Blondel's Two Reaction Theory (Theory of Salient Pole Machine)

Blondel's Two Reaction Theory (Theory of Salient Pole Machine)

It is known that in case of nonsalient pole type alternators the air gap is uniform. Due to uniform air gap, the field flux as well as armature flux vary sinusoidally in the air gap. In nonsalient rotor alternators, air gap length is constant and reactance is also constant. Due to this the m.m.f.s of armature and field act upon the same magnetic circuit all the time hence can be added vectorially. But in salient pole type alternators the length of the air gap varies and the reluctance also varies. Hence the armature flux and field flux cannot vary sinusoidally in the air gap. The reluctances of the magnetic circuits on which m.m.f.s act are different in case of salient pole alternators.

Hence the armature and field m.m.f.s cannot be treated in a simple way as they can be in a nonsalient pole alternators.

The theory which gives the method of analysis of the disturbing effects caused by salient pole construction is called Two Reaction Theory. Professor Andre Blondel has put forward the two reaction theory.

Key Point : According to this theory the armature m.m.f. can be divided into two components as,

1. Component acting along the pole axis called direct axis.

2. Component acting at right angles to the pole axis called quadrature axis.

The component acting along direct axis can be magnetising or demagnetising. The component acting along quadrature axis is cross magnetising. These components produce the effects of different kinds.

The Fig. 3.10.1 shows the stator m.m.f. wave and the flux distribution in the air gap along direct axis and quadrature axis of the pole.

(a) Direct axis       (b) Quadrature axis

The reluctance offered to the m.m.f. wave is lowest when it is aligned with the field pole axis. This axis is called direct axis of pole i.e. d-axis. The reluctance offered is highest when the m.m.f. wave is oriented at 90° to the field pole axis which is called quadrature axis i.e. q-axis. The air gap is least in the centre of the poles and progressively increases on moving away from the centre. Due to such shape of the pole-shoes, the field winding wound on salient poles produce the m.m.f. wave which is nearly sinusoidal and it always acts along the pole axis which is direct axis.

Let Ff be the m.m.f. wave produced by field winding, then it always acts along the direct axis. This m.m.f. is responsible to produce an excitation e.m.f. E_f which lags F_f by an angle 90°.

When armature carries current, it produces its own m.m.f. wave PAR- This can be resolved in two components, one acting along d-axis (magnetising or demagnetising) and one acting along q-axis (cross-magnetising). Similarly armature current Ia also can be divided into two components, one along direct axis and one along quadrature axis. These components are denoted as,

F_d = Component along direct axis

F_AR :

F_q = Component along quadrature axis

I_d = Component along direct axis

I_a

I_q = Component along quadrature axis 

The positions of F_AR, Fd and F, in space are shown in the Fig. 3.10.2.

The instant chosen to show these positions is such that the current in phase R is maximum positive and is lagging E_f by angle ψ.

The phasor diagram corresponding to the positions considered is shown in the Fig. 3.10.3. The I_a, l_ags Ey by angle ψ.

It can be observed that F_d is produced by I_d which is at 90° to E_f while F_q, is produced bzy I_q, which is in phase with E_f

The flux components of ϕ_AR which are ϕ_d and ϕ_q along the direct and quadrature axis respectively are also shown in the Fig. 3.10.3.

It can be noted that the reactance offered to flux along direct axis is less than the reactance offered to flux along quadrature axis. Due to this, the flux ϕ_AR is no longer along F_AR or I_a. Depending upon the reluctances offered along the direct and quadrature axis, the flux ϕ_AR lags behind I_a,

 

1. Direct and Quadrature Axis Synchronous Reactances

We know that, the armature reaction flux ϕ_AR has two components, ϕ_d along direct axis and ϕ_q along quadrature axis. These fluxes are proportional to the respective m.m.f. magnitudes and the permeance of the flux path oriented along the respective axes.

ϕ_q = P_dF_d

where P_d = Permeance along the direct axis

Permeance is the reciprocal of reluctance and indicates ease with which flux can travel along the path.

But    F_d = m.m.f. = K_ar I_d in phase with I_d

The m.m.f. is always proportional to current. While K_ar is the armature reaction coefficient.

ϕ_q = P_d K_ar I_d

Similarly ϕ_q = P_d K_ar I_d

As the reluctance along direct axis is less than that along quadrature axis, the permeance P_d along direct axis is more than that along quadrature axis, (P_d > P_q).

Let E_d and E_q be the induced e.m.f.s due to the fluxes ϕ_d and ϕ_q respectively. Now E_d lags ϕ_d  by 90° while E_q lags ϕ_q by 90°.

