Capacitors 4: Low Pass Filters

by gaussmarkov

Resisting Voltage Change

Capacitors are often said “to oppose voltage change.” When the current is finite, the change in the voltage across the plates of a capacitor generally lags behind the change in the voltage of the source. Also, the shape of voltage changes affects how the capacitor charges, the resulting shape of voltage across the plates of the capacitor, and the pattern of the current through the capacitor.

To illustrate a capacitor opposing voltage change, we will simulate a simple circuit with a resistor and a capacitor in series in LTSpice. As in a voltage divider, we will treat the junction between the resistor and capacitor as the output of the circuit. The voltage at this point is the voltage across the capacitor because the other lead of the capacitor is tied to ground. The resistor has a 10K ohms resistance and the capacitor has a 100µF capacitance. The resistor forces current to be finite.

charging a capacitor schematic

If we apply a rectangular pulse of positive voltage to a capacitor, then a capacitor charges and then discharges as shown in this figure. Its voltage does not change instantly with the supply voltage. Instead it rises or falls behind the supply. Note that the capacitor’s voltage never quite reaches the maximum voltage. This reflects the increasing difficulty of adding more electrons to one of its plates.

charging a capacitor schematic

If we apply a triangle pulse, then the capacitor voltage is smoother. After the supply voltage has peaked, the capacitor continues to gain voltage. At the peak of the triangle, the capacitor voltage path switches from a convex shape to a concave shape, rather than changing direction. In other words, as long as the supply voltage is rising the capacitors voltage is rising faster and faster or accelerating. But when the supply voltage starts to fall, then the capacitor voltage starts to decelerate, at first slowing its rate of increase until the capacitor voltage eventually stops growing and starts to fall. The capacitor voltage cannot rise above the supply voltage, so it peaks where the two voltages are equal.

charging a capacitor schematic

There is one input AC shape that is not changed by a capacitor and that is the sine wave. Nevertheless, if we supply a sine wave pulse, then we get a delayed sine wave pulse. In this figure, we have a few cycles of the sine wave so that the capacitor voltage settles into its sine response. Because the capacitor resists voltage change, it also takes a few cycles to reach its steady-state response.

charging a capacitor schematic

All of these examples show the symptoms of a capacitor resisting a voltage change. First, the output voltage swing is smaller. Second, the output voltage changes lag behind the input voltage changes.

Low Pass Filter

Our simple circuit is actually a passive low pass filter. That is, if we compare the output for input AC with the same amplitude but different frequencies, the low frequency AC has a relatively higher amplitude at the output node. To illustrate how this occurs, we simulated the circuits shown in this figure in LTspice/SwitcherCAD III.

charging a capacitor schematic

There are two identical circuits with AC voltage sources that differ only in the length of time a square wave has a positive voltage. First, compare the path of voltage across the two capacitors for a single pulse. Because the initial conditions are identical, the paths are identical until the shorter pulse turns off. After that point, one capacitor keeps charging while the other discharges. As a result, the shorter pulse reaches a lower peak in voltage than the longer pulse. This is why higher frequencies are have smaller amplitudes at the output: the capacitor has less time to charge over each cycle.

charging a capacitor schematic

When there is AC, this pattern is repeated. The next figure shows a 10-second portion after the pattern has been repeated for 65 seconds. The short pulse becomes the higher frequency signal, oscillating four times as fast as the long pulse. And its amplitude is about half the amplitude of the lower frequency signal. This phenomenon is the basic character of a low pass filter: low frequencies are attenuated less than high frequencies.

charging a capacitor schematic

The square input wave is useful for highlighting the differences in charge times for high and low frequency inputs. However, analyzing this low pass filter is more convenient when the source AC is a sine wave. In such a case, the output AC voltage is also a sine wave with the same frequency. The only differences between the input and output AC are amplitude and relative phase. There are no additional shape differences to consider. In the next figure, the previous calculations are repeated after replacing the square waves with sine waves having the same amplitudes and frequencies.

charging a capacitor schematic

Once again, the lower frequency input yields the output with the greater amplitude. Also, in both cases one can interpret the output as lagging behind the input. For example, look at the peaks of the sine waves. The output peaks appear to the right of the input peaks, indicating that the peaks occur later in time. And note that the peak of the lower frequency output is closer to the peak of the input (as a fraction of one cycle) than it is for the higher frequency case. The high frequency output is almost one quarter cycle behind its input. This is a general outcome for sine waves as well.

