D.  Six-pulse  bridge rectifier
 
Fig. BR-1 shows the schematic of a six-pulse, controlled bridge rectifier connected to an ideal three-phase source with commutating inductances included in each phase.  We'll start by first considering the operation of an uncontrolled rectifier without commutating inductances; i.e., the same circuit as shown except the thyristors are replaced by diodes and the inductors are removed.  Without commutation, only two diodes will conduct at any time, one on the top half of the bridge and one on the bottom half of the bridge.  Also, in order to have a voltage across the load, the two conducting diodes must be in different legs of the bridge; e.g., diodes 1 and 4 cannot be on at the same time.  Thus the voltage applied to the DC load consists of a portion of a line to line voltage from the three phase source.
 

Fig. BR-1:  Schematic of six-pulse, controlled bridge rectifier with commutating inductance

 
Fig. BR-2 shows the waveforms of the six line-line voltages that are available from the three-phase source.  There are six waveforms because polarity must be considered; i.e., the voltage from a to b is the opposite of the voltage from b to a.  The voltage of a to b would be delivered to the DC load if diodes 1 and 6 are conducting, while the voltage of b to a would be delivered when diodes 3 and 4 are conducting.  The right side of Fig. BR-2 shows the phasor representations of the six line to line voltages.

The diodes that are on at any given time are determined by which line to line voltage has the highest magnitude at that point in time.  Thus at t=0, the voltage from c to b, Vcb, is the highest.  Referring back to Fig. BR-1, diodes 5 and 6 would be conducting.  At 1.389 msec (30 degrees), the voltage Vab becomes higher than Vcb, causing diode number 1 to become forward biased and diode 5 to be reversed biased.  As a result, the current transfers from diode 6 to diode 1.  Since we are neglecting commutation, the change is instantaneous.  The DC output voltage, for a resistive load, is comprised of the tips of the six line to line voltages.

 

 
Fig. BR-2:  Waveforms and phasors of voltages that make up the output DC voltage
 

Fig. BR-3 shows the Graphic Modeller simulation for an ideal, controlled rectifier; i.e., one with no commutation inductance in either the source or the thyristors.  It consists of a three-phase source and the bridge rectifier.  This simulation can be used to demonstrate to students how the output voltage of the rectifier varies as the delay angle for the thyristors is changed.

As an example, Fig. BR-4 shows the DC output voltage, calculated by the model, for two cases.  The top trace shows the DC voltage for zero firing delay, meaning the thyristors are operating like diodes.  Only 0.1 seconds are shown, so slightly over one-half of a 60 hz cycle is shown.  Thus there are approximately three pulses in the DC voltage.  The lower trace shows the DC voltage when the firing angle of the thyristors was set to 30 degrees, meaning that thyristor commutation does not begin until the line voltage reaches its peak value.  Each interval of voltage is still 60 degrees, so the DC voltage follows the line voltage past the uncontrolled commutation point, resulting in a lower average value.
 
 

Fig. BR-3:  Graphic Modeller simulation of an ideal rectifier with an ideal source
 
Fig. BR-4:  DC output voltage for zero delay (top) and 30 degree delay (bottom)
 

In fact, the DC output voltage can be calculated for both cases.  Equation BR-1 gives the average DC value for the uncontrolled case (zero firing delay).  In this equation, Vrms is the rms value of the line to neutral phase voltage in the Wye-connected source.  Thus, there is a factor of the square root of two accounting for the peak value of the sinusoidal waveform, while the square root of three factor accounts for the difference in magnitude of the line-line voltage as compared to the line-neutral voltage.

Equation BR-2 provides the average DC voltage when the thyristor is operated with a delay angle.  Clearly if the delay is zero, equation BR-2 reduces to equation BR-1.
 

 
Of course, in reality, it is impossible for the current in one phase and thyristor to instantaneously transfer the current to another phase and thyristor because there is inductance in both the source and in each thyristor.  Thus a second simulation is available to demonstrate the effect of non-ideal commutation, as shown in Fig. BR-5.  At the time of the writing of this paper, this simulation does not allow phase angle control, so the switching elements are shown as diodes.  The effect of the commutating inductance in each phase of the source is that the diodes cannot turn on and off instantaneously.  Thus when one diode turns on, the one it is replacing stays on until the current through its source inductance goes to zero.  The result is that two phases are shorted together during the commutation and the voltage is the average of the two phases.
 
Fig. BR-5:  Simulation of uncontrolled bridge rectifier with commutating inductance
 

Referring back to Fig. BR-2, consider the first commutation, when Vcb (the yellow curve) is replaced by Vab (the blue curve).  Since the two are shorted together during the commutation, the output voltage becomes the average of the two voltages until the commutation is complete.  This has the effect of lowering the average DC voltage.  Fig. BR-6 shows the output voltage calculated by the simulation with an inductance of 1.0 mH in each phase and a resistive load of 10 ohms on the DC bridge.  Again a 0.1 second interval is shown and the averaging effect of the source voltages is clearly seen at the beginning of each conducting interval. By varying the commutating inductance and/or the load current, students can observe how the length of the commutation is changed.
 
 

Fig. BR-6:  DC output of uncontrolled bridge rectifier with commutating inductance
 

Also of interest is the effect the bridge rectifier has on the source voltages.  Since the phases are being shorted together, the voltage waveforms at the bridge side of the inductors suffer notching, as shown in Fig. BR-7.  This results in harmonics in the voltages of any devices that are connected in parallel with the bridge rectifier.  This helps demonstrate to students how power electronics can contribute to power quality problems in the power system.
 

Fig. BR-7:  Notching of phase voltages due to commutation
 
 
 

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