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# Addition of Alternating Quantities

If two sinusoidal alternating quantities like the voltages u1 and u2 considered on the previous page need to be added, their instantaneous values at any point in a line diagram are added algebraically (refer to the right-hand side of the diagram below). A total voltage u is thus obtained. A line diagram also permits alternating quantities of different frequencies to be added in this manner.

**Alternating quantities of any frequency in a line diagram can be added algebraically in terms of their instantaneous values.**

If sinusoidal alternating quantities of the same frequency are added, the resultant voltage is also sinusoidal. In this case, a vector diagram permits much more elegant addition as demonstrated on the left-hand side of the illustration below. The two vectors to be added (each displayed as a dotted line) are drawn as two sides of a parallelogram, similar to the way that forces acting in different directions are added using a parallelogram. The vector along the parallelogram's diagonal represents the cumulative voltage u and has a peak value of u0. Such addition of directional quantities is termed geometric addition.

It can be seen that the phase angle of this cumulative voltage lies between that of the two added vectors.

**Sinusoidal alternating quantities of identical frequency are added via geometric composition of the corresponding vectors in terms of value and direction.**

The following animations illustrate geometric addition of voltage vectors (top) and rotation of vector groups in a vector diagram (bottom).