Deducing the Machine Performance - Electrical Machines - Electrical Engineering

Deducing the machine performance

The per-phase equivalent circuit of an induction machine allows us to deduce many steady-state performance quantities. Important among these are the torque-speed (or torque-slip) characteristic, the starting torque, the maximum (stalled) torque and the operating point when the machine is coupled to a load. In the approximate per-phase equivalent circuit the rotor branch is represented by the resistance R_r′ in series with the reactance referred to the stator and with the effective resistance scaled by slip, R_r′/s. The total power transferred across the air-gap per phase is the power consumed in the rotor branch R_r′/s. Of this, the portion dissipated in R_r′ is the rotor copper loss and the remainder is the mechanical power developed by the machine. Neglecting mechanical losses, the mechanical power available at the shaft per phase is therefore the power in R_r′(1 - s)/s. The electromagnetic torque is obtained by dividing the developed mechanical power by the mechanical angular speed of the shaft.

The complete torque-speed characteristic

To obtain an expression for the torque as a function of slip, consider the per-phase approximate equivalent circuit excited by a sinusoidal phase voltage. If the magnetising current is neglected and the Thevenin simplification about the rotor terminals is not invoked, the stator current phasor I_s is given by the applied phase voltage phasor divided by the total per-phase impedance of the circuit:

where V_s is the phase voltage phasor and I_s is the current phasor. The current flowing in the rotor branch produces the power transferred across the air gap; since this current flows through the resistive element R_r′/s, the air-gap power per phase is given by 

The mechanical power developed per phase, neglecting mechanical losses, was shown above to be the portion of the air-gap power corresponding to R_r′(1 - s)/s. The electromagnetic torque per phase is the developed mechanical power divided by the rotor mechanical speed ω_m. Using ω_m = (1 - s)ω_s, where ω_s is the synchronous angular speed, we obtain the torque per phase as

Multiplying by the number of phases gives the total electromagnetic torque:

This expression (Eqn. 17) is written for a two-pole (one pole-pair) machine; for a machine with p pole-pairs the expression must be multiplied by p. The torque obtained by evaluating this expression as s varies is the torque-slip (or torque-speed) characteristic - one of the most important characteristics of the induction machine.

A typical torque-speed characteristic for a 3 kW, 4-pole, 60 Hz machine is shown in Figure 22. The rated operating speed cited there is 1780 rpm. The derivation above used the approximate equivalent circuit; using the exact equivalent circuit changes the detail of the curve slightly. Readers with access to MATLAB, Octave or Scilab may compare approximate and exact predictions numerically; such a comparison for a 3 kW, 4-pole, 50 Hz machine (rated speed 1440 rpm) is shown in Figure 23. The approximate equivalent circuit gives a good match in the normal operating speed range, but differences appear near starting and under heavy slip conditions. The slope and shape of the characteristic depend intimately on the machine parameters (resistances and leakage reactances) and on the terminal voltage waveform being sinusoidal.

Figure 22: Induction machine speed-torque characteristic

The torque-slip curve is obtained by varying slip while holding the applied phase voltage constant and waiting until steady state (all transients damped out). Thus experimental determination requires measuring steady torque at given speeds; one cannot obtain the steady torque-speed curve by simply starting the motor from rest with full voltage and recording instantaneous values while the machine accelerates.

Figure 23: Comparison of exact and approximate circuit predictions and measuring the torque and speed dynamically as it runs up to steady speed.

A complete torque-speed characteristic spans negative and positive slip values. Positive slip (0 < s ≤ 1) corresponds to motoring operation where the rotor turns in the same direction as the stator rotating field and torque is positive. negative slip corresponds to the machine delivering mechanical power back to the supply (generation), and slip values greater than unity (s /> 1) correspond to braking or counter-rotation conditions. A schematic complete characteristic for a four pole machine with synchronous speed 1500 rpm is shown in Figure 24. In that plot negative speeds correspond to slip values greater than unity and speeds greater than synchronous correspond to negative slip.

Figure 24: Complete speed-torque characteristic

The torque curve in the motoring region (0 ≤ s ≤ 1) has a single peak - the maximum or stalled torque. If the load torque requirement exceeds this maximum torque the machine will stall. The slip at which maximum torque occurs is commonly denoted s_max. For the example shown in Figure 24 the maximum occurs at s ≈

(value 0.38 for that machine). For slips smaller than s_max the torque drops steeply and reaches zero at synchronous speed (s = 0). For slips larger than s_max the torque falls more slowly towards the value at s = 1, which is the starting torque.

