B- H loop and demagnetization characteristics The starting-point for understanding magnet
3. Induction machines
3.2 E QUIVALENT CIRCUIT
The induction motor rotor is similar to the secondary of a transformer, with two major differences: (a) the rotor is rotating relative to the primary, so the frequency in the secondary is s f ; and (b) the airgap decreases the coupling between the primary and secondary, so that the leakage
inductances are higher than in a typical power transformer. Consequently the magnetizing reactance is lower and the magnetizing current is higher.
In all other respects the equivalent circuit is the same as for the transformer, shown in Fig. 3.4 for one phase. The rotor frequency s f is usually in the range 0.5!5 Hz in normal operation.
To simplify analysis we refer all parameters to the primary (stator) winding. To do this we need the transformation ratios for the voltage, current, and impedance. By Faraday's Law, the sinusoidal induced voltage in each winding is proportional to the turns and the frequency. In a transformer the voltage ratio is n1f/n2f = n1/n2, the turns ratio, where n1 and n2 are the effective numbers of turns in
the stator and rotor windings, respectively: the frequency cancels because it is the same in both windings. In the induction motor the voltage ratio is n1 f/n2 sf = n1/sn2.
The current ratio is n2/n1, exactly as in the transformer : current balance is a magnetostatic effect
controlled by Ampere's Law, and is unaffected by frequency.
Pgap ' 3 I2
2R 2
s (3.6)
Pmech ' Pgap(1 & s). (3.7)
Tgap ' Pmech (1 ! s)T '
p
TPgap [Nm]. (3.8)
Tshaft ' T & Tfric. (3.9)
Fig. 3.5 Equivalent circuit of one phase of polyphase induction motor. The frequency is the line frequency in both primary and secondary, and the turns ratio is eliminated. The EMF Erb is used to represent leakage flux in saturated rotor slot- bridges, and does not contribute to torque production. Boxed variables appear in the phasor diagram, Fig. 3.9.
R2
s ' R2 % R2
1
s ! 1 . (3.5)
It is now possible to refer all voltages, currents, and impedances in Fig. 3.4 to the stator winding, as in Fig. 3.5. All voltages and currents in Fig. 3.5 are at line frequency f, and all reactances have their line- frequency values. While the secondary EMF and reactance lose their dependence on slip, the rotor resistance becomes R2/s, and this is further divided into two components in Fig. 3.5, using
The first term represents I2R loss, while the second term represents electromechanical power
conversion. The referred equivalent circuit (Fig. 3.5 ) shows that any increase in rotor current appears as an increase in the stator current I1 drawn from the supply. It can be used to calculate all the
important currents and voltages in the induction motor, as well as the torque, power, and efficiency.
Power, torque and efficiency
The current I2 flows in the referred rotor resistance R2/s and causes an apparent power dissipation of
where the 3 accounts for all three phases. This is the electromechanical power or airgap power, i.e. the power transferred across the airgap from the stator to the rotor. The actual copper loss in the rotor, however, is only 3I22R2, and since s < 1 (in motoring operation), the airgap power exceeds the rotor
copper loss by 3I22R2/s ! 3I22R2 = Pgap (1 ! s). The electromechanical power is therefore
Since the actual rotor speed is (1 ! s) T/p rad/s, electromagnetic torque must be
The airgap power Pgap divides into two. A fraction s goes as rotor copper loss (heating the rotor). The
remaining fraction (1 ! s) is converted into mechanical power at a speed that is equal to (1 ! s) times the synchronous speed, producing the electromagnetic torque Tgap. The shaft torque Tshaft is less than the electromagnetic torque because of friction torque Tfric:
0 ' Tm × Tshaft 3V1I1 cos N1 × 100%, (3.10) I2 ' V1 (R1 % R2/s) % j XL (3.11) Tgap ' 3p T × R2 s × V12 (R1 % R2/s)2 % XL2 [Nm] . (3.12) T
6
3p T × V12 R2 × s, (3.13) T6
3p T × V12R 2 XL2 × 1 s, (3.14)The efficiency is now calculated as
where cos N1 is the power factor and Tm = Tr/p is the speed in mechanical radians/sec. Note that V1 is the phase voltage, since the equivalent circuit is on a per-phase basis. Likewise, I1 is the phase current.
If the motor is connected in wye and the phases are balanced, V1 = VL//3 and I1 = IL, where VL is the
line-line voltage and IL is the line current. If it is connected in delta, V1 = VL and I1 = IL//3.
Effect of slip on torque, efficiency and power factor
Induction motors are usually designed to operate with low values of slip, typically 0.01!0.05. This increases the fraction of airgap power that is converted to mechanical power, and decreases the fraction that is dissipated as rotor I2R loss, helping to maximise the efficiency. A small value of s
increases the ratio (R2/s)/X2, which also improves the power factor.
The speed during normal motoring operation is just below synchronous speed, and it changes only slightly as the load torque changes. For this reason the induction motor is often described as a "constant speed" machine, provided that the supply frequency is fixed. From the equivalent circuit (Fig. 3.5), if we neglect the no-load current Inl and lump the leakage reactances together as XL = X1 + X2,
If this is substituted into eqn. (3.6) for Pgap, then from eqn. (3.8) the airgap torque is given by
This equation relates the electromagnetic torque to the slip, when the supply voltage and frequency are fixed. It can therefore be used to plot a graph of torque vs. slip. Since the speed in rev/min is given by (1 ! s) Ns, the graph also shows the variation of speed with torque. Fig. 3.6 shows a typical example.
