# Two Reaction Theory – Salient Pole Synchronous Machine

**Two Reaction Theory** was proposed by** Andre Blondel**. The theory proposes to resolve the given armature MMFs into two mutually perpendicular components, with one located along the axis of the rotor of the salient pole. It is known as the **direct axis** or **d axis** component. The other component is located perpendicular to the axis of the rotor salient pole. It is known as the **quadrature axis** or** q axis** component.

The d axis component of the armature MMF F_{a }is denoted by F_{d }and the q axis component by F_{q}. The component F_{d} is either magnetizing or demagnetizing. The component F_{q }results in a cross-magnetizing effect.

If Ψ is the angle between the armature current I_{a} and the excitation voltage E_{f }and F_{a} is the amplitude of the armature MMF, then

## Salient Pole Synchronous Machine Two Rection Theory

In the cylindrical rotor synchronous machine, the air gap is uniform. The pole structure of the rotor of a salient pole machine makes the air gap highly non-uniform. Consider a 2 pole, salient pole rotor rotating in the anticlockwise direction within a 2 pole stator as shown in the figure below.

The axis along the axis of the rotor is called the direct or the d axis. The axis perpendicular to d axis is known as the quadrature or q axis. The direct axis flux path involves two small air gaps and is the path of the minimum reluctance. The path shown in the above figure by ϕ_{q} has two large air gaps and is the path of the maximum reluctance.

The rotor flux B_{R} is shown vertically upwards as shown in the figure below.

The rotor flux induces a voltage E_{f} in the stator. The stator armature current I_{a} will flow through the synchronous motor when a lagging power factor load is connected it. This stator armature current I_{a} lags behind the generated voltage E_{f }by an angle Ψ.

The armature current produces stator magnetomotive force F_{s}. This MMF lags behind I_{a} by angle 90 degrees. The MMF F_{S} produces stator magnetic field B_{S} long the direction of Fs. The stator MMF is resolved into two components, namely the direct axis component F_{d} and the quadrature axis component F_{q}.

If,

- ϕ
_{d }is the direct axis flux - Φ
_{q}is the quadrature axis flux - R
_{d}is the reluctance of the direct axis flux path

Therefore

As, R_{d} < R_{q}, the direct axis component of MMF F_{d} produces more flux than the quadrature axis component of the MMF. The fluxes of the direct and quadrature axis produce a voltage in the windings of the stator by armature reaction.

Let,

- E
_{ad}be the direct axis component of the armature reaction voltage. - E
_{aq }be the quadrature axis component of the armature reaction voltage.

Since each armature reaction voltage is directly proportional to its stator current and lags behind by 90 degrees angles. Therefore, armature reaction voltages can be written as shown below.

Where,

- X
_{ad}is the armature reaction reactance in the direct axis per phase. - X
_{aq}is the armature reaction reactance in the quadrature axis per phase.

The value of X_{ad} is always greater than X_{aq}. As the EMF induced by a given MMF acting on the direct axis is smaller than for the quadrature axis due to its higher reluctance.

The total voltage induced in the stator is the sum of EMF induced by the field excitation. The equations are written as follows:-

The voltage E’ is equal to the sum of the terminal voltage V and the voltage drops in the resistance and leakage reactance of the armature. The equation is written as

The armature current is divided into two components; one is the phase with the excitation voltage E_{f }and the other is in phase quadrature to it.

If

- I
_{q}is the axis component of I_{a}in phase with E_{f}. - I
_{d}is the d axis I_{a}lagging E_{f}by 90 degrees.

Therefore,

Combining the equation (4) and (5) we get

Combining the equation (6) and (7) we get

The reactance **X _{d}** is called the

**direct axis synchronous reactance**, and the reactance

**X**

_{q }is called the

**quadrature axis synchronous reactance.**

Combining the equations (9) (10) and (11), we get the equations shown below.

The equation (12) shown above is the final voltage equation for a salient pole synchronous generator.