Circuit Model of Single phase Induction Motor

Circuit Model of Single phase Induction Motor:

Circuit Model of Single phase Induction Motor – The winding unbalance and the fact that both the main and auxiliary windings are fed by the same supply result in unbalanced main and auxiliary fields. The phasor along the winding axes can be split into symmetrical components F̅f and F̅b as given by Eqs (10.17a) and (10.17b). The forward component set F̅f and jF̅f produces a forward rotating field; similarly the backward component set F̅b and -jF̅ results in a backward rotating field.

The rotor slips with respect to the two rotating fields are s and (2 — s) respectively as given by Eqs (10.5a) and (10.5b) and as a result the magnetizing and rotor circuits as seen by the two rotating fields with reference to the main winding are different and are shown in Figs 10.22(a) and 10.22(b).

Circuit Model of Single phase Induction Motor

It is noted that no-load losses have been neglected and therefore the core-loss conductance has not been shown in both the circuits. The impedance seen by Emf, the forward field induced emf of the main winding, is

Circuit Model of Single phase Induction Motor

and the impedance seen by the Emb, backward field induced emf in the main winding

Circuit Model of Single phase Induction Motor

Circuit Model of Single phase Induction Motor

Equations (10.16a) and (10.16b) will now be converted  into current form.

Let

Circuit Model of Single phase Induction Motor

Then from Eqs (10.16a) and (10.16b)

Circuit Model of Single phase Induction Motor

From Eqs (10.32a) and (10.32b)

Circuit Model of Single phase Induction Motor

The current in the auxiliary winding is nut since the turns of auxiliary and main winding are different , the auxiliary winding current as seen from the main winding is equal to

Circuit Model of Single phase Induction MotorFrom Eqs (10.33a) and (10.33b) the symmetrical components of mail and auxiliary winding currents with reference to the min winding can be expressed as

Circuit Model of Single phase Induction Motor

The forward field reaches the auxiliary winding 90° elect. ahead of the main winding and vice versa for the backward rotating field. Thus the emf s in the auxiliary winding induced by the two fields are:

Circuit Model of Single phase Induction Motor

Also let the main and auxiliary winding terminal voltages be V̅m and V̅a respectively. The auxiliary winding voltage is equal to (Va/a) as seen from the main winding. This set of voltages can also be split into symmetrical components as

Circuit Model of Single phase Induction Motor

or alternatively

Circuit Model of Single phase Induction Motor

Now consider the main winding terminal voltage V̅m. It comprises three components: the emf induced by the forward rotating field, the emf induced by backward rotating field and the voltage drop in the winding impedance Z̅1m owing to current Im flowing through it. Thus

Circuit Model of Single phase Induction Motor

Substituting for E̅mf and E̅mb from Eqs (10.29a) and (10.29b)

Circuit Model of Single phase Induction Motor

which is represented by the circuit of Fig. 10.23(a).

Similarly the auxiliary winding terminal voltage V̅a comprises three components,

i.e.

Circuit Model of Single phase Induction Motor

where Z̅1a is the winding impedance of the auxiliary winding which in general has a  capacitor included in it (starting/running capacitor). Using Eqs (10.36a) and (10.36b),

Circuit Model of Single phase Induction Motor

whose circuit representation is given in Fig. (10.23b). Substituting Im from Eq. (10.33a) in Eq. (10.39b),

Circuit Model of Single phase Induction Motor

Similarly substituting Ia from Eq. (10.33b) in Eq. (10.40c),

Circuit Model of Single phase Induction Motor

Circuit Model of Single phase Induction Motor

With V̅m and V̅a as expressed in Eqs (10.41a) and (10.41b), one obtains from Eqs (10.38a) and (10.38b)

Circuit Model of Single phase Induction Motor

Circuit Model of Single phase Induction Motor

Circuit Model of Single phase Induction Motor

Equations (10.42a) and (10.42b) can be written as

Circuit Model of Single phase Induction Motor

Circuit Model of Single phase Induction Motor

Equations (10.44a), (10.44b) and (10.37a) are represented by the circuit model of Fig. 10.24.

It is further noted that

Circuit Model of Single phase Induction Motor

From these the circuit model of Fig. 10.24 can be drawn in the form of Fig.10.25. In Fig. 10.25 disconnecting the auxiliary winding under running condition is equivalently represented as the opening of switch S.

Once the auxiliary Winding is disconnected.

Circuit Model of Single phase Induction Motor

By doubling the current and reducing the impedance to half the circuit model of Fig. 10.26 is obtained. It may be seen that this is the same circuit model as already presented in Fig. 10.5(c) on a heuristic basis.

Circuit Model of Single phase Induction Motor

Circuit Model of Single phase Induction Motor

From Eqs. (10.42a) and (10.42b),

Circuit Model of Single phase Induction Motor

The winding currents are given by

Circuit Model of Single phase Induction Motor

Circuit Model of Single phase Induction Motor

The developed torque and mechanical power are given by

Circuit Model of Single phase Induction Motor

Capacitor Impedance for Balanced Operation at Specific Speed

For balanced single-phase operation

Circuit Model of Single phase Induction Motor

or from Eq. (10.48b)

 Circuit Model of Single phase Induction Motor

From Eq. (10.38b)

Circuit Model of Single phase Induction Motor

For single-phase operation

Circuit Model of Single phase Induction Motor

Substituting Eq. (10.54) in Eq. (10.53)

Circuit Model of Single phase Induction Motor

With Z̅12 defined in Eq. (10.43b)

Circuit Model of Single phase Induction Motor