At speeds near synchronous speed, s 6 0 and
i.e., the torque is proportional to slip. Since the slip is generally small for torques which are less than or equal to the rated load torque, the speed remains near synchronous speed as the torque varies. The actual torque is determined by the load, and the operating point is at the intersection of the motor's torque/speed characteristic with the torque/speed characteristic of the load, Fig. 3.6.
If the speed rises above synchronous speed, the slip becomes negative and the torque changes sign. This is called regeneration or braking. If the induction machine is driven (for example, by a wind turbine) at speeds above synchronous speed, it generates and feeds power into the AC supply system.
At low speed, s 6 1 and XL tends to exceed (R1 + R2/s), so that
i.e., the torque is inversely proportional to slip at low speed. The locked-rotor torque is the value at standstill, with s =1. This is the torque when the motor is first switched on.
sTmax ' R2 R12 % X L2 (3.15) Tmax ' 1 2 × 3p T × V12 R1 % R12 % XL2 [Nm]. (3.16)
Fig. 3.6 Torque/speed characteristic
Breakdown torque
Fig. 3.6 shows that there is a maximum torque Tmax, occurring at a value of slip slightly beyond the linear section of the torque/slip curve. This value of slip can be estimated by differentiating the torque equation with respect to slip, and setting the derivative to zero. The result is
The corresponding maximum torque is called the breakdown torque:
If the load is gradually increased above the rated load, the slip increases and the motor produces more torque. The operating point moves up the torque/slip curve until the slip reaches sTmax. Any increase
of load torque then causes the motor to slow down still more, increasing the slip. Beyond sTmax the
motor torque decreases, and the motor rapidly decelerates and stalls; this is called breakdown. Induction motors are normally rated such that at rated voltage and rated frequency the rated torque is roughly half the breakdown torque. This provides a margin of safety against stalling due to transient changes of load torque, or undervoltage or underfrequency conditions. Since Tmax is proportional to
V12, a 10% reduction in voltage produces a 20% reduction in breakdown torque.
Phasor diagram
The phasor diagram (Fig. 3.9) represents the steady-state operation of the equivalent circuit of Fig. 3.5. The equivalent-circuit parameters are normally calculated for AC sinewave operation with constant supply voltage and balanced conditions. Iterative solution of the phasor diagram makes it possible to allow for nonlinearities such as saturation of the magnetizing reactance and leakage reactance, the effect of slip on the rotor bar resistance, and the variation of core loss and stray loss with the flux level. However, it should be remembered that the phasor analysis is based on the fundamental space- harmonic MMF and is limited to sinusoidal waveforms of voltage and current.
Fig. 3.7 Varying R2 to achieve high starting torque Fig. 3.8 Varying the supply frequency at constant V/Hz
Speed control
Fixed supply: If the supply voltage and frequency are fixed, the speed can be controlled by varying the rotor resistance by means of an external resistor connected into the rotor circuit by slip-rings and brushes. This technique is used in large wound-rotor induction motors, especially for controlling the rate at which they start up. The effect is to change the torque/speed characteristic as shown in Fig. 3.7. A high resistance R2 maximizes the starting torque. As the rotor accelerates the external resistance
is shorted out and the characteristic changes to a "low-slip, high-efficiency" characteristic. Wound-rotor machines are expensive and are relatively less common than cage-rotor machines.
Pole amplitude modulation: With a fixed supply, the speed of a cage-rotor machine can be changed by reconnecting the stator windings in such a way as to change the pole number. For example, reconnecting the stator from 6 poles to 8 poles reduces the synchronous speed by 25%. Otherwise there is no practical way to control the synchronous speed of cage-rotor machines on a fixed supply.
Variable voltage: Changing the voltage causes the torque to be scaled in proportion to V12, so the
operating point moves to a higher or lower speed as the slope of the torque/slip curve changes. The range of speed variation is small, unless the motor is designed with a high rotor resistance, but this makes the motor inefficient. This technique is used with single-phase and inexpensive triac controllers.
Variable frequency: The ideal way to control the speed of an induction motor is by varying the supply frequency. This causes the torque/slip curve to be translated along the speed axis, Fig. 3.8. If the voltage/frequency ratio is kept constant ("constant volts/Hz"), the breakdown torque remains constant over most of the speed range; at lower speeds it tends to fall as the stator resistance begins to become significant compared with the leakage reactance. With this type of drive the slip for a given torque can be held constant while the speed is varied (almost proportional to frequency).
Modern field-oriented drives are capable of extremely rapid torque response. In principle they operate by orienting the stator MMF distribution at an optimal angle relative to the flux trapped by the rotor currents, and under transient conditions they are not limited to sinusoidal current. However, the equivalent circuit model is still the basis of analysis and design of these drives.
Double-cage and deep-bar rotors: Some induction motors are designed with a double cage. The inner cage has a high leakage reactance and a low resistance, and the outer cage has a low reactance and a high resistance. The resulting torque/speed characteristic is similar to the sum of the high-resistance and low-resistance curves in Fig. 3.7, providing high starting torque and low operating slip (therefore high efficiency) in the one motor. As an alternative to the double cage, skin-effect is used in the deep-bar rotor to increase the rotor R/X ratio as the slip (and therefore the rotor frequency) increases.
1The classical references are Alger, Veinott, Richter and others. To understand these calculations it is virtually essential to
study the original references, but some idea of the procedures can be seen in §3.6 which describes the calculation of slot permeance. PC-IMD provides a range of optional methods for calculating the most important equivalent-circuit parameters, including most of the classical methods and a number of original ones.
V1 VR1 VX1 VZ1 VR2 VX2 VZ2 Erb E1 I1 I2 ER2 Irc Imag Inl φ
Fig. 3.9 Phasor diagram for one phase of a balanced polyphase induction motor. The phasor Erbrepresents rotor slot bridge leakage.