IDEAL NO-LOAD OPERATION

The ideal no-load operation mode corresponds to zero rotor current. From (7.11), for Ir0 = 0 we obtain

S0E2 VR = 0; S0 = VER2 (7.17)

The slip S0 for ideal no-load depends on the value and phase of the rotor applied voltage VR. For VR in phase with E2: S0 > 0 and, with them in opposite phase, S0 < 0.

The conventional ideal no – load-synchronism, for the short-circuited rotor (VR = 0) corresponds to S0 = 0, n0 = f1/p1. If the rotor windings (in a wound rotor) are supplied with a forward voltage sequence of adequate frequency f2 = Sf1 (f1 > 0, f2 > 0), subsynchronous operation (motoring and generating) may be obtained. If the rotor voltage sequence is changed, f2 = sf1 < 0 (f1 > 0), supersynchronous operation may be obtained. This is the case of the doubly fed induction machine. For the time being we will deal, however, with the conventional ideal no-load (conventional synchronism) for which S0 = 0. The equivalent circuit degenerates into the one in Figure 7.2a (rotor circuit is open).

Building the phasor diagram (Figure 7.2b) starts with Im, continues with jX1mIm, then I0a

Ioa = jXR1m1mIm (7.18)

and Is0 = Ioa + Im (7.19)

Finally, the stator phase voltage Vs (Figure 7.2b) is

Vs = jX1m Im + Rs Is0 + jXslIs0 (7.20)

The input (active) power Ps0 is dissipated into electromagnetic loss, fundamental and harmonics stator core losses, and stator windings and space harmonics caused rotor core and cage losses. The driving motor covers the mechanical losses and whatever losses would occur in the rotor core and squirrel cage due to space harmonics fields and hysteresis.

image

Figure 7.2 Ideal no-load operation (VR = 0):

a.) equivalent circuit b.) phasor diagram c.) power balance d.) test rig

For the time being, when doing the measurements, we will consider only stator core and winding losses.

Ps0 ≈ 3R1m 0aI 2 + 3R Is s02 = 3R1m X1mX2 1+mR2 1m2 + 3Rs Is02 = (7.21)

= piron + 3R Is s02

From d.c. measurements, we may determine the stator resistance Rs. Then, from (7.21), with Ps0, Is0 measured with a power analyzer, we may calculate the iron losses piron for given stator voltage Vs and frequency f1.

We may repeat the testing for variable f1 and V1 (or variable V1/f1) to determine the core loss dependence on frequency and voltage. The input reactive power is

Qs0 = 3X1m X1mR2 1+mR2 1m2 + 3Xsl Is02 (7.22)

From (7.21)-(7.22), with Rs known, Qs0, Is0, Ps0 measured, we may calculate only two out of the three unknowns (parameters): X1m, R1m, Xsl.

We know that R1m >> X1m >> Xsl. However, Xsl may be taken by the design value, or the value measured with the rotor out or half the stall rotor (S = 1) reactance Xsc, as shown later in this chapter.

Consequently, X1m and R1m may be found with good precision from the ideal no-load test (Figure 7.2d). Unfortunately, a synchronous motor with the same number of poles is needed to provide driving at synchronism. This is why the no-load motoring operation mode has become standard for industrial use, while the ideal no-load test is mainly used for prototyping.

Example 7.1 Ideal no-load parameters

An induction motor driven at synchronism (n = n1 = 1800rpm, f1 = 60Hz, p1 = 2) is fed at rated voltage V1 = 240 V (phase RMS) and draws a phase current Is0 = 3 A, the power analyzer measures Ps0 = 36 W, Qs0 = 700 VAR, the stator resistance Rs = 0.1 Ω, Xsl = 0.3 Ω. Let us calculate the core loss piron, X1m, R1m.

