Block E - Rectifiers

Objectives

1. Measure peak, average, and rms voltage across the resistive load of a rectifier circuit and the average current in the load.

2. Measure the peak to peak ripple voltage when a capacitor is added to the load.

Background

Electric power is delivered to our homes and factories in the form of 60 Hz ac, but many end uses require dc. This conversion from AC to DC is accomplished by one or more diodes in a rectifier. A diode is a solid state device that allows current to flow in only one direction. The voltage drop across the diode when current is flowing is about 0.7 V. If the other voltages in the circuit are sufficiently high, then this 0.7 V can be ignored without significant loss of accuracy. In this case, we are assuming the diode to be ideal, that is, with zero voltage drop when conducting and infinite impedance when reverse biased.

The simplest rectifier circuit is the single-phase half wave rectifier shown in Fig. 1. The secondary voltage is v = V_m cos t. When v is positive, the diode is forward biased and conducting so that the load voltage v_L is the same as v (assuming an ideal diode). When v is negative, the diode is reverse biased, no current flows, and the output voltage is zero. We have an output wave consisting of a half cycle of a cosine wave and a half cycle of zero voltage. The load voltage is positive or zero, thus making it DC rather than AC, although it is still quite variable. This waveform is used to drive some small household motors, but many applications require a smoother output.

The next step in improving the smoothness of the output is the single-phase full wave rectifier shown in Fig. 2. This particular circuit requires a transformer with a center-tapped secondary. Our analysis will be for the case where each half of the secondary has an instantaneous voltage v = V_m cos t., so we have twice as many turns in the secondary in Fig. 2. as in Fig. 1. When v is positive, the top diode will conduct, and when v is negative, the bottom diode will conduct, yielding the voltage waveform shown. We have filled in every other half cycle from the half wave rectifier case. This produces a much smoother waveform but the voltage still varies all the way from zero to V_m.

A smoother waveform is obtained by using a three-phase half wave rectifier, as shown in Fig. 3. The voltage of the first winding is v_a = V_m cos t , with v_b and v_c lagging v_a by 120^o and 240^o respectively. Each diode conducts one third of the time. Each `hump' of the output extends for 120^o. Since Cos 60^o = 0.5, the output voltage only varies from V_m to 0.5 V_m, rather than from V_m to zero.

This process can be continued by going to a three-phase full wave rectifier, then to a six-phase full wave rectifier, and so on. Hopefully the progression will be obvious after examining the circuits of Figs. 1-3, so we will go no further in this lab.

When we apply the formulas for average and rms voltages (given in Block A) to these waveforms, we get the following results.

Rectifier circuit	Fig. 1	Fig. 2	Fig. 3
Peak voltage across load	Vm	Vm	Vm
Average voltage across load	Vm/	2Vm/	0.827 Vm
rms voltage across load	Vm/2	Vm/	0.841 Vm

There are many applications where the `raw' rectified waveforms are not smooth enough. The next step in smoothing the output waveform is to add a capacitor, as shown in Fig. 4. Whenever v is more positive than the capacitor voltage, the diode conducts and the capacitor charges. When v drops below the capacitor voltage, the capacitor tries to discharge through the diode, but is unable to do so. The diode becomes an open circuit and the capacitor discharges through the load resistor until v again becomes more positive than the capacitor voltage. The difference between the maximum and minimum voltages across the load is called the ripple voltage V_r. The time t_r at which the minimum voltage occurs is found by solving the equation

............(1)

This transcendental equation does not have a closed form solution. However, it can be solved reasonably quickly by an iterative process. Just assume a value for t between 1.5 and 2, and solve for t. Substitute this value of t in both sides of (1) and evaluate. If the cosine function is smaller than the exponential function for this value of t, assume a larger value of t. When the two sides of (1) have the same value, then you have assumed the correct value of t. Either side of (1) is then the minimum voltage. Subtracting this from V_m then yields V_r.

Discussion and Calculations

1. Calculate the expected results for peak voltage, average voltage, and rms voltage across the load, and the average current through the load, for the circuits in Fig. 1, 2, and 3. The load resistance is 400 . Assume that the rms value of v or v_a is 60 volts in each figure.

2. For Fig. 4 with C = 42 µF and R = 400 , plot the load voltage variation from 0 to 1/60 second. Assuming V_m = 60, estimate the minimum value the load voltage will be, using (1).

Instructional Activity in Class

1. Measure the series resistance of the two approximately 200 resistors on your bench with the digital multimeter. Record it in your notebook.

2. Check the fuse in your digital multimeter. The next activity will not work properly if the fuse is blown. Replace it only with a 2 A fast blow of the correct length.

3. Assemble the circuit in Fig. 5. Use x10 probes to the oscilloscope. Turn the 120 volt breaker off when connecting or changing the circuit. Remember that 120 volts can be fatal. Treat it with respect. Use the 150 DC VOLTS panel meter to measure average volts and the 150 AC VOLTS panel meter to measure rms volts. Use the digital multimeter to measure average current through the load by setting it on DC mA. Set the oscilloscope input on DC coupling. Sketch the channel 1 and 2 waveforms on the same set of axes in your notebook with S₂ open. Record peak volts from the oscilloscope, average and rms volts from the panel meters, and average current from the digital multimeter and compare with theoretical values.

   Close S₂ and sketch the output voltage waveforms. Record the maximum and minimum voltages across the load. Compare with the results of Prelab Activity 2.

   Use the Fluke Harmonic Analyzer to measure the harmonic content of the current entering the primary of the transformer (both with and without the capacitor in the circuit). Record the THD (Total Harmonic Distortion) and the amplitudes of the first 10 harmonics.

4. Assemble the circuit in Fig. 6. Do not connect the oscilloscope ground to more than one point in the circuit. Measure peak volts, average volts, rms volts, and average current with switch S₂ open. Sketch the waveforms. Record maximum and minimum voltages across the load. Close S₂ and repeat, including a sketch of the waveforms. Record maximum and minimum voltages across the load. Compare with theory.

   Use the Fluke Harmonic Analyzer to measure the harmonic content of the current entering the primary of the transformer (both with and without the capacitor in the circuit). Record the THD and the amplitudes of the first 10 harmonics.

5. Assemble the circuit in Fig. 7. Adjust the three phase variable autotransformer until you have the same peak voltage as in the previous two activities. Sketch the waveform across the load. Measure peak volts, average volts, rms volts, and average current with S₂ open. Record maximum and minimum voltages across the load. Close S₂ and repeat. Record maximum and minimum voltages across the load. Compare with theory.

   Use the Fluke Harmonic Analyzer to measure the harmonic content of phase "a" of the current entering the primary of the transformer (both with and without the capacitor in the circuit). Record the THD and the amplitudes of the first 10 harmonics.

Conclusion

1. Estimate the percentage error involved in this experiment by assuming ideal diodes rather than real diodes with a 0.7 V drop while conducting. Is this likely to be a problem (introduce significant error) in an experiment using analog meters accurate to about 2 %?

2. Is there any potential problem involved with making the capacitor C arbitrarily large in order to make the ripple voltage arbitrarily small?

3. Each diode in a three-phase full wave rectifier will conduct for one sixth of the time. By analogy with the three-phase half wave case, what will the minimum voltage be, with no capacitor in the circuit?

4. Prepare a table comparing the THD and harmonic contents of the currents in the three cases from Lab Activities 3-5. Does including the capacitor have a consistent effect on the amplitude of the harmonics?