Topic 8 Filter Circuits

At the end of this topic you should be able to:

Calculate the reactance of a capacitor with the formula Xc = 1/2pfC.
Draw and explain passive filters using RC circuits
Draw and explain first order active filters including treble cut, treble boost, bass cut and bass boost.
Calculate the break point of active filter circuits.

The Reactance of a Capacitor

The following demonstration shows how a capacitor “blocks” direct current, but “allows” alternating current to flow.

Circuit A is connected to a DC supply, whereas Circuit B is connected to an AC supply.
In Circuit A the lamp might flash momentarily, but would then remain unlit. In Circuit B the lamp remains glowing at normal brightness.

Question 1. Use your knowledge of capacitor action to explain these observations. Click HERE to revise from Module 1. ANSWER

It seems that in Circuit A the current is being blocked, while in Circuit B the current is being allowed to pass.

We see this because the capacitor is being continually charged and discharged. No current is flowing through the capacitor. It cannot because there is an insulating layer. However we will treat the capacitor as if it were allowing a current to flow.

If we replace the 1000 mF capacitor with one of 100 mF, the charging and discharging currents are less. Therefore the bulb will glow less brightly. We can make the bulb brighter again by increasing the frequency.

From this we can conclude:

Large capacitors offer “less opposition” to AC.
Increasing the frequency increases the current in a circuit containing a capacitor since the same charge flows on and off the plates in a shorter time.

We can sum this up in the next diagram:

The reactance of a capacitor is the ratio of the voltage to the current. The symbol for reactance is X_C and the units are ohms (W). It is the “effective” resistance of the capacitor at a particular frequency. The reactance is linked to the capacitance, C, and the frequency, f, in the following relationship:

X_C = __1__

2pfC

where:

f is the frequency in Hertz
C is the capacitance in farads. Watch out for this bear trap.

We can also define the reactance in terms of the rms voltage and the rms current of a sinusoidal waveform:

X_C = V_rms

I_rms

Question 2. Why do we use the term reactance to describe the ratio of voltage to current, in preference to resistance? ANSWER

Notice that in calculations we treat reactance like resistance as in Ohm’s Law.

Worked Example

A 400 mF capacitor is connected to a 56 Hz supply. If the supply voltage is 14 V rms, what is the rms current through the capacitor?

Use X_C = __1__ to work out the capacitative reactance.

2pfC

X_C = __1__ = __________1_____________ = 7.11 W

2pfC 2 ´ p ´ 56 Hz ´ 400 ´ 10^-6 F

Current = 14 V ¸ 7.11 W = 1.97 A

Question 3: What is the reactance of a 1000 mF capacitor at a frequency of 1000 Hz? ANSWER

We can plot a graph of how the reactance varies as frequency. As X_C varies as 1/f, the graph will be a hyperbola.

To get a straight line, we would need to plot X_C against 1/f, the reciprocal of the frequency, which is T, the period. The gradient will be 2pC.

The essential difference between a reactive and a resistive circuit is that:

Power is dissipated in a resistive component
No power is dissipated in a reactive component.

At the very simplest level:

In a resistive circuit, energy is dissipated as heat.
In a reactive circuit with a capacitor, the energy is used to build up the electric field.

Now go on to RC Filters.