Electrical Machines and Energy Systems: Operating Principles (Part 2) SYED A Rizvi

Electrical Machines and Energy Systems: Operating Principles (Part 2) SYED A Rizvi

AC Machines Operating Principles: Synchronous Motor In synchronous motors, the stator of the motor has a rotating magnetic field. The rotor is wound to produce a stationary magnetic field through the rotor windings powered by an external D.C. source. However, the rotor is free to rotate with respect to its axle. Since the opposite magnetic poles attract each other, the rotor will rotate in an attempt to lock its magnetic poles with the opposing poles of the stator s rotating magnetic field (not as simple, as discussed later). However, the rotor will lag the stator field by a mechanical angle α. In this way, the rotor continuously chases stator s magnetic field rotating with the same speed as that of the stator (in synchronism of the stator field). 2

AC Machines Operating Principles: Synchronous Motor The mechanical angle α depends on the torque on the rotor s shaft and also referred to as the torque angle. The rotor continues to rotate in synchronism of the stator field regardless of the load (torque) on the rotor s shaft (within certain limits, of course). However, as the torque on the rotor s shaft increases, the torque angle α increases as well until it reaches 90 o. At that point, the rotor falls out of synchronism with the stator s magnetic field and eventually stops rotating. The maximum torque that the machine can provide (the torque corresponding to α = 90 o ) is called pullout torque (typically three times the full load torque of the machine). 3

AC Machines Operating Principles: Synchronous Motor The figure on the right shows the interaction of the magnetic fields of the stator and rotor of the machine, which results in an induce torque τ induced in the rotor. Mathematically, τ induced = k(br BBs) = k(bb rr BB ss ) ssssss(α) (12) Direction of rotation of the induce torque τ induced? CCW! 4

AC Machines Operating Principles: Synchronous Motor An increase in the load on the rotor s shaft results in a momentary slow down in rotor s speed causing an increase in the torque angle α. However, an increase in α results in increasing the induced torque (Eq. 12), which in turn, increases rotor s speed to synchronous speed. As can be seen from Eq. (12), the maximum induced torque is obtained at α = 90 o. For α > 90 o, the induced torque decreases as α increases causing a further slow down in rotor s speed and eventually stopping the motor. What happens when α > 180 o? Before we develop an equivalent circuit for the synchronous motor, we need to consider the impact of the rotating magnetic field of the rotor on the windings of the stator. Before we do that, let s consider the operation of the synchronous generator (we ll come back to motor). 5

A synchronous motor or generator is essentially the same machine except that they have opposite direction of power flow. That is, in the synchronous motor the input is electrical power and the output is the mechanical power (torque to drive a load). On the other hand, in the synchronous generator the input is mechanical power (a prime mover to rotate the rotor) that creates a time varying magnetic flux ( ). The output is the electrical power (induced voltage in stator windings based on Faraday s law of electromagnetic induction). The induced voltage in each phase of the stator winding is given by EE AA = K ddd dddd (13) where K depends upon machine construction. 6

It can be seen from Eq. (13) that magnitude of the generated voltage is proportional to magnetic flux of the rotating field and thus to the field current (assuming a fixed rotational speed of the PM). With the generator under no load condition, there is no current in the stator windings and, therefore, B s = 0. Accordingly, the induced torque, τ induced = 0. When a load is connected to the generator, the current through the stator windings create B s. Note that B s will lag B R by an angle α. The interaction of the magnetic field now produces τ induced, according to Eq. (12). What is direction of the induced torque τ induced? Opposite the direction of rotation of the rotor (Prime Mover)! 7

Figure below shows the equivalent circuit of the generator (per phase). 8

E A = induced voltage in the armature (in volts) V P = phase voltage available at the terminals (in volts) V F = field voltage (D.C.) applied to the field windings (volts) I A = armature current (in amps) I F = field current (in amps) R A = armature resistance (in ohms) X A = armature reactance (in ohms) R F = field resistance (variable, in ohms) X A = field reactance (in ohms) 9

In the following analysis, we will use the phase voltage as the reference with a phase angle of 0 o. That is, V P = V P 0 o E A = E A α o I A = I A ɵ o and EE AA = VV PP + II AA (R A + jjx A ) (14) Note that the figure on the right shows the phaser diagram of a generator with a lagging power factor. 10

The output power of the generator (per phase) is given by P out = VV PP II AA cos θθ (15) In general R A << X A and therefore, can be ignored. Rewriting Eq. (14) under this condition would yield EE AA = VV PP + II AA (jjx A ) (16) or or EE AA = VV PP 0 o + II AA ɵ XX AA 90 o (17) EE AA = VV PP 0 o + II AA XX AA ɵ + 90 o (18) 11

or EE AA = VV PP + (II AA XX AA ) cos θθ + 90 o + jj(( II AA XX AA ) sin θθ + 90 o ) (19) noting cos θθ + 90 o = sin θθ (20) and therefore, Eq. (19) can be rewritten as sin θθ + 90 o = cos θθ (21) EE AA = VV PP II AA XX AA sin θθ + jj( II AA XX AA ) cos θθ (22) 12

but comparing Eqs. (22) and (23), we get EE AA = EE AA cos αα + jjee AA sin αα (23) ( II AA XX AA ) cos θθ = EE AA sin αα (24) or II AA cos θθ = EE AA sin αα XX AA (24) Substituting LHS of Eq. (24) into Eq. (15), we get P out = VV PPEE AA sin αα XX AA (25) 13

The figure below shows the phaser diagram of a generator with a unity power factor. 14

The figure below shows the phaser diagram of a generator with a leading power factor. 15