Tutorial 11 - Capacitors and Alternating Currents

The content of this tutorial is not on the AQA syllabus (or, for that matter, the OCR or EDEXCEL).  It is on the SQA Advanced Higher syllabus.  It is part of the option A (Alternating Currents) on the Welsh Board (WJEC).  Other Option A content can be found in Tutorials 9 and 10.

I have included it here because students studying the AQA Electronics Option need to know about the concepts of AC Theory to understand the ideas of filters.  Also students studying a syllabus that is not AQA will find these notes helpful.

Before you study these notes, you should be familiar with the basic concepts of capacitors.

Phase in resistive AC circuits

Up to now, we have studied resistive components.  If we connect a resistor across an alternating current, we see a sinusoidal graph like this:

We notice that both the current and the voltage are in step, or in phase.  We know that resistance is the opposition to the flow of current, and the resistance in this circuit remains constant whatever the frequency.

You will be familiar with the idea of phase from your study of simple harmonic motion.  You may well have been told or worked it out for yourself that SHM and circular motion are inextricably linked.  They use the same terms like:

• angular velocity (w);

• displacement (x);

• time (t).

Alternating current is generated by rotating machines.  The voltage is linked with the rotation using the relationship:

### Phasors

Have a look at this graph in which the current and voltage are 90o (p/2 rad) out of phase:

Analysing a graph like this can be quite tedious.  It is easier to use phasors.  Quantities in alternating currents can be represented by phase vectors or phasors.  A phasor is representation of a sinusoidal wave form as a rotating vector.  Phasors are particularly useful when you have two or more alternating electrical quantities that are a fixed amount out of phase.  This happens when we put reactive components like capacitors and inductors into an electrical circuit.

In our study of phasors, we will assume that:

·         The amplitude, A, remains constant;

·         The angular velocity, w, remains constant;

·         The phase relationship, f, remains constant.

(The curious looking symbol, f, is “phi”, a Greek lower-case letter ‘f’ or ‘ph’.  It is the physics code for phase angle.)

Remember from circular motion and SHM that:

w = 2pf

(Note that some syllabuses use w rather than 2pf)

Let’s look at a phasor that represents a sinusoidal waveform.  It has frequency f, and an angular velocity of w rad s-1.

This diagram is showing how there is a rotating vector that is projected onto a moving piece of paper.  It traces a sine wave.

The rotating vector is turning at a constant angular velocity of w rad s-1.  By convention it turns anticlockwise.  By convention, the zero point is the 3 o’clock position.  Angles are measured in radians.

In this tutorial we will consider the phase vectors are 90o (p/2 radians) apart.

For a resistive circuit, the phase vectors are like this:

Note that for clarity they are shown parallel.  Strictly speaking they should lie on top of each other.

Reactance
In AC circuits, there is another kind of opposition to the flow of current, called reactance.   Reactance is defined as:

The ratio of the alternating voltage to alternating current.

The physics code for reactance is X and the units ohm (W).

It is different to resistance in that the voltage vector and the current vector are 90o apart.

There are two main components that have a reactance, which is frequency dependent.  The reactance in a capacitor is the opposition to the change in voltage across the capacitor.  The reactance in an inductor is the opposition to the change in current through the inductor.  There is no reactance in purely resistive components.

Reactance is dependent on the frequency.

To make sense of reactance, we need to know something about the capacitor and the inductor.

Reactance of a capacitor

Suppose we connect a capacitor in series with a light bulb.  If we connect it to a DC supply, the bulb will (if the capacitor is large enough) momentarily flash and go out.  If we turn off the supply and short out the terminals, the bulb will flash momentarily and then go out.  We know that the capacitor has stored charge, and the charge passing through the bulb makes the filament glow for a very short time.

Now we set this circuit up with an alternating source:

If we connect a capacitor in series with a bulb:

• If connected to a low frequency, the bulb doesn't come on.