The resultant e.m.f. is the phasor sum of E_f, E_d and E_q.

Substituting expressions for ϕ_d and ϕ_q

Now  X_ard = Equivalent reactance corresponding to the d-axis component of armature reaction

= K_e P_d K_ar

and    X_arq = Equivalent reactance corresponding to the q-axis component of armature reaction

= K_e P_d K_ar

For a realistic alternator we know that the voltage equation is,

Where V_t = Terminal voltage

X_L = Leakage reactance

But 

Substituting in expression for

where          X_d = d-axis synchronous reactance = X_L + X_ard    ...(3.10.2)

and    X_q = q-axis synchronous reactance = X_L + X_arq    ...(3.10.3)

It can be seen from the above equation that the terminal voltage V_t is nothing but the voltage left after deducting ohmic drop I_a R_a, the reactive drop I_d X_d in quadrature with Id and the reactive drop I_q X_q  in quadrature with I_q, from the total e.m.f. E_f.

The phasor diagram corresponding to the equation (3.10.1) can be shown as in the Fig. 3.10.4. The current I_a lags terminal voltage V_t by Then add Ia Ra in phase with I_a to V_t. The drop I_d X_d leads I_d by 90° as in case purely reactive circuit current lags voltage by 90° i.e. voltage leads current by 90° . Similarly the drop I_q X_q leads X_q by 90°. The total e.m.f. is E_f.

 

2. Detail Analysis of Phasor Diagram

In the phasor diagram shown in the Fig. 3/10.4 the angles ψ and δ are not  known, though Vt, I_a and ϕ values are known. Hence the location of E_f is also unknown. The components of I_a , I_d and I_q can not be determined which are required to sketch the phasor diagram.

Let us find out some geometrical relationships between the various quantities which are involved in the phasor diagram. For this, let us draw the phasor diagram including all the components in detail.

We know from the phasor diagram shown in the Fig. 3.10.4 that,

I_d = I_a sin ψ  …(3.10.4)

I_q = I_a cos ψ  …(3.10.5)

cos ψ = I_q / I_a … (3.10.6)

The drop I_a R_a has two components which are,

I_d R_d = Drop due to R_a in phase with I_d

I_q R_a = Drop due to R_a in phase with I_q 

The I_d X_d and I_q X_q can be drawn leading I_d and I_q by 90° respectively. The detail phasor diagram is shown in the Fig. 3.10.5.

In the phasor diagram,

OF = E_f

OG = V_t

GH = I_d R_a and H_A = I_q R_a

GA = I_q R_a

AE = I_dX_d and EF = I_q R_q

Now DAC is drawn perpendicular to the current phasor I_a and CB is drawn perpendicular to AE.

The triangle ABC is right angle triangle,

But from equation (3.10.6),

Thus point C can be located. Hence the direction of  E_f  is also known.

Now triangle ODC is also right angle triangle.

As I_a X_q is known, the angle ψ can be calculated from equation (3.10.10) As ϕ is known we can write,

δ = ψ –  ϕ  for lagging p.f.

E_f = V_t cos δ + I_q + R_a + I_d X_d   …(3.10.11)

Hence magnitude of Ef can be obtained by using equation (3.10.11).

Note In the above relations, ϕ  is taken positive for lagging p.f. For leading p.f., ϕ must be taken negative.

 

Example 3.10.1 For a salient pole synchronous machine, prove the d-axis synchronous reactance X_d, can be obtained from its OCC and SCC. Neglect armature resistance. AU : Dec.-05, Marks 8

Solution : The phasor diagram for a salient pole synchronous machine with zero armature resistance is shown in the Fig. 3.10.6.

Under steady state, during short circuit conditions, the terminal voltage V_t, is zero.

I_q X _q = 0

I_q = 0

The armature current I, during short circuit conditions is given as,

From the above phasor diagram it can be seen that

E_t = V_t cos δ + I_d X_d

But V_t = 0 during short circuit conditions .:

E_f = I_d X_d

and l_asc = I_d

The phasor diagram is modified as shown in the Fig. 3.10.6 (a).

The excitation voltage E_f can be obtained from open circuit characteristics while I_asc can be obtained from short circuit characteristics for a given field current.

Here K' is constant while Ld is d-axis synchronous inductance of the synchronous machine. From the above equation it can be seen that the armature short circuit current remains substantially constant over wide range of frequency or alternator speed. So during the short circuit test, the speed of alternator should not be necessarily the synchronous speed. But at low speeds, the armature resistance of alternator is comparable with the reactance and thus the change in armature current is obvious.

Review Question

1. Explain the two reaction theory for the synchronous machines. AU : May-06, 07, 09, Marks 10