Electrical Machines

Induction motor torque-speed characteristic

The torque-speed curve of a three-phase induction motor starts at zero at synchronous speed, rises to a peak called the breakdown (maximum) torque, and then falls to the starting torque at standstill. If the stator resistance and magnetizing branch are neglected, the slip at maximum torque is s_m = R_2'/X_2', which is the same as R_2'/s_m = X_2'. The peak torque does not depend on rotor resistance, but the speed at which it occurs does. Starting torque is maximum when the total rotor resistance referred to the stator equals the standstill reactance, so the extra resistance to insert is the required total resistance minus the existing rotor resistance. Below the peak, the torque is nearly proportional to slip, T \propto sV^2/f. For a constant-torque load at fixed voltage and frequency, R_2'/s stays constant, so adding series resistance R_{ext} raises the slip by the ratio of the total to the original rotor resistance. Under constant V/f control, the flux stays constant. The maximum starting torque is obtained by choosing a frequency at which the rotor reactance equals the rotor resistance, which makes s_m = 1. A double squirrel cage has a high-resistance outer cage and a low-resistance inner cage with higher leakage reactance. At starting the rotor frequency equals the supply frequency, so the high reactance of the inner cage forces most of the current into the outer cage. At rated speed the rotor frequency is small, and the low-resistance inner cage carries most of the current. On a speed-torque plot, the accelerating torque is the developed torque minus the load torque, so a path that keeps the developed torque high and above the load line reaches the final speed in the shortest time.

Electrical machine losses and efficiency

Efficiency is the ratio of output power to input power, and the losses are the difference between the two. For an induction motor, the stator copper and core losses are subtracted from the input to obtain the air-gap power P_g. The mechanical power developed is (1-s)P_g, and the rotor copper loss is sP_g. After friction and windage losses are subtracted, the shaft output is found, and dividing by the input gives the efficiency. If the stator copper, core and rotational losses are all neglected, the efficiency reduces to 1-s, which depends only on slip. Iron loss in a transformer or machine is the sum of hysteresis loss and eddy current loss. Hysteresis loss follows the Steinmetz relation P_h \propto fB_m^{1.6}, and eddy current loss follows P_e \propto f^2B_m^2. Since B_m \propto V/f, the two components change differently when voltage and frequency are both changed, so two measurements at different V and f can be used to separate them. When a transformer is used at a different voltage and frequency from its design values, the peak flux density changes as V/f, which changes the core loss. The copper loss depends only on the load current, so for the same current it does not change. For a DC shunt motor, the constant core and rotational losses can be found from the no-load input, and the best efficiency occurs when the variable copper loss equals the constant loss.

Transformer winding polarity and ratio

The polarity of a transformer winding is marked by a dot, which indicates the instantaneous direction of the induced voltage. Two windings with the dots at the same end are in phase, and this determines the sign of the induced voltage and the phase relation in three-phase connections. The turns ratio fixes the voltage ratio, and the EMF in each winding follows E = 4.44fN\Phi_m in rms form. In a three-phase bank, the phase shift between the primary and secondary line voltages is a multiple of 30^\circ, and the vector group is named by a clock number. Each hour on the clock corresponds to 30^\circ, and the secondary line voltage is taken as the reference at the hour hand position. A lag of 300^\circ, which is the same as a lead of 60^\circ, corresponds to the clock number 10, so the group is Dd10. For a star-delta transformer with a turns ratio of N_1:N_2=2:1, the ampere-turn balance gives N_1 I_1 = N_2 I_2 per phase. The delta line current equals \sqrt{3} times the delta phase current, so the magnitudes of the two line currents differ by a factor of 2\sqrt{3}, and the phase displacement is 30^\circ. A transformer with a complex, unit-magnitude ratio and a series admittance y_t referred to one side has equal diagonal terms of its nodal admittance matrix, Y_{11}=Y_{22}=y_t. Its off-diagonal terms are -y_t/a and -y_t/a^*, so they are conjugates of one another.

DC motor speed-torque characteristic

The back emf of a DC motor is E_b = V_t - I_aR_a, and at constant flux it is proportional to speed. In a separately excited or shunt motor, the flux is nearly constant, so the armature current is proportional to torque. If the load torque increases with speed, the armature current rises with speed, and the speed can be found by solving the voltage equation with the new total resistance. When the load torque is constant, the armature current stays the same for the same field. If the supply voltage falls, the back emf must fall by the same amount, and the speed falls in the same ratio as the back emf, provided the field current is unchanged. In a series motor with a linear magnetic circuit, the flux is proportional to the armature current, so the torque is proportional to I_a^2 and the back emf is proportional to N I_a. If the load torque varies as a power of speed, the new armature current is found by matching the torque, and the external series resistance follows from the voltage balance at the new speed.

Single-phase induction motor types

A single-phase induction motor has no starting torque on its own, so an auxiliary winding is used to create a rotating field during starting. Split-phase (resistance-start) motors use a high-resistance auxiliary winding that is disconnected after starting, and shaded-pole motors use shading rings to produce a weak rotating field. Capacitor-start motors place a capacitor in the auxiliary winding for starting and disconnect it at speed, which gives a higher starting torque than resistance start. A capacitor-start, capacitor-run motor keeps the capacitor in circuit during running. Its running capacitor makes the auxiliary winding current nearly in quadrature, which gives a more uniform rotating field and the best running power factor of these types. The rotor of a single-phase motor is subject to both a forward and a backward rotating field. For a motor running at slip s at frequency f, the forward field induces rotor currents at sf, and the backward field induces rotor currents at (2-s)f.

Induction motor slip and speed

Slip is defined as s = (N_s - N)/N_s, where the synchronous speed is N_s = 120f/P and N is the rotor speed. The frequency of the rotor currents is equal to the slip times the supply frequency, f_r = sf, so the slip can be found directly from the measured rotor frequency. The mechanical speed follows from N = N_s(1-s). Because the synchronous speed depends on the number of poles, the same slip gives different speeds for motors with different pole counts. The rotor frequency of 1 Hz in a motor supplied at 40 Hz gives slip 0.025, and the rotor speed then follows from the synchronous speed. A 10-pole motor at 50 Hz has N_s = 600 rpm, so a running speed of 540 rpm corresponds to a slip of 0.1. Such a motor has a forward rotor frequency of sf and a backward rotor frequency of (2-s)f.