Solution

From (7.21), the core loss piron is

piron = Ps0 −3R Is s02 = 36 −3 0.1 3⋅ ⋅ 2 = 33.3W (7.23)

Now, from (7.21) and (7.22), we get

XR1m12m+X1Rm12m2 = Ps0 3I3s0R I2s s02 = 36 3 33 0.1 32 2 = 3327.3 =1.233Ω (7.24)

XR1m 2 +XR1m1m2 = Qs0 3I3s0X I2sl s02 = 7003 33 0.3 32 2 = 25.626Ω (7.25)

Dividing (7.25) by (7.26) we get

R1m = 25.626 = 20.78 (7.26)

X1m 1.233

From (7.25),

X1m2 = 25.626; (20.X781m)2 +1 = 25.626 ⇒ X1m = 25.685 (7.27)

 XR11mm  +1

R1m is calculated from (7.26),

R1m = X1m ⋅20.78 = 25.626 20⋅ .78 = 533.74Ω (7.28)

By doing experiments for various frequencies f1 (and Vs/f1 ratios), the slight dependence of R1m on frequency f1 and on the magnetization current Im may be proved.

As a bonus, the power factor cosϕoi may be obtained as

 −1 QPs0s0  = 0.05136 (7.29) cosϕ0i = costan 

The lower the power factor at ideal no-load, the lower the core loss in the machine (the winding losses are low in this case).

In general, when the machine is driven under load, the value of emf (E1 = X1mIm) does not vary notably up to rated load and thus the core loss found from ideal no-load testing may be used for assessing performance during loading, through the loss segregation procedure. Note however that, on load, additional losses, produced by space field harmonics,occur. For a precise efficiency computation, these “stray load losses” have to be added to the core loss measured under ideal no-load or even for no-load motoring operation.

SHORT-CIRCUIT (ZERO SPEED) OPERATION

At start, the IM speed is zero (S = 1), but the electromagnetic torque is positive (Table 7.1), so when three-phase fed, the IM tends to start (rotate); to prevent this, the rotor has to be stalled.

First, we adapt the equivalent circuit by letting S = 1 and Rr′ and Xsl, Xrl′ be replaced by their values as affected by skin effect and magnetic saturation

(mainly leakage saturation as shown in Chapter 6): Xslstart, R′rstart, X′rlstart (Figure 7.3).

image

image

 

Figure 7.3 Short-circuit (zero speed) operation:

a.) complete equivalent circuit at S = 1, b.) simplified equivalent circuit S = 1, c.) power balance, d.) phasor diagram, e.) three-phase zero speed testing, f.) single-phase supply zero speed testing

For standard frequencies of 50(60) Hz, and above, X1m >> R′rstart. Also, X1m >> X′rlstart, so it is acceptable to neglect it in the classical short-circuit equivalent circuit (Figure 7.3b).

For low frequencies, however, this is not so; that is, X1m <> R′rstart , so the complete equivalent circuit (Figure 7.3a) is mandatory.

The power balance and the phasor diagram (for the simplified circuit of Figure 7.3b) are shown in Figure 7.3c and d. The test rigs for three-phase and single-phase supply testing are presented in Figure 7.3e and f.

It is evident that for single-phase supply, there is no starting torque as we have a non-traveling stator mmf aligned to phase a. Consequently, no rotor stalling is required.

The equivalent impedance is now (3/2)Zsc because phase a is in series with phases b and c in parallel. The simplified equivalent circuit (Figure 7.3b) may be used consistently for supply frequencies above 50(60) Hz. For lower frequencies, the complete equivalent circuit has to be known. Still, the core loss may be neglected (R1m ≈ ∞) but, from ideal no-load test at various voltages, we have to have L1m(Im) function. A rather cumbersome iterative (optimization) procedure is then required to find R′rstart, X′rlstart, and Xslstart with only two equations from measurements of Rsc, Vsc/Isc.

Psc = 3R Is sc2 + 3R’rstart I’rsc2 (7.30)

Zsc = Rs + jXslstart + R’jXrlstart1m(R’+j Xrlstart( 1m++jX’X’rlstartrlstart)) (7.31)

This particular case, so typical with variable frequency IM supplies, will not be pursued further. For frequencies above 50(60) Hz the short-circuit impedance is simply

Zsc ≈ Rsc + jX ; Rsc sc = Rs + R’rlstart ; Xsc = Xslstart + X’rlstart (7.32) and with Psc, Vssc, Isc measured, we may Rsc = 3PIscsc2 ; Xsc =(7.33)

for three phase zero speed testing and

Rsc = 23 IPsc~sc~2 ; Xsc = 23 (7.34)

for single phase zero speed testing.

