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This chapter provides an overview of h operator. When performing calculations on single-phase circuits an operator was used, which, when multiplied by a vector, had the effect of rotating the vector through an angle of 90°, and for this purpose the j operator was used. In dealing with three-phase quantities, and in particular when applying the method of symmetrical components, there are definite advantages to be gained by having an operator that is capable of rotating a vector through an angle of 120°, because of the angular displacement of balanced three-phase vectors. The chapter further discusses symmetrical three-phase voltages, positive-phase sequence, negative phase sequence, zero-phase sequence, and Star-delta impedance transformations.

1 Representation of sinusoidal quantity by a vector

The instantaneous value v of sinusoidal voltage of frequency f Hz is given by

υ=Vsinωt, (1)

where V = maximum value and ω = 2 π f.

This may be represented by a rotating vector of magnitude V having a constant angular velocity ω rad/s as shown in Fig. 1. The vector OV rotates anticlockwise about O starting from Ox when t = 0. The angle turned through, at any time t, is then given by ωt and the instantaneous value v at any time is given by V sin ωt which is the projection of OV on the Y-axis, Oy.

If the vector V is now shown on an Argand diagram as in Fig. 2 where θ = ωt, it may be represented by (a + jb).

a=Vcosθ,

where

b=Vsinθ,a+jb=V(cosθ+jsinθ). (2)

Another way of expressing the vector is as V/θ, where V is known as the modulus and would correspond to the peak value of the alternating voltage wave of Fig. 1, and θ is known as the argument and represents the angular displacement of the vector V from the real axis X’OX.

Now

cosθ+jsinθ=ejθ, (3)

and

V(cosθ+jsinθ)=Vejθ. (4)

Thus the expression Vejθ may be used to represent a vector of modulus V and argument θ.

2 The h operator

When performing calculations on single-phase circuits use was made of an operator, which, when multiplied by a vector, had the effect of rotating the vector through an angle of 90°, and for this purpose the j operator was used. In dealing with three-phase quantities, and in particular when applying the method of symmetrical components, there are definite advantages to be gained by having an operator which is capable of rotating a vector through an angle of 120°, due to the angular displacement of balanced three-phase vectors.

First let us consider the multiplication of two complex quantities which may be represented by and , then we may write

r1/θ1_r2/θ2_=r1(cosθ1+jsinθ1)r2(cosθ2+jsinθ2)=r1r2(cosθ1cosθ2+jsinθ1cosθ2+jcosθ1sinθ2−sinθ1sinθ2)=r1r2(cosθ1cosθ2−sinθ1sinθ2)+j(sinθ1cosθ2+cosθ1sinθ2)=r1r2[cos(θ1+θ2)+jsin(θ1+θ2)]=r1r2/θ1+θ2_. (5)

Thus it may be observed that the modulus of the product of two complex quantities is equal to the product of their moduli, and that the argument of their product is equal to the sum of their arguments.

Suppose now that is replaced by a voltage vector V/θ, and is used to denote an operator such that it will rotate the voltage vector through an angle of 120° to give a new vector V/θ + 120°, then

r1/θ″1_,V/θ_=V/θ+120°_.

Comparing the moduli and arguments with the results of eqn. (5) gives

r1V=V.

Therefore

r1=1;

also

θ+θ1=θ+120°

Therefore

θ1=120°.

The operator r1 /θ1 is then given by

r1/θ1_=1/120°_=ej2π/3. (6)

Various symbols, including a, λ, and h, have been assigned to this operator by different writers on the subject, but the latter is now used in accordance with standard practice, as recommended by the British Standards Institution. Hence,

h=ej2π/3,=cos2π/3+jsin2π/3,=−12+j12√3. (7)

Considering again the results of eqns. (5) and (6) it may be stated that, in general, a vector multiplied by ejπ/n will cause an angular rotation of 2π/n in the vector, and so it is a simple matter to produce an operator capable of rotating a vector through any desired angle. For example, the j operator which rotates a vector through an angle of 90° would be represented by ejπ/2, where

ejπ/2=(cosπ/2+jsinπ/π)=j. (8)

A further significance may be attached to the h operator if we consider a vector represented by V as shown in Fig. 3 and which is shown drawn on the reference axis OX. If this vector is multiplied by h, giving hV, it will be rotated through 120°. A second multiplication by h, giving h2V, will give a further rotation of 120° and a third multiplication by h, giving h3V, will cause a further rotation of 120° bringing us back to the original position of vector V on the reference axis. It is now apparent that

h3V=V,h3=1. (9)

In other words, h...