Blocks of forward path

Figure 6.3 demonstrates the main blocks that were used to realize the forward path in the given LSRPE model. Their title definitions are as highlighted in table 6.1. They are briefly illustrated as given below:

Setting speed Block

This block, which is labelled by number 1 in figures 6.2 and 6.3, includes a modelling facility to set the desired mechanical reference speed, ωm,in rpm. It also

includes simple mathematical expressions for conversion the mechanical speed from rpm to rad/sec and electrical speed, ωe, in rpm. Therefore, it has three outputs to

represent these three speed forms.

𝜔𝑚(𝑟𝑎𝑑/𝑠𝑒𝑐) =₃₀𝜋 ∙ 𝜔𝑚 (𝑟𝑝𝑚) (6. 1)

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Block of reference torque quadrature current

This block, which is labelled by number 2 in figures 6.2 and 6.3, is to extract the quadrature current, iq , which could adjust the output torque. The block was designed to

include two main parts. The first part is to compare the motor actual speed, ωm_actual with

the setting reference speed, ωm_ref, to produce a speed error signal, espeed:

𝑒_{𝑠𝑝𝑒𝑒𝑑} = 𝜔_{𝑚_𝑟𝑒𝑓}− 𝜔_{𝑚_𝑎𝑐𝑡𝑢𝑎𝑙} (6. 3) This error was passed through a proportional-integral speed controller to convert the error into torque component, from which an error quadrature current component, iqe, was

extracted. The electromagnetic torque which always produces by the SM-PMSM is given by [183] as follow:

𝑇𝑒 = 𝑘 ∙ 𝑖𝑞 , 𝑤ℎ𝑒𝑟𝑒 𝑘 =3𝑝₂ 𝜆𝑚 (6. 4)

The goal of the proportional-integral speed controller is to produce a reference torque component Tref according to the following equation:

𝑇𝑟𝑒𝑓= 𝐾𝑝∙ 𝑒𝑠𝑝𝑒𝑒𝑑 + 𝐾𝑖 ∙ ∫ 𝑒₀𝑡 𝑠𝑝𝑒𝑒𝑑𝑑𝑡 (6. 5)

Combining (6.4) and (6.5) yields iqe:

𝑖_𝑞𝑒 = 𝑇𝑟𝑒𝑓

𝑘 =

𝐾𝑝∙𝑒𝑠𝑝𝑒𝑒𝑑+ 𝐾𝑖 ∙∫ 𝑒0𝑡 𝑠𝑝𝑒𝑒𝑑𝑑𝑡

𝑘 (6. 6)

In the underlying LSRPE model, the PI parameters KP and Ki, were adjusted to reach

the acceptable output responses from the motor model. The adjusting procedure of the PI parameters was based on a manner similar to that adopted by bi-section method in numerical analysis [186].

The second part is to obtain a reference quadrature current iq_ref from two current

components. They are the error quadrature current and the actual quadrature component motor current iq_motor. Thereby, the iq_ref was obtained through comparing these two

current components and passing the yielded error signal through a current proportional- integral controller.

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Reference current Block

This block, which is labelled by number 3 in figures 6.2 and 6.3, is to obtain a reference for the three-phase motor currents, iabc_ref. To conduct this, a reference direct

current component, id_ref, was obtained from combining the high frequency injected

signal with the direct current component which was assumed to be zero. At this point, there are two options to continue. First is by monitoring the current variations. Therefore, the iq_ref and id-ref should be converted from the rotary reference frame to the three

variables machine frame Iabc_ref by an embedded user defined function based on recalling

the following matrix from chapter 2:

[ 𝐼_𝑎 𝐼𝑏 𝐼_𝑐] = [ cos (𝜃) sin (𝜃) cos (𝜃 − 120) sin (𝜃 − 120) cos (𝜃 + 120) sin (𝜃 + 120)] [ 𝐼𝑑 𝐼_𝑞] (6. 7)

The second is by monitoring the voltage variations. Therefore, the iq_ref and id-ref should

be converted into the corresponding voltages Vq_ref and Vd_ref. By this option the control

modelling scheme should continue as illustrated in figure 6.1. In this work, the first option was adopted just to cover all the possible controlling implementations.

As it is clear from the conversion matrix in equation (6.7), rotor position information is an essential term to achieve the task. Therefore, the estimated position angle, in the feedback path, was fed through this block input which is labelled by “theta”.

Power management Block

This block, which is labelled by number 4 in figures 6.2 and 6.3, was constructed by two main blocks, a model for sine pulse width modulation SPWM and a model for power inverter. Two inputs exist in this block, the first is for the actual motor currents, Iabc, and the second is the reference currents, Iabc_ref. The corresponding currents in the

two inputs were compared with each other. The differences were used to produce the pulse width modulation signals. The last were exploited to drive the inverter, which accordingly provided their phase sinusoidal voltages to the next block, motor model. Figure 6.9 illustrates the current waveforms at the input terminals of this block and the resultant PWM signals.

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PM motor model

This motor model is represented by block number 5 in figures 6.2 and 6.3. Basically, the motor model, which was designated in chapter 4, was employed in this modelling scheme. Some modifications and additions were made to meet the requirements of the LSRPE modelling scheme.

Applied torque Block

This block is labelled by number 12 in figures 6.2 and 6.3, To verify the capability of the proposed model to deal safely with the load variations, three levels for the applied torque were chosen, Three step function models were embedded to generate the required step levels at different specified times. These levels simulated three torque values 1, -3 and 2 Nm as shown in figure 6.4 which are labelled by T1, T2, and T3 respectively. Then, the applied torque was obtained from the summation of torque levels T1, T2 and T3. This applied torque, which has transitions 0-1-3-0, is represented in figure 6.4 by label (Σ). The torque levels T1, T2 and T3 were satisfied through three step function models whose parameters were set to verify the amplitude and timing of torque application.

135 Fig. 6. 4, Variation of the applied torque load in the LSRPE modelling system