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Performance under variable-frequency VSI drive

The torque equations 6.28, 6.33 and 6.34 of the foregoing sections indicate the motor behaviour of the synchronous motor under variable voltage and variable frequency supply. The analyses below assumes that three-phase sinusoidal ac supply of contnuously variable rms output voltage and frequency is available from an inverter, as indicated in the block diagram of figure 6.17.

Nonsalient-pole synchronous motor driven with a VSI drive

Neglecting stator resistance R, the torque expression of 6.28 can be written as

6.34

where ω₁ = variable angular frequency, 2πf, (electrical) of the ac supply to the motor

V₁= rms ac supply voltage to the motor at frequency f1

X_s1= synchronous reactance, 2πf₁ L_s of the motor at the supply frequency.

Equation 6.34 implies that the developed torque is proportional to the applied rms voltage and its maximum value occurs for δ=90°. Since the mechanical speed is , Å_fω₁ and also, X_s1ω₁

This means that the same maximim torque is developed if the or ratio is maintained constant at all frequencies (speed). Note that this is very similar to the case of the induction motor. Near base speed, the motor back emf E_f and V₁ are much larger than the voltage drop in the stator resistance, so that the above conclusion is quite valid near the base speed (base frequency). At low frequency, this drop may not be negligible.

Typical T-ω characteristics with variable voltage and variable frequency are indicated in the figure 6.18.

–– Synchronous motor T–ω characteristic with constant V/f ratio

---- Induction motor T–ω characteristic at base frequency

····· Induction motor T–ω characteristic with constant V/f ratio characteristic at ½ base frequency

Equation 6.37 states that if V/f ratio is kept constant, the same torque is developed at a given δ and that the same maximum torque is developed at all speed. Note that this result is based on the assumption of negligible stator resistance.

1. Open Loop Control

From the foregoing analysis it follows that an open loop speed controller should consist of a V/f controller as indicated in figure 6.19. The input frequency f₁ of the supply frequency to the motor is selected according to the desired speed and the input voltage V₁ is made proportional to f₁. A sinussoidally modulated (SPWM) inverter may be used for this. The change of the speed reference must be such that the limiting δ(= 90°) is not be exceeded at any time. A filter in the speed reference input may be used for this purpose



Normally, V₁ is increased proportionately with f₁, until the base frequency (speed) is reached. Thereafter, V₁ is maintained constant at V₀ which is the rated voltage of the motor. For operation above base speed, the frequency f₁ only is increased. This, However, weakens the stator flux linkage, according to equation 6.38.

2. Operation with limited V and δ.

When a shaft mounted position sensor is available, the rotor flux linkage phasor of each phase winding can be reconstructed in phase and amplitude. (There are also other ways measuring the angle δ.). The maximum angle of the stator voltages V_1a, V_lb, and V_1c(i.e., δ₁), with respect to the rotor flux linkage phasors for each phase can then be restricted to predermined values. The maximum developed torque is then given by

6.39

The mechanical speed for any input frequency f₁ is given by rad/sec. The maximum developed torque is determined by the limiting δ angle. If the supply voltage V₁ (rms value) is also kept constant as is the case with operation above base speed, the motor will then develop a constant maximum-power characteristic given by

6.40



3. Variable-frequency performance at low frequency

Refer to the phasor diagram of figure below in which R is not leglected.



From the phasor diagram

6.41

6.42

Solving for I_d and I_q from 6.41 and 6.42

6.43

6.44

The developed power is given by

Watts/phase

Watts/phase

Watts/phase 6.45

The total developed torque for all three phases is given by

Nm

Nm 6.46

We now express the input frequency f₁ in terms of the base frequency f₀, as



so that E_f1 = λE_f0 =λe₀V₀, where , is the per unit excitation voltage at the base frequency (speed). The phase reactance at frequency f₁ is given by



where and X_s0 is the synchronous reactance of the motor at the base frequency (speed).

From 6.46, the developed torque is given by



Nm 6.47

Nm 6.48

By differentiating the numerator of 6.48 with respect to δ and equating to zero, it can be shown that maximum torque will occur for a δ_max given by

6.49

By using this δ_max for a given λ from equation 6.49, the maximum torque the motor will develop at any speed can be found from equation 6.48. Even with a constant V-f drive, the maximum torque the motor will develop falls at low speed because of the voltage drop in the stator resistance. The T-ω characteristic of the motor in figure 6.22 clearly indicates the drop in the maximum torque the motor will develop at low-speed when the stator resistance is not negligible.



Note that near the base frequency, λX_SO >>R and R≈0, so that the torque expression of 6.48 becomes the same as in 6.34.

At low frequencies, where λ≈0, X_s1 and E_fl are both negligible. From 6.41 and 6.42

6.50

6.51

so that

Equation 6.52 implies that the zero-frequency voltage boost should be equal to

6.53

From 6.48, the developed torque at low frequency is given by

6.54

Consequently, at low speed, T_max will occur when δ= 0°

The zero-speed voltage boost (= R x l_rated) the motor develops full rated torque at zero speed. The boost voltage has to be gradually reduced to zero so that at base speed, no voltage boost is applied.

As before, for any input frequency given by f₁ = λf₀,

; ; ;

An open-loop speed controller will be similar to the block diagram indicated in figure 6.19. If on the other hand, the load angle is limited to some arbitrary value δ₁ and the supply voltage is also kept constant, say at the rated value V₀, the torque equation 6.57 may then be expressed as

Where

Equation 6.58 represents a constant power like T–ω characteristics similar to those of section 6.4.1.2 (figure 6.20).

1. Low frequency performance of the salient pole motor.

   At low frequency, the voltage drop in the stator resistance may not be negligible. Refer to the phasor diagram of the salient pole motor with R≠0, as drawn in the figure below.

   

   From the phasor diagram of figure 6.25

   6.61

   6.62

   Solving for I_d and I_q from 6.61 and 6.62

   6.63

   6.64

   The developed power is given by

   Watts/phase

   Watts/phase

   6.65

   Note at low speed, where λ≈0, since X_d, X_q, and E_f are small. These

   results are expected and imply that the zero-frequency voltage boost should be

   6.66

   The stator voltage VS frequency should thus be adjusted at low frequency as indicated by equation 6.66. This is also the case for the non salient-pole motor, as indicated by equation 6.53.

   The developed torque from 6.65 is

   Nm

   6.67

   For arbitrary supply frequency f_l=λf₀, and input voltage V₁

   Nm 6.68

   From equation 6.68, it may be seen that the maximum torque the motor can develop falls as the input frequrency (f₁) or speed decreases, if the stater resistance R is not negligible. As before, the condition (δ_max) for the maximum developed torque at any frequency (speed) can be found by differentiating 6.68 and then to zero. T_max is then found by using the δ_max in equation 6.68. The expression for δ_max will not be attempted here for the sake of brevity. The T-ω characteristics for constant a V-f drive for various frequencies (speed) will be similar in shape as indicated in figure 6.26.

   The maximum developed torque capability of the motor can be maintained by using voltage boost at low frequency, tapering it to zero in some fashion near base speed, as indicated in the block diagram of figure 6.23.

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