If the test is done at rated voltage, the starting current Istart (Isc)Vsn is much larger than the rated current,

Istart ≈ 4.5 ÷8.0

In

for cage rotors, larger for high efficiency motors, and

(7.35)

Istart ≈10 ÷12

(7.36)

In

for short-circuited rotor windings.

The starting torque Tes is:

Tes = 3R’rstartω1Istart p1

with

(7.37)

Tes = (0.7 ÷ 2.3)Ten for cage rotors and

(7.38)

Tes = (0.1÷0.3)Ten for short-circuited wound rotors.

(7.39)

 

image

Figure 7.4 Stator voltage versus short-circuit current

Thorough testing of IM at zero speed generally completed up to rated current. Consequently, lower voltages are required, thus avoiding machine overheating. A quick test at rated voltage is done on prototypes or random IMs from, production line to measure the peak starting current and the starting torque. This is a must for heavy starting applications (nuclear power plant cooling pump-motors, for example) as both skin effect and leakage saturation notably modify the motor starting parameters: Xslstart, X′rlstart, R′rstart.

Also, closed slot rotors have their slot leakage flux path saturated at rotor currents notably less than rated current, so a low voltage test at zero speed should produce results as in Figure 7.4.

Intercepting the Isc/Vsc curve with abscissa, we obtain, for the closed slot rotor, a non-zero emf Es.[1] Es is in the order of 6 to 12V for 220 V phase RMS, 50(60) Hz motors. This additional emf is sometimes introduced in the equivalent circuit together with a constant rotor leakage inductance to account for rotor slot−bridge saturation. Es is 900 ahead of rotor current Ir′ and is equal to

Es 2f 2W1 1 K ; Bsbridge hor Le (7.40)

Bsbridge is the saturation flux density in the rotor slot bridges (Bsbridge = (2 – 2.2)T). The bridge height is hor = 0.3 to 1 mm depending on rotor peripheral speed. The smaller, the better.

A more complete investigation of combined skin and saturation effects on leakage inductances is to be found in Chapter 9 for both semiclosed and closed rotor slots.

Example 7.2 Parameters from zero speed testing

An induction motor with a cage semiclosed slot rotor has been tested at zero speed for Vssc = 30 V (phase RMS, 60 Hz). The input power and phase current are: Psc = 810 kW, Isc = 30 A. The a.c. stator resistance Rs = 0.1Ω. The rotor resistance, without skin effect, is good for running conditions, Rr = 0.1Ω. Let us determine the short-circuit (and rotor) resistance and leakage reactance at zero speed, and the start-to-load rotor resistance ratio due to skin effect.

Solution

From (7.33) we may determine directly the values of short-circuit resistance and reactance, Rsc and Xsc,

Rsc = 3PIscsc2 = 3810⋅302 = 0.3Ω;

2 2 (7.41)

Xsc = VIscssc  − Rsc2 =  3030 − 0.32 = 0.954Ω

The rotor resistance at start R′rstart is

 

R’rstart = Rsc − Rs = 0.3− 0.1= 0.2Ω

(7.42)

So, the rotor resistance at start is two times that of full load conditions.

R’rstart = 0.2 = 2.0 (7.43)

R’r 0.1

The skin effect is responsible for this increase.

Separating the two leakage reactances Xslstart and X′rlstart from Xsc is hardly possible. In general, Xslstart is affected at start by leakage saturation, if the stator slots are not open, while X′rlstart is affected by both leakage saturation and skin effect. However, at low voltage (at about rated current), the stator leakage and rotor leakage reactances are not affected by leakage saturation; only skin effect affects both R′rstart and X′rlstart. In this case it is common practice to consider

X X’rlstart (7.44)

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