{C} = 100R, the phase angle will be just under 89.5°. If R = 100X_{C}, the phase angle will be a bit over 0.5°.
If we drop to a closer range, we find that if X_{C} = 10R, our phase angle becomes about 84.3°, and if R = 10X_{C}, it becomes about 5.7°. Clearly, most of the transition zone occurs when the impedances of the two components are within a factor of 10 of each other, and essentially all of it occurs when these impedances are within a factor of 100 of each other. This squares with practical electronics: If two resistors in the same circuit differ in value by more than a factor of 100, one of them will have negligible effect on the operation of the circuit; the other will have full control. Only if extreme precision is required will it be necessary to account for both.
At f_{CO}, when X_{C} = R, the phase angle becomes arctan(X_{C}/R) = arctan(1) = 45°.
The output voltage depends on both the signal frequency and the input voltage amplitude. To eliminate the signal amplitude from consideration, it is convenient to plot the ratio of V_{OUT}/V_{IN} against frequency, rather than just the value of V_{OUT}. The graph to the right shows the result, relative to f_{CO} for the circuit.
As with the phase shift we examined above, we find that most of the change in amplitude occurs within the range 0.1 f_{CO} to 10 f_{CO}. At higher frequencies, we see the full input signal appearing at the output at a frequency just about 16 f_{CO}. However, at lower frequencies, the rapid drop in V_{OUT}/V_{IN} tapers off and becomes much more gradual below about 0.1 f_{CO}.
In theory, V_{OUT} never actually reaches zero; it just approaches that value ever more closely as the frequency gets farther away from f_{CO}. In practice, once the frequency differs by a factor of about 100, V_{OUT}/V_{IN} has become so small that it may be treated as negligible.


 
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