Blocks of feedback path

Figure 6.5 demonstrates the main blocks that were used to realize the feedback path in the given LSRPE model. Their numbering definitions are as highlighted in table 6.1.

136 These feedback blocks are briefly illustrated as given below:

Block of magnetic saliency detection

This block, which is labelled by number 6 in figures 6.2 and 6.5, was designated to input the motor stator currents, iabc, and to use the mathematical relation of converting

the stator variables into the two dimensions stationary reference frame 𝑖_𝛼 and 𝑖_𝛽. Thereby, the iabc currents were converted into the αβ- reference frame by employing the

following matrix transferring function:

[ 𝑖𝛼 𝑖_𝛽 𝑖_𝑜] = 2 3 [ 1 −12 −1 2 0 √3₂ −√3₂ 1 2 1 2 1 2 ] ∙ [ 𝑖_𝑎 𝑖𝑏 𝑖_𝑐] (6. 8)

As the key function of this block is to track the machine magnetic saliency, so one of these two component was band-pass filtered, iβ, and processed through a heterodyning

process to extract the envelop of the current component, iα, whose envelope variation

determined the amount of machine salience. The result for this saliency detection is shown in figure 6.11.

Unity peak normalization Block

This block is labelled by number 7 in figures 6.2 and 6.5. Its main function is to normalize the value of input current so as to provide an output oscillated between +1A and -1A. Figure 6.11, in the modelling results section, 6.4, demonstrates the required function, to be conducted by this block, through showing its input and output signals. This block function was achieved through three mathematical operations. Firstly, the input was offset to remove the dc component and thereby the input waveform sinusoidally vibrated between positive and negative parts. Secondly, the peak of each part was individually detected, as it will be illustrated in the next paragraph, because it was found that the saliency effect does not yield a pure sinusoidal waveform, |𝐼𝑝+| ≠

|𝐼𝑝−|. Eventually, the positive and negative parts of the saliency effect impact were

divided by the positive and negative peaks respectively. Figure 6.6 highlights this proposed concept.

137 The first input, in this block, is to input the sinusoidal waveform of the saliency detection which has to be normalized. While, the second input reads the motor setting speed, ωm. The frequency value (f) of the second input signal and the speed value (ωm)

of the first input relate to each other by the fundamental equation of the synchronous motors, which is given by:

𝜔𝑚 = 60 ∗𝑓_𝑝 (6. 9)

Therefore, this block uses the first input value to determine the expected frequency value of the saliency signal at first input. From which, the block could calculate the time- value of one fourth of the full wave period (T/4) of the sinusoidal saliency waveform. Fundamentally, the (T/4) value is corresponding to the peak value of any sinusoidal waveform. Thereby, the block detects the peak current (IP) of the first input through

employing this (T/4) time-value.

Figure 6.6 presents an illustrative a block diagram for the concept of implementation for this block function.

Fig. 6. 6, Structure of the unity peak normalization block

138

Position estimation Block, θe

This block, which is labelled by number 8 in figures 6.2 and 6.5, has a single input by which it receipts the output of the unity peak normalized block, and a single output, which represents the estimated value of the rotor position at low speed. This block achieved its task by simply applying the inverse trigonometric sine function, arcsine, for the input signal. As the range of the MATLAB function (asin) is -90o to 90o, so the rotor positions in the second and third quarters will not be detected. Therefore, a certain algorithm was designated and embedded in this block to overcome this obstacle. The proposed algorithm was based on detecting the positive and negative slope ranges of the input signal, from which it had to determine the right quarter of the rotor position.

Block of speed estimation, ωe

This block, which is labelled by number 9 in figures 6.2 and 6.5. The basic method to determine the mechanical speed is by dividing the difference between two cascaded angular rotor positions by the time interval between the readings of these positions [187]. This is mathematically expressed by:

𝜔

=

∆𝑡 (6. 10) A main drawback involved by applying this method is a noticeable vibration that appears on the speed curve. To tackle this problem, a smooth speed is obtained by adopting a process of low pass filtering for the instantaneous changes in rotor position. The transfer function F(s) of this filter has been determined by reference [187] as given below:

𝐹(𝑠) =

𝜏∗𝑠+1 (6. 11) .where τ is an adjustable interval which is chosen by try and error. Anyhow, reference [187] has mentioned that this interval is a trade-off between the introduced lag and the noise of filtering process.

139 In this work, the proposed speed estimator was based on equation (6.10) with an addition of a smoothing sub-block to reduce the drawback of speed vibration. Figure 6.7 demonstrates a block diagram for the speed estimation block.

Fig. 6. 7, Block diagram for speed estimation

Position recovery Block

This block, which is labelled by number 11 in figures 6.2 and 6.5, is continuously observes the change in rotor position estimation. The general trend for position estimation is an increment in the value of rotor position detections. Therefore, this block was designated to detect and recover any drop in the present reading for position estimation comparing with the previous reading. If this condition occurs, the recovery block ignores the present false reading and estimates the existing position by incrementing the previous position by one degree.

Block of Combining the initial and low speed positions

At start-up period, the model of low-speed rotor position estimation was designated to read the initial rotor position from the zero speed estimator. The start-up period was roughly estimated to be 0.05 second. Figure 6.8 is a block diagram to illustrate the concept of this block, which is labelled by number 10 in figures 6.2 and 6.5.

140 Fig. 6. 8, Block of combining the zero and low speeds rotor positions