3.2 Dynamic and Static Characteristics of Induction Machine Drives Drives
3.2.4 Static Characteristics of an Induction Motor
Static characteristics concern the steady state of a drive and give in an analytic or graphic form the functional relations between the parameters characterizing motor regimes. Typical static characteristics can for instance indicate the relations be- tween electromagnetic torque, current and the capacity of a motor or between effi- ciency and the slip, voltage supply and the power output of the drive and the like.
One can note that static characteristics constitute a set of constant points along a
trajectory {qi∈q} for selected variables of the state qi or their functions, that illus- trate the values that are interesting from the point of view of the specific regimes of a machine, for example electromagnetic torque Te. Static characteristics collect the end points of trajectories for which the system reaches a steady state. They do not provide information regarding the transfer from a specific point on the charac- teristics to another one, how much time it will take and whether it is attainable.
Hence, in static characteristics we do not have to do with such parameters as mo- ment of inertia J, and the electromagnetic torque Te and load torque Tl are equal since the drive is in the state of equilibrium, i.e. it does not accelerate or brake (see 3.12). For example, very relevant characteristics are presented using functions (3.120, 3.122). They illustrate the electromagnetic torque for an induction machine depending on a number of parameters. A typical task involves the study of the re- lation between the characteristics of the electromagnetic torque and the slip Te(s) for constant remaining parameters, since it informs of the driving capabilities of the motor in the steady state. The relation between machine’s torque and slip Te(s) gives the maximum of this function for two slip values called break torque slip or pull-out slip.
χ
σ σ
1 1
2 2
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛
±
=
s s
s s
s r s r b
X R X R
X R k
s k (3.123)
or in standardized parameters
( )
χ
ω σ α
ω α σ ωα
2 2
2 2
/ s
s s s s
r
sb
+
± +
= (3.124)
The root term χ in formulae (3.123, 3.124) is the factor for correction of the value of the break torque slip as a result of the of stator windings resistance Rs influence.
Since leakage coefficient is σ =0.08…0.03 (see Table 3.1) the following ine- quality is fulfilled
⎟⎟⎠
⎜⎜ ⎞
⎝
>>⎛
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛
s s s
s
X R X
R
σ (3.125) In addition, these relations are inversely proportional to the square root of the fre- quency of the supply source. Two degrees of simplification that are applicable in the development of static characteristics of an induction motor result from the pre- sented estimates. The first of them is not very far-reaching and involves disregard- ing of resistance Rs in the terms denoting torque (3.120, 3.122) and break torque
slip (3.123, 3.124), in which this effect is smaller in accordance with the estima- tion in (3.125). In this case we obtain:
1 1
1 ) (1
1 ) (
2
2 2
2
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛
±
=
+ +
=
s s s r r b
s r r s s r
r
s s e
X X R R k s k
k R s R X k
k
s R T pU
s
σ σ ω σ
(3.126)
The most extensive simplification concerns the case when the resistance of the stator windings is completely disregarded, i.e. Rs = 0. In this case we obtain:
s r s r b r
s s r
r
s s
e X
R k s k s
R X k
k
s R T pU
σ σ
ω + =±
= and
1 ) ( 1 )
( 2 2
2
(3.127)
Fig. 3.28 presents static characteristics of the motor’s torque in the function of the slip for a small power induction motor for the three examined variants of simplifi- cation regarding resistance Rs. One can note the small difference between the curve for the torque marked with solid line (i.e. the one presenting relations with- out simplifications (3.120,3.122)), and dotted line (i.e. the one presenting the re- sult of calculations on the basis of formulae (3.126) involving the first degree of simplification). However, when the resistance of stator windings is totally disre- garded (Rs = 0) in accordance with formulae (3.127), the error in the characteris- tics of torque Te is considerable, as the relative involvement of resistance Rs in the stall impedance of small power motor is meaningful.
Fig. 3.28 Torque-slip characteristics for the small power induction motor illustrating sim- plifications concerning stator resistance Rs: _____ Rs taken into consideration completely, according to (3.120, 3.122); ▫▫▫▫▫▫▫▫ into consideration taken only the most significant com- ponent containing Rs, according to (3.126); --- Rs totally disregarded (3.127)
It is noteworthy that for Rs = 0 the characteristic of motor torque becomes an odd function of the slip s, so it is symmetrical in relation to the point of the idle run s = 0. Accounting for resistance Rs torque waveform on the side of the motor regime (s>0) is considerably smaller in terms of absolute values than for the case of generating regime, i.e. for s<0. In addition, on the side of the generator regime the effect of the first degree of simplification accounting for resistance Rs is more clearly discernible than for the case of the motor regime, which can be simply in- terpreted by analyzing relations (3.120, 3.122). The presented effect of the resis- tance of stator windings on the characteristics of the torque increases along with the reduction of the pulsation of the supply voltage ωs and becomes very high for small frequencies. This subject will be covered in more detail later. This effect is graphically presented in Fig. 3.29 in the range of the supply frequencies 1 < fs ≤ 50 [Hz].
