Tutorial 5 B - Inductor Reactance

Learning Objective

To understand the reactance of an inductor;

To recognise and use the equation XL = 2pfL

To measure the reactance with different frequencies

Understand the phase relationship between voltages;
Analyse simple LR circuits;
Discuss the phase change in an LR circuit.

Key Questions

What is the reactance of an inductor?

How do we demonstrate the reactance of an inductor?

What is the phase relationship with an inductive circuit?
How do LR circuits behave?
What is the impedance in the LR circuit?
What happens at radio frequencies?

## Reactance of an Inductor

In the last tutorial, we saw that if an inductor was put in series with a bulb:

• At low frequencies, the bulb in series with the inductor was bright;

• As we increased the frequency, the bulb became dimmer.

• It allowed DC to flow freely, but opposed the flow of AC

It was as if the inductor had a kind of “resistance”.

Electrically, the inductor is simply a wire coiled up.  A perfect inductor has zero resistance.  In reality there is resistance, because copper wire has resistance, but it is very low.  In this section we will assume that the inductor is perfect.  The equation for reactance is:

 Worked Example What is the reactance of a 3.5 mH inductor connected to a 12 V AC supply that has a frequency of 500 Hz? Answer Use   XL = 11 W

If we plot reactance against frequency, the graph is a straight line of positive gradient going through the origin.  This means that the reactance is directly proportional to the frequency for a perfect inductor.

The graph is like this:

You can determine the value of the inductance by taking the gradient, and dividing by 2p.  Note that the line, taken from an actual experiment, is not quite a perfect straight line, but a line of best fit has been added.

The derivation of the reactance is given in the extension page.

 Discuss whether an inductor would allow a constantly changing unidirectional current to pass. Question 7 A 10 mH inductor has a reactance of 320 W.       a)      Show that the frequency of the AC supply is about 5100 Hz;      b)  Calculate the reactance at 1000 Hz.

## Phasor diagrams and inductor reactance

We know that the current in an inductor is dependent on the frequency of the alternations.  We use this equation for the reactance of an inductor:

In the derivation for this equation, we started off with the relationship for instantaneous voltage:

In the derivation we used this expression in the argument  (Click HERE to see the derivation):

There is something that is much more significant about the result above.  The voltage is related to time through the sine function, while the current is related through the negative cosine function.  So what?

Let’s keep w the same and see how the voltage and current vary across the same time period.  If we plot the graphs we will see the significance:

The current graph is lagging the voltage graph by 90o or p/2 rad.  So the voltage phase vector is leading the current phasor by p/2 rad.

We can show this in a phasor diagram as this:

### Simple LR circuits

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

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

This is a simple series LR circuit.

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

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.

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

Draw the phasor diagram here:

The voltage across the inductor is at 90o and is leading the current, so its phase vector points vertically upwards.  The resultant voltage is shown by the phasor Vres.  We can work out Vres by simply using Pythagoras.

The sum of the two voltage is a vector sum, not an arithmetical sum.  This is because the vectors are 90o apart.

Now add in the phase angle:

We can easily work out the phase by which the current leads the resultant voltage:

 What quantity, current or voltage, is always the same in a series circuit? Question 9 Explain why the current and voltage vectors are parallel. Question 10 At a certain frequency, the voltage across a inductor is found to be 3.5 V while the voltage across the resistor is found to be 4.0 V. a.    Why is the total voltage not 7.5 V? b.    What is the resultant voltage? Question 11 Using the data from question 10, calculate the phase angle between the current (the voltage across the resistor) and the resulting voltage.

### Impedance

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

When we studied capacitors, we introduced a new quantity, impedance, which was given the Physics code Z and had the units Ohms (W).  Impedance takes into account the resistive and reactive elements in a circuit.  The same applies to reactance of an inductor.

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 an inductor:

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:

We can work out the phase angle between the resistance and impedance quite easily, just like we measured it between the voltage across the resistor and the voltage across the inductor.

We can write other equations for the phase angle:

The reactance and resistance do NOT add up arithmetically; you have to do a vector sum

The phase angle is in radians.  If you want to convert to degrees, you have to multiply by p and divide by 180.

 Question 12 At a certain frequency, a inductor has a reactance of 20 ohms.  It is in series with a resistor of 10 ohms.  What is the impedance?  What is the phase angle between the resistance and the impedance?