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Equivalent Circuit of a Synchronous Motor
A synchronous motor is a doubly-excited system, i.e., it is connected to two electrical systems −
A 3-phase supply is connected to the armature winding.
A DC supply is connected to the rotor field winding.
The figure shows the equivalent circuit for one phase of a three-phase synchronous motor
Let
V = Terminal voltage per phase applied to the motor
Ef = Excitation voltage
Ia = Armature current per phase drawn by the motor from the supply
Ra = Effective armature resistance per phase
XS = Synchronous reactance per phase of the motor armature winding
ZS = Synchronous impedance per phase of the armature
By applying KVL in the equivalent circuit of the motor, we have,
$$\mathrm{V=E_{f}+I_{a}Z_{s}\:\:\:\:\:\:...(1)}$$
The synchronous impedance of the motor is
$$\mathrm{Z_{S}=R_{a}+jX_{S}\:\:\:\:\:\:...(2)}$$
From eqns. (1) & (2), we get,
$$\mathrm{V=E_{f}+I_{a}(R_{a}+jX_{S})\:\:\:\:\:\:...(3)}$$
$$\mathrm{\Longrightarrow\:E_{f}=V-I_{a}R_{a}-jI_{a}X_{S}\:\:\:\:\:\:...(4)}$$
Where,
IaRais the voltage drop per phase in the armature resistance.
IaXSis the reactive voltage drop per phase due to armature reactance and armature reaction effect.
Also, the armature current per phase is given by,
$$\mathrm{I_{a}=\frac{V-E_{f}}{Z_{S}}\:=\frac{E_{r}}{Z_{S}}\:=\frac{E_{r}}{R_{a}+jX_{S}}\:\:\:\:\:\:...(5)}$$
Here, Eris the resultant voltage in the armature circuit.
The equivalent circuit of the synchronous motor helps considerably in understanding the operation of the motor as follows −
If the field excitation is such that Er = V, then the synchronous motor is said to be normally excited.
If the field excitation is such that Er < V, then the synchronous motor is said to be under-excited.
If the field excitation is such that Er > V, then the synchronous motor is said to be over-excited.
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