7.10 Alternating Currents & Transformers

Figure 1 below shows the graph of the voltage against time for a mains supply in the UK. 

Figure 1

Using Figure 1, draw a line to show the dc voltage that gives the same power as produced by the alternating waveform in the mains supply.

Sketch a graph which shows how the power supplied by this voltage to a resistor with resistance of 300 Ω varies with time. 

Label the vertical axis as power and mark on both axes any significant values.

An oscilloscope has a screen of eight vertical and ten horizontal divisions. 

Describe how you would use the oscilloscope to display the alternating waveform in Figure 1 so that four complete cycles are visible. 

An electric oven is connected to root mean square (rms) mains supply using a cable with a non-negligible resistance. The cable connects the heating element in the oven to the mains supply. 

Explain, using the resistance of the cable, why the rms voltage across the heating element in the oven will be less than the rms voltage calculated in part (a).

A step-up transformer has 500 turns on the primary coil and 1200 turns on the secondary coil. The secondary coil provides an rms current of 0.40 A to a 230 V ac supply. 14% of the energy transferred by the transformer is wasted as heat. 

Calculate the peak current in the primary coil.

A DVD player that uses 0.2 A of current is provided with a mains adapter. The adaptor uses a transformer with a turns ratio of 5:3 from a mains supply of 230 V. The DVD player works when supplied with a power of 46 W. 

State the type of transformer and calculate the efficiency of the transformer in the adaptor.

Copper wires each with a length of 5000 m and a diameter of 1.3 mm is used to conduct electricity from a power station to an office building. Aluminium wires of half the length and twice the diameter is used to conduct electricity from a power station to a school.  

(i)         Deduce, with a calculation, which wire is more efficient at transmitting electrical energy. 

(ii)        Explain why this then leads to the use of alternating current rather than direct current when transmitting electrical energy. 

            Resistivity of copper = 1.7 × 10^–8 Ω m

            Resistivity of aluminium = 2.7 × 10^–8 Ω m

A student is provided with apparatus shown in Figure 1a. 

Figure 1a

A coil of wire is connected to a signal generator and the search coil is connected to an oscilloscope. When a sinusoidal alternating current is passed through the coil an alternating emf is induced in the search coil. 

An emf is induced in a coil of wire displayed on an oscilloscope with 8 vertical and 10 horizontal divisions as shown in Figure 1b.

Figure 1b

Draw the trace of the oscilloscope on Figure 1b if the frequency of the emf in the coil is 2500 Hz and the time-base setting is 0.2 ms cm^–1.

The student clamps the rectangular frame so that it remains in a vertical plane. Without changing the position of the search coil she rotates the circular frame about a vertical axis so that it is turned through an angle, as shown in Figure 2a. 

Figure 2a

She turns off the time−base on the oscilloscope so that a vertical line is displayed on the screen. The y−voltage gain is at 10 mV cm⁻¹ and she records the length L of the vertical line and the angle θ through which the circular frame has been rotated. 

The trace on the oscilloscope appears as shown in Figure 2b. 

Figure 2b

Describe two adjustments the student should make to the trace to reduce the uncertainty in L. 

You should refer to specific controls on the oscilloscope. You may use Figure 2a to illustrate your answer.

The student adjusts the signal generator so that the frequency of the emf in the circular coil is tripled. 

State and explain any changes she should make to the settings of the oscilloscope in part (b) so that she can repeat the experiment.

A transformer is required to produce an r.m.s output of 5.0 kV when it is connected to the 230 V r.m.s mains supply. The secondary coil has 900 turns. 

Calculate the number of turns on the primary coil, assuming the transformer is ideal.

The transformer suffers from eddy current losses. 

Explain how eddy currents arise.

The current in the primary coil is 0.8 A. The secondary coil provides a power of 160 W. 

Calculate the efficiency of the transformer.

(i)         Suggest one reason why the transformer may be less than 100% efficient. 

(ii)        Explain one feature of the core that would improve the efficiency of the transformer.

An oscilloscope is connected to a sinusoidal AC source. The frequency and the voltage output of the AC source can be varied. 

At a certain frequency the AC signal has an r.m.s output of 9.3 V. Figure 1 shows the trace obtained on the screen of the oscilloscope which has a frequency of 12.5 Hz. 

Figure 1

Show that the time for one horizontal division on the screen of the oscilloscope is around 20 ms.

Calculate the peak-to-peak voltage of the signal. 

Calculate the y-gain of the oscilloscope settings.

The settings on the oscilloscope are now changed to a time base of 10 ms per division and y-gain of 6.6 V per division. 

Draw the grid in Figure 2, which represents the screen of the oscilloscope, the trace that would be seen with these settings. 

Figure 2

A step-down transformer is used to supply electricity from the national grid. An e.m.f is induced in the secondary coil of the transformer.

Explain, in terms of electromagnetic induction, how a current in the primary coil gives rise to this induced e.m.f to step down voltage.

Show that for an ideal transformer,

A step-up transformer has 200 turns on the primary coil and 1500 turns on the secondary coil. The secondary coil provides an r.m.s current of 0.30 A to a 240 V ac supply. The efficiency of the transformer is 85%. 

Calculate the r.m.s current in the primary coil.

Hence, calculate the peak current in the primary coil.

The mains supply is a.c and is transmitted through copper wires in the national grid. 

Explain why alternating current, rather than direct current, is used when transmitting electrical energy in the national grid.

In Japan, the mains supply in the western half of the country runs at a frequency of 60 Hz whilst the eastern half is at 50 Hz. A string of power stations across the middle of the country steps the frequency up and down as it flows between grids intersecting both regions. 

Figure 2 shows an a.c signal on an oscilloscope set at a y-gain of 50 V and time division of 2.8 ms. 

Figure 2

Determine whether the signal is from the mains supply in the eastern or western part of Japan.