
In the previous section we defined the generated EMF from Faradays Law. In this section we will define the equation set for commutation of the PMB motor. In the first analysis the source energy was some type of mechanical force turning the motors rotor, hence creating a generator effect. This section the energy source turning the motor will be a three phase voltage applied to the phase windings of the motor. In a PMB motor commutation is only controlled electrically by stimulating the phase coils in the appropriate sequence. Since commutation has two states, 1 the induced phase voltages that cause commutation and 2 the generated EMF produced from the commutation, the two states must have some type of interaction either to increase or decrease the motors performance. We will show the interaction of this generated EMF to the applied voltage. Another factor for consideration is that PMB Motors have a step angle assigned to it relating to the actual assignment of the rotor coils which is position 0. In the initial development of these types of motors the average number of steps were small meaning that the motor would have large angular steps for each state and the alignment was course. Today with the industries refined manufacturing processes motors with steps above 1000 along with the addition of microstepping, we are able to set the motors initial alignment more accurately. The other advantages of PMB motors that we stated earlier is that the RPM is directly proportional to the applied voltage. This means that we would control the amplitude of the applied voltage digitally to control the speed of the motor. This is a big improvement over brush type motors. Since this is true, then we are also able to control the direction of the motor by reversing the applied voltages sequence. We now have both speed and direction therefore allowing torque control as well under the applied voltage control. We now have a motor that we can control and there is no arcing as with motors that require brushes and contacts to commutate. The conditions for commutation stated means that the applied voltage has to be generated and controlled by some mechanism, preferably a digitally generated control set. Therefore we will have to generate the applied voltage in steps. As we will show this fits in nicely with the actual design of the PMB motor. From the previous analysis we see that the generated EMF is speed dependent, therefore there is a direct relationship between speed and applied voltage. Since we will be using a digital generated waveform, the amplitude of the waveform will be broken down in steps. The number of steps is called the vertical resolution and is represented as a number from 1 to N. Hence if we set the number of steps vertically to 20 then we will only have 20 speeds to select from. Therefore the higher the number the more vertical sensitivity we have to control the motor. So, why do we care about this variable ? The larger the number of steps the more we can control the speed variations and the less ripple transition we get when changing speeds. This in turn, results in a better performance of the motor. There are other factors that also effect ripple performance one is torque ripple which is directly proportional to the number of poles of the motor. By now you can see that there is some kind of mechanical to electrical performance ration emerging. What we would like to do is to just get the system performance characteristics and not go into each internal mechanism that contributes to the losses and other motor parameters. Remember this is a system performance that we will be able to relate to the individual sections of the system at a later time. The compromise of the number of poles and amplitude resolution to get the best performance from the motor is one of the concerns that we will be able to analyze. For our analysis we will use 1024 steps or points for the vertical resolution, which relates to 10 bits or a 60.1975 db dynamic range vertically. This vertical resolution in voltage derived from the equation , where B is the number of binary weighted bits, hence: . This means that we will breakup the amplitude of the sine wave equation set into 1024 parts for each point over the 2 period, producing a 10 bit vertical resolution. For low voltage motors,less than 20 volts dc, anything more than ten bits would be difficult to see a performance improvement since the motors field coils harmonics would average this out. However, for higher applied voltages of 100 or 200 volts then 12 bits (4096) or 14 bits(16,384) vertical resolution would be recommended in general. Also for very high resolution applications where special motors are design that are greater than six poles then a high vertical resolution is also recommended. Generally the mechanical analysis of the application will dictate the motor requirements and a compromise can be analytically obtained. In either case, this general analysis can be applied to the Bridge Test System TDA to complete the characterization of the power module used to drive the motor. We will now add two more parameters to the Universal Symbol Assignments table, "n" for the number of points to generate the waveform. Each point of the waveform will have a amplitude resolution of 1024 points. We will also require an index "i" to position the point over the period.
The integer n has been determined for us by the sequencer in the Bridge Test System TDA. which is 260 points for a 360° period. This number was determined by the memory capability of the sequencer in the Bridge Test System. The vertical resolution was selected at 1000 vertical steps for a 0 to 100% dutycycle allowing a 0.1% dutycycle step since the memory capabilities of the sequencer is only 262,144 bytes of memory we would require 260,000 bytes to produce one full period of the waveform. The FET onoff control requirement is one byte per waveform point. This is explained in the Bridge Test System TDA Summary. Therefore each generated point on the waveform has a vertical accuracy of 0.001 of the full scale amplitude and there are 260 points to the 360° or 2 period waveform. Therefore if we used a 10 volt DC supply we would get an absolution maximum vertical resolution of 10 millivolt steps. This is why 10 bits of vertical resolution is adequate for most applications below 20 volts. The following equation set still does not take into consideration the generated EMF. That is added on at the end of the equations on the next page when we get to the load requirements. Top 
The equation set derived for the phasor equation set defines the Line to Line voltages as:
The sinewave voltage equations for Line to Neutral also from phasor equation set is defined as:
APPLIED VOLTAGE EQUATION SET WAVEFORMS

Observation of the phasor vector representation and analysis can show that we will only get about 86.6%, ( ), effectiveness of the total applied voltage using a sinewave equation set. Therefore in order to recover this 14% loss of applied voltage to the fields a concept called Triplen Injection is used. This is accomplished by adding a third harmonic into the phase voltages. Since Line to Line voltages are applied the triplen is cancelled via the phase coils and we are left with the original sinewave at a much higher amplitude phase to phase. At this time we are able to add a scale factor to recover the lost amplitude. The voltage equation set and the waveforms are shown below. Keep in mind that these equation sets are applied with no phase shift added to the equation set. We will discuss the phase shift this later as well. Observing the resultant waveform for the line to line voltage we see that we still get the sinewave but twice the amplitude although the Line to Neutral is not a sine wave. The cancellation of the triplen injection is in the phase coils. Since the cancellation is performed in the actual phase to phase coils, the impedance of these coils must be matched. Any mismatches phase to phase would create some third harmonic design difficulties when working with high torque motors of milliohm field coil resistance along with the added imbalances of the control loop impedances. Therefore, this triplen injection concept requires that all three phase impedances be balanced in order to prevent third harmonic ripple being induced in the fields and reduce motor performance. We also observe that the phase excitation is not at the same applied voltage of the triplen waveform as well as not being a sinewave. We also observe that the resultant wave form is in phase with the generated EMF by definition of the equation sets starting point and the triplen waveforms have a phase shift to them. Top PHASE A TRIPLEN INJECTED WAVEFORMS
PHASE B TRIPLEN INJECTED WAVEFORMS
PHASE C TRIPLEN INJECTED WAVEFORMS

Since the initial application of this analysis is not an actual PMB motor but the equivalent inductive load of the field coils, we are not concerned at this time with the generated EMF, however, we will explain the addition of this parameter in the next page of analysis. Our analysis shows the results as a bipolar system. Since these are DC motors and generally operate from a single supply voltage, we will normalize the equation set to insure a unipolar resultant waveform. In order to do this normalization we will introduce a concept call Phase Grounding. This is simply the grounding of one phase while the remaining two are excited and is accomplished by taking the phase that has the lowest value and setting it to zero, then subtracting that phase from the remaining phases This is shown in the Bridge Test System TDA. It is also apparent that even with the static inductive load we have introduced some type of a phase depending on the frequency the tests are conducted at. This also becomes apparent when a motor is used as the load and the phase shift has to be added to the equation set depending on the speed of the motor. This is an easy addition since we are generating the equation set digitally. 
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