# F6-2: Mutual Inductance - Transformers¶

NB: Students are advised to perform experiment F6-1 Self Inductance before attempting F6-2.

## Apparatus¶

Coils 300 turns & 150 turns; wire for 20 turn & 10 turn coils; 2 iron C-cores; C-core clip; $$10\Omega$$ and $$5\Omega$$ resistors; CRO (oscilloscope); AC power supply; connecting leads (5 short).

NB: This experiment requires mains electricity.

## Procedure¶

1. Construct the following:

1. Using $$N_p$$ = 150 turns, $$N_s$$ = 10 turns. Measure and set $$V_{in} = V_p = 4 \text{V}$$ peak. Measure $$V_s$$ peak. Calculate $$\frac{V_p}{V_s}$$ and $$\frac{N_p}{N_s}$$ and compare their values.

2. Repeat a) with the following numbers of turns:

3. Keeping $$N_p = 150$$ turns and $$N_s = 300$$ turns, reduce $$V_{in} = V_p$$ to $$2\text{V}$$ peak. Measure $$V_s$$ peak, and calculate and compare $$\frac{V_p}{V_s}$$ and $$\frac{N_p}{N_s}$$.

4. Remove the clip from the iron cores, and remove one C-core. Place two coils, one on each arm of a single C-core. Again use $$N_p = 150$$ turns and $$N_s = 300$$ turns. Set $$V_{in} = V_p = 4\text{V}$$ peak. Measure $$V_s$$.

1. Construct the following with $$N_p = 300$$ turns, $$N_s = 150$$ turns, and an extra test coil of 10 turns as shown:

1. Connect the CRO to measure $$V_p$$ peak, and set $$V_{in}$$ so that $$V_p = 4$$V peak.
2. Measure $$V_R$$ peak, and thus calculate $$I_p$$ peak, the current in the primary circuit.
3. Measure $$V_T$$ peak, across the 10 turn test coil.
4. Connect a $$5\Omega$$ resistor between A and B, across the 150 turn secondary coil. Connect the CRO to measure $$V_p$$ and adjust $$V_{in}$$ so that $$V_p = 4$$V peak.
5. Measure $$V_R$$ peak, and thus calculate $$I_p$$ peak.
6. Measure $$V_T$$ peak again. This should be about the same size as the value measured in procedure 2 c) above.

## Theory¶

1. Flux $$\Phi$$ and induced emf $$E$$ are related by the following:

$E_p = -N_p \frac{d \Phi_s}{dt} \label{eqnA} \tag{equation A}$
$E_s = -N_s \frac{d \Phi_p}{dt} \label{eqnB} \tag{equation B}$

The unit of flux is the weber - Wb. When $$\Phi_s$$ (the flux through the secondary coil) $$=\Phi_p$$ (the flux through the primary coil), then:

$\frac{d \Phi_s}{dt} = \frac{d \Phi_p}{dt}$

therefore:

$\frac{E_s}{N_s} = \frac{E_p}{N_p}$

thus:

$\frac{E_p}{E_s} = \frac{N_p}{N_s}$

If the resistances of the primary coil and the secondary coil are both low and the currents flowing through them are not too large, then:

$V_p \approx E_p \quad \text{and} \quad V_s \approx E_s$
2. The 10 turn coil is used to detect if the flux $$\Phi$$ in the iron core changes in the experiment. If $$\Phi = \Phi_{peak} \sin \omega t$$, then $$\ref{eqnB}$$ can be used to show that $$V_T$$ peak $$\propto \Phi_{peak}$$, provided that $$\omega$$ is constant.

## Analysis¶

1. Why, in experiment 1a) to 1c), are the two calculated ratios not exactly equal (hint: use the theory, and the fact that the primary coil has some resistance)?
2. Use the theory to explain the result of procedure 1 d).
3. According to Lenz’s Law, the induced current in the secondary coil in the procedure 2 d) is in such a direction so as to reduce the flux in the core. However procedure 2 f) shows that the flux remains approximately constant. How is this possible (hint: consider the primary coil)?
4. Give an explanation in terms of power flow for the change in $$I_p$$ produced as a result of connecting the $$5\Omega$$ resistor to the secondary coil.
1. These coils are simple electrical transformers. What are the causes of power loss in a transformer, and how can they be minimised?
2. All electricity supply companies use transformers in their power distribution systems. Explain, giving reasons, how they are used.