Fig. 3.29 Characteristic of the break-torque slip sb versus pulsation of the supply voltage ωs
for the small power motor
The formulae for the break-torque slip and motor torque accounting for simpli- fications concerning the resistance can be additionally presented in formulae con- taining standardized parameters. The equivalent of the formulae (3.126) takes the form:
2 2
2
) (
) ( ) 1 (
s r s
r
s s e
s s X
T pU
α α ω σ σ α
+ +
= − and
( )2 2
2
/ s
s s s
r
sb
ω σ α
ω σ
ωα
± +
= (3.128) Concurrently, formulae (3.127) are replaced with the form which disregards resistance Rs, by introducing αs = 0:
2 2
2
) ( )
( s
s X
k k T pU
r s
r
s r s s
e σω α
α +
= and
σ ωα
s r
sb =± (3.129) In the latter case it is easy to calculate the value of the break-torque:
2 2
2 ) 2
( ⎟⎟
⎠
⎜⎜ ⎞
⎝
= ⎛
=
=
s s
L s r s s X
s r s s b b e
U L
k pk X
k k T pU s T
z z
ω σ ω
σ (3.130) Formula (3.130) constitutes the basic rule applicable for adjusting the RMS value of sinusoidal supply voltage Us of the motor to the frequency of this voltage fs in such a manner, that guarantees a constant break-torque value of Tb. Hence, the re- lation takes the form:
sn n s
s U
U ω
ω = , which, subsequently gives:
s n n
s f
U f
U = (3.131) During the course of action that follows in the discussion of frequency based con- trol of motor’s rotational speed it will become evident that this rule is completely insufficient within the range of small supply frequencies. This is so due to the ris- ing share of the resistance Rs in the impedance of the motor stall along with the decrease in the frequency of supply. The relation denoting the break-torque with- out simplifications, in which resistance Rs is not disregarded, is much more com- plex than the one in (3.130). The greater complexity of the relation results from the substitution of the break-torque slip sb (3.123) in the expression denoting the electromagnetic torque of the motor. As a result we obtain:
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛ +
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛ −
= 2 2
2 1
) (
χ ω σ ω α
χ χ α σ
ω
s s s
s s r s s s
b L
k k pU
T sign (3.132)
where coefficient χ results from (3.123) and is given by the relation:
1 1
2 2
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛
=
s s s s
X R X R
σ
χ (3.133)
Under the simplifying assumption that Rs = 0, we have αs = 0 and χ = 1 and, as a consequence, break-torque expression (3.132) is reduced to this form (3.130). The relation in (3.133) is applied to indicate the effect of resistance Rs on the break- torque Tb more clearly. The following illustrations in Fig. 3.30 show voltage- frequency relations required to provide constant value of nominal break-torque Tbn
in the function of stator voltage pulsation ωs. For the motor regime of operation the required voltage is clearly higher than for the generator regime. From Fig. 3.30 we can also see that smaller motors, within low frequency range, require much higher supply voltages than large motors to sustain the nominal level of Tbn. A close inspection of Fig. 3.30b indicates that for higher pulsations ωs the differ- ences between motors disappear, but still there is constant discrepancy between the symmetrical ‘ideal’ V-line for αs = 0 and the curves, for which stator resis- tance Rs was accounted for. For the motor operation the required voltages are higher while for generator operation they are lower in comparison to the ‘ideal’ V- line. One might say that the actual V-line for which resistance Rs is included is shifted in the direction of lower pulsations ωs in respect to the ‘ideal’ V-line for which Rs is completely ignored.
a) b)
Fig. 3.30 Voltage-pulsation curves indicating the a stator voltage level required to sustain a nominal break-torque Tbn while ωs pulsation changes. The curves are presented for different induction motors with αs = 18.8, 5.4, 1.7, 1.2, 0.0 : a) for full range of stator voltage pulsa- tion ωs, b) range of ωs limited to low values
Subsequently, Fig. 3.31 presents the characteristics of the motors in the func- tion of the slip in two versions: completely accounting for parameter Rs - smaller characteristic in each pair, and the one totally disregarding resistance, i.e. for Rs = 0 - with the above presented characteristic. For nominal value of ωs = 2πfs, the distinctive difference between the two versions take place for the small power motor.
Fig. 3.31 Torque-slip curves (relative values) for the three induction motors: small, medium and high power. The effect of Rs = 0 simplification is illustrated for fs = 50 [Hz]
Subsequently, Fig. 3.32 presents the characteristics of stator current for the three motors accounting for resistance Rs. The relation (3.134) is applied in this case, which comes as a consequence of (3.118):
⎥ ⎥
⎦
⎤
⎢ ⎢
⎣
⎡ ⎟⎟ ⎠ +
⎜⎜ ⎞
⎝ + ⎛
⎟ +
⎠
⎜ ⎞
⎝ + ⎛
⎟ ⎠
⎜ ⎞
⎝ + ⎛
=
1 2
2 2
2 2
2
2 2
s r s r s r s r
s
r s
s s s
k s s k s
s X
I U
ω α α α α
ω α σ
ω α
(3.134)
When the resistance of stator windings is disregarded (αs = 0), the relation which defines the current in the stator windings takes a considerably more succinct form, which is additionally easy to verify for the two extreme motor states, i.e. for s = 0 and s = ∞.
2 2
2
2 2
⎟⎠
⎜ ⎞
⎝ +⎛
⎟⎠
⎜ ⎞
⎝ +⎛
=
s s X
I U
r s
r s
s s
s σ ω α
ω α
(3.135)
Self reactance of the stator windings Xs is encountered in a multitude of relations concerning induction motors. The value of this parameter can be easily determined from calculations or manufacturers’ data for idle run. From the equivalent diagram (Fig. 3.26) of the motor it results that
0 0 0
0
0 Z R jX X Z sinϕ
I U
s s
s+ =
=
= (3.136)
Fig. 3.32 Stator current-slip curves (relative values) for the three exemplary motors accord- ing to (3.134) presented in relative values, for fs = 50 [Hz]
where U0,I0,Z0,ϕ0 denote voltage, current, impedance and phase angle for the idle run of the motor. If the phase angle during idle run is not familiar, it is possi- ble to use assessment relevant for the rated frequency: Rs << Xs and calculate in an approximated way:
0 0
I
Xs =U (3.137)