Decoupling Capacitors

You may want to jump to the high pass filter discussion, but this is a good place to introduce the notion of a decoupling capacitor (as opposed to the coupling capacitor described on the dc and ac page). Decoupling capacitors appear in DC power supply filters to remove possible AC. In a stompbox circuit, the power supply is supposed to be DC. If there is any AC in the signal, it will appear in the output of the stompbox along with the guitar signal. So care is taken to protect the guitar signal.

The simplest decoupling arrangement is to connect one lead of a capacitor to the power supply and the other lead to ground. Because the capacitor blocks DC, the desired DC power supply is unaffected. But the capacitor transmits (some) AC to ground, effectively acting as a low pass filter. You can think of DC as AC with a frequency of 0Hz. In the schematic for the TS808 clone for example, a 100μF polarized capacitor is used, so that the negative lead is connected to ground and the other lead to the power rail. Such capacitors are also commonly called bypass capacitors.

How do AC signals get into the power supply of a stompbox? One common source is a 50Hz or 60Hz cycle that comes from transformers converting AC electricity imperfectly into DC electricity. The bypass capacitor acts as part of a low pass filter to reduce this source of hum. Now 60Hz is a pretty low frequency and this is why bypass capacitors have large values, like 100μF. As I will explain later, the larger the capacitor value is the more AC signal a low pass filter attenuates.

There is another source of AC signals in a power supply and that is the operation of the stompbox ciruit itself. In a simple amplifier, for example, the current drawn by a transistor depends on the AC signal going through it. The valleys in the signal correspond to moments when most current is drawn. The internal resistance of the power source–this is especially an issue with old batteries–combines with cycles in current consumption to create cycles in the supply voltage. This is Ohm’s law in action: voltage is proportional to current, where the factor of proportionality is constant resistance.

When this second source of AC in the power supply occurs, the various parts of the circuit that draw on the power supply are said to be coupled. The current drawn by one part of the circuit causes the power supply seen by another part of the circuit to fluctuate. Usually this is undesirable. Look again at the schematic for the Tubescreamer clone and you will see two transistors, Q1 (near the input) and Q2 (near the output), both with +9V power connections above them. Or return to the simpler Basic Fuzz Face circuit, where there are two transistors fed by the same -9V power supply. These are both cases where the two transistors are coupled.

A bypass capacitor in the power supply section decouples the two transistors by providing the additional momentary current needed to smooth out the voltage supplied. The effect is not perfect. Voltage does not become constant. But the effect is sufficient in most cases to make the voltage supply effectively (or approximately) constant. So bypass capacitors are also called decoupling capacitors. And this is a second way in which that capacitor attenuates AC in the power supply.

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7 Responses to “Capacitors 4: Low Pass Filters”

  1. Paul N. said:

    I have a question. You say:

    the larger the capacitor value is the more AC signal a low pass filter attenuates.

    So with a power supply cap like this, wouldn’t you always want the largest cap possible, to remove more AC from the DC voltage? What are the downsides of using a capacitor too large – what else is there to take into consideration for a cap in this position?

    Amazing site – I have learned so much. Thanks!!!

    Posted 23.07.2008 at 6:30 am

  2. gaussmarkov said:

    hey Paul,

    there are probably more reasons than i know. a leading constraint is the physical size of the capacitor. as cap values grow so does size. cost, too.

    so folks end up balancing those things against what’s needed to get satisfactory performance. you do want your circuit in a 1590B don’t you? 🙂

    and thanks for the encouraging words. they motivate me to finish writing up some new stuff.

    Posted 23.07.2008 at 1:18 pm

  3. Sander de Groot (Asfastasdark) said:

    Really awesome site! But about the comment above: So basically you would want to get a decoupling capacitor of the largest value possible that fits in an enclosure? Or are there downsides to this?

    Posted 06.09.2008 at 6:17 pm

  4. Umino said:

    great site, but one question – can you also use that low pass filter for audio – like when you want to run a full range signal into your subwoofer amp and need to cut out the highs ? would that work with this circuit ?? and if so – wich cap and resistors should i use to have it cut at around 100hz ? PEACE – and keep up the good work

    Posted 04.10.2008 at 9:16 am

  5. Matt said:

    The LTSpice drawings are a bit confusing… It would be helpful if they had more clear inputs and outputs.

    Posted 03.04.2009 at 12:51 pm

  6. Anonymous said:

    what is the t-filter with two capacitor and ressistance?

    Posted 30.11.2009 at 9:30 am

  7. preethi said:

    The page design is very beautiful

    Posted 05.08.2010 at 3:47 am