From the waveform analysis of the triplen injection equation set, we see that the Y Amplitude result was bipolar and we introduced the grounded phase philosophy to normalize the equation set. We also showed that the "Y" amplitude may also be labeled as the duty cycle from 0 to 100% to control the motor speed and torque as mentioned earlier. Observing the previous triplen injected waveforms, we see that in order to normalize the waveforms, we will require an offset and a gain factor to utilize the full dynamic range of the excitation voltage. The parameters selected are a Gain factor 1601 and a offset factor 1601. The peak of phase A of the triplen shows to be at 90 () and shows the peak amplitude of the triplen is still limited to the same 86.6% of its dynamic range as with the sine wave. However, the resultant waveform is not therefore we will control the triplen waveform to adjust the resultant to give us the full dynamic range. In order to maintain a positive dutycycle we will calculate the gain and the offset to utilize 100% of the dynamic range.
is the gain factor to regain 100% of the range. is the offset to insure the waveform positive values.
Combining the gain and offset into the
equation set we get: The peak amplitude of the triplen is now defined as with a base of zero allowing the full dynamic range of the waveform. We have now recovered the losses from a sine wave equation set. As we see the triplen injection waveforms are not sinewaves, but the resulting waverforms from Line to Line is still a sinewave. We still have to keep in mind that we require matching phase coils in order to maintain the canceling out of the triplen injection.
NORMALIZED WAVEFORM WITH GAIN AND OFFSET

GROUNDED PHASE CONDITIONS

LINE to LINE CONDITIONS

As we can see the final waveform is still a sinewave and the driver voltage for each phase is 0 to 1 for the duty cycle OF 0 TO 100%. This meets our requirements to drive the power module. The next step is to apply this to a motor system or some type of similar inductive load to be able to compare the theoretical and empirical data. The actual FETs are turned onoff depending on the dutycycle at each point of the waveform. Each point has 1000 steps to it as shown in the FET drive waveforms. This controls the vertical amplitude or applied voltage for each point on the generated waveforms.
We will now show how the generated EMF contributes to the equation set when we use a real motor for testing the power bridge module. . With no phase shift the generated EMF will add to the applied commutation voltage as by definition of the equation set starting at position 0. Therefore to insure that we are applying the required voltage to get the desired speed and torque desired we weight the equation set to take into consideration the generated of the motor. Top
The diagram to the left shows the circuit for a single phase Line to Line of the motor. Since we already require that the phases be equal then only one phase has to be shown as a reference. Using the parameters setup in the first page of this analysis we will begin to tie all of this together to the application. We will start with a motor with the following parameters as shown to the left in the table. Since the actual load used in the Bridge Test System is static, we will model the circuit similar as the load used to collect the empirical data. The Equivalent load circuit shown on the left simulates our test conditions. 

MOTOR EQUIVALENT CIRCUIT PHASE VOLTAGE Vab

EQUIVALENT CIRCUIT LOAD DIAGRAM

The objective here is to drive the inverter FETs with a PWM pulse that simulates the 3 phase waveform to a controlled current. There will be one phase where the upper FET is on and the following phase where the lower FET will be on. There will be not condition where the upper and lower FET of the same phase is on at the same time. As described in the Bridge Test System TDA. We will now calculate the voltage on the terminals based on the selected control current. Then, calculate the duty cycle required to generate the waveform based on the parameters. The calculation of the generated EMF Vk is set to zero because we are using a static load. The frequency or RPM the three phase waveform is running at is 1384.6. This is calculated from the sequencer operation in the Bridge Test System TDA. The following values were obtained from the Bridge Test System and are as follows: The Field Inductor DC Resistance is defined as: ohms The FET Rds ON resistance is given as: ohms The Field coil inductances is defined as: Henry The DC supply voltage is give as: volts The total circuit resistance is defined as: The circuit reactance component is defined as: The motor impedance is defined as: ohms The phase shift that would be used to align the motors' generated EMF, if a motor was used as the actual load and it was commutating, is defined as: . This phase shift would allow the generated EMF to be added to the applied EMF. From the initial analysis of polyphase systems we defined the voltage across the phase coils in a Delta configuration is 1/. Therefore the maximum voltage that is obtainable across the coils with is defined as volts. We may now define the motor voltage as . Since we are not using a motor in this test the term is zero. At this point we are now ready to apply the PWM part of the equation. previously stated that the speed of the motor is directly proportional to the voltage applied. Therefore applying a duty cycle to each point of the waveform generated a waveform the desired voltage. The duty cycle is defined as . Deriving the motor Current as a function of the Duty Cycle will allow us to control the motor speed as a digitally generated control function. Actually what we would like to do is compare the empirical data obtained from the Bridge Test System to the current theoretical analysis. Solving for the motor current as a function of the Dutycycle and setting the generated EMF to zero we get. The empirical data collected from the test system is shown below. The actual PWM rate used in the Bridge Test System to collect the empirical data is explained in the Bridge Test System Presentation. The waveform generated has 1000 steps per waveform point which yields a10 bit vertical resolution and is clocked at 50 ns per step. There are 260 points for 2 waveform. That computes to 1000*50 ns*260 = 260,000 bytes of memory and a period of 13 milliseconds. Since our application is for a six pole motor and requires three electrical periods for one mechanical revolution, then the RPM as previously defined is 60 seconds x 3 cycles per mechanical revolution for a 6 pole motor13 milliseconds period = 1384.6 RPM. The data and graphs below show the results of this test. Top

As we can see the theoretical and empirical are within a 5% window. Considering the parameters, The wire, sensors, the coil inductance, the loop resistance, etc. a 5% error window, is easily adjusted out. With a few adjustment within the 5% window we could easily obtain a 2% error margin. Test setup connections and extended length of cable on the load box could also account for some of the error. Also temperature rise of the FETs since we were only using a static heatsink with just the ambient air to extract the heat from the heatsink, therefore thermal equilibrium of the mass of the heatsink is also a contributing factor to the errors, although the measurements were taken and captured within a few seconds for each of the currents and dutycycles measured. In the next section we will discuss other circuit characteristics that will include the FET switching characteristics and the reactive components time constants and the determining of the actual PWM frequency. Since the phase angle varies with the motor speed as defined by , the phase current of the applied voltage also changes. This means that we have to define the current function with respect to a specific speed. This will allow us to define the motor states for any load condition as well. We will also cover the transfer of electrical energy to mechanical torque in another section of this analysis. Current and Voltage control loops with respect to controlling the load and torque of the motor will be presented at a later date as time permits. 
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