The slip corresponding to maximum torque may be found by differentiating the torque expression with respect to slip and setting the derivative to zero. Carrying out that standard analysis yields the condition for maximum torque

and substitution of this s into the torque expression gives the value of the maximum (stalling) torque:

The algebraic steps that lead to (18) and (19) are standard and follow from differentiating the torque expression and solving for s. The result shows two useful properties: the maximum torque is independent of R_r′ while the slip at which it occurs is proportional to R_r′ (i.e., s_max ∝ R_r′). This observation is exploited when rotor resistance can be changed (for example, in slip-ring machines) to alter the starting behaviour without changing the maximum torque. In particular, if R_r′ is chosen equal to

then s_max becomes unity and the maximum torque occurs at starting (s = 1) - a useful adjustment when high starting torque is required.

For negative slip (generator action) the magnitude of the maximum torque may be higher than in the motoring region depending on the circuit parameters; the sign of torque simply reverses sign with slip.

Operating point

The steady operating speed of an induction machine connected to a given load is the point at which the machine torque-speed characteristic and the load torque characteristic intersect. Figure 25 shows a motor torque characteristic and a constant torque load characteristic superimposed. The intersection points are possible steady operating points.

Figure 25: Machine and load characteristics

Whether an intersection is a physically attainable operating point depends on the local stability of that point. In practice both machine and load characteristics vary slightly with time and operating conditions; small perturbations therefore reveal the stability. Consider a point 1 where the two curves meet. If the load torque increases slightly, the intersection shifts to 1′. At 1′ the developed torque is insufficient to balance the increased load so there is a positive torque difference ΔT_e that accelerates the machine. As the speed increases the torque produced by the motor may increase further (depending on the local slope of the motor characteristic), producing a runaway and moving the operating point away from 1. Thus point 1 is an unstable operating point.

Figure 26: Stability of operating point

Now consider point 2. If the operating point is displaced slightly to 2′ the motor torque minus the load torque will drive the speed back towards point 2. Small disturbances therefore decay and the system returns to 2; thus point 2 is a stable operating point. By inspecting the torque-speed curve one finds that the region from s = 0 up to s = is unstable while the region from s =

down to s = 0 is stable. Consequently a practical induction machine will settle to an operating slip within the stable region (i.e., between s = 0 and s = for the case shown).

Modes of operation

The torque-slip curve partitions operation into distinct modes according to the value of slip.

Motoring region (0 < s ≤ /> When the stator is supplied with balanced three-phase voltage and the rotor is initially at rest, slip s = 1. The rotating magnetic field set up by the stator induces currents in the rotor conductors. Interaction of rotor currents with the rotating field produces a torque in the direction of the field rotation; the rotor accelerates and the slip reduces toward the steady operating value. The motoring region is where the machine consumes electrical power and delivers mechanical power to a passive load. The rotor rotates in the same direction as the stator field.

Braking / counter-rotation region (s > 1): If the rotor is forced to rotate opposite to the stator rotating field (for instance if the phase sequence of supply is reversed while the rotor is still turning), the slip exceeds unity and the stator field tends to oppose the rotor motion. This region produces a retarding torque and is used for braking. If the supply is not removed when the rotor speed passes through zero, the machine will pass into the motoring region with reversed direction of rotation.

Generating region (s < 0): If an external prime mover drives the rotor faster than synchronous speed (for example, a wind turbine which is mechanically driving the induction machine shaft), slip becomes negative. In this case the machine develops a negative electromagnetic torque and returns active power to the supply - the machine operates as an induction generator. The generated electrical power equals the mechanical power delivered to the shaft minus losses. Modern wind-energy systems commonly use induction machines (often with power electronic converters or rotor-resistance control) to exploit this generating mode. Equation (17) indicates torque changes sign with slip so that torque is negative for negative values of slip.

Summary remarks

The torque-slip characteristic derived from the equivalent circuit summarises the steady-state electromechanical behaviour of the induction machine. The location and magnitude of the starting torque, maximum (stalled) torque and the stable operating region are determined by the machine electrical parameters and the applied voltage. Adjusting rotor resistance (in slip-ring machines) can change the slip at which maximum torque occurs and can therefore improve starting behaviour without changing the maximum torque magnitude. For experimental or numerical study, the characteristic must be obtained under steady-state sinusoidal excitation and with transients allowed to decay; comparisons between the approximate and exact equivalent circuits are instructive and show that the approximate circuit is satisfactory over the normal operating range.