• If the frequency is increased, the filament starts to glow.  It gets brighter as the frequency is increased.

The capacitor seems to block DC, but allows AC to "flow".  It's as if the bulb has some kind of resistance to AC.  We say that the capacitor has a reactance.  The resistance of a capacitor is infinite because there is a layer of insulating material that prevents current from passing between the plates.  The reason that the current appears to flow is because the capacitor is constantly charging and discharging.

The higher the frequency the bigger the current that flows.  This is because current is the rate of flow of charge.

 The capacitor does NOT conduct electricity. Current does NOT flow through the capacitor. The "flow" of a.c. is due to the charge and discharge of the capacitor.

From the statement above, we can say that the reactance decreases with frequency.   Reactance is formally defined as:

the ratio of the potential difference to the current in a capacitor circuit.

In  a capacitor circuit, the equation is written as:

Reactance is measured in ohms (W).

 A student writes defines the reactance of a capacitor as: the ratio of the potential difference to the current flowing through a capacitor in a circuit. Comment on this answer.

We can use this circuit to measure the voltage and current as we change the frequency in a capacitor circuit:

We can process the data to show us how reactance varies with frequency:

If we plot reactance against the reciprocal of frequency, we get a straight line:

These data were processed from an experiment, so the straight line is not quite perfect.  The dotted blue line is the line of best fit and extrapolated.  This tells us that the reactance is inversely proportional to the frequency.

For a capacitor of capacitance C connected to a voltage source with a frequency f, the reactance, XC is given by:

 Worked Example What is the reactance of a 47 F capacitor connected to a 12 V AC supply that has a frequency of 500 Hz? Answer Use:

 A 10 F capacitor has a reactance of 320 . a) Show that the frequency of the AC supply is about 50 Hz; b) Calculate the reactance at 1000 Hz.

Simple CR Circuits

A pure capacitor connected to an alternating source is simply an electrical curiosity.  In reality there are resistive elements, such as the internal resistance of the source, and the resistance of the wires.  With resistive elements in the circuit, it becomes more interesting.

Let’s use the same circuit as above, but this time we add a resistor, R.

This is a simple series CR (or RC) circuit.

We will measure the voltage across the resistor as well as the capacitor.

We need to draw the current phasor first.  By convention we always draw the quantity which is the same in a circuit first, i.e. at the zero position.  The current into the capacitor leads the voltage by 90o (

 What quantity, current or voltage, is always the same in a series circuit?

So our current vector goes from left to right at the 3 o’clock position.  Parallel to that is the voltage across the resistor.

 Explain why the two vectors are parallel.

The phasor diagram looks like this:

 The voltage across the capacitor and the resistor do NOT add up arithmetically.  If VR = 3 V and VC = 4 V:    VR + VC ‡ 7 V

The voltage across the capacitor is at 90o and is lagging the voltage across the resistor, so its phase vector points vertically downwards.  The resultant voltage is shown by the phasor Vres.  We can work out Vres by simply using Pythagoras.

Since the voltage vector is pointing downwards, strictly speaking it should be negative.  However, since the voltage is squared, the negative sign becomes positive, so don't worry about it.

Impedance
In a reactive circuit, we cannot talk about resistance as such.

 What would be the resistance of a series RC circuit? Explain your answer.

So we have to introduce a new quantity, impedance, which is given the Physics code
Z and has the units Ohms (). Impedance takes into account the resistive and reactive elements in a circuit.

The formal definition of impedance is:

The ratio between resultant potential difference and the current in a reactive AC circuit.

We can write this as:

We know that for the resistive elements:

We also know that for the reactance of a capacitor:

Since the current is the same, we can redraw our phasor diagram as:

So we can say that the impedance is the vector sum of the resistance and the reactance. So by using Pythagoras again, we can write:

 At a certain frequency, a capacitor has a reactance of 20 ohms. It is in series with a resistor of 10 ohms. What is the impedance?