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Vector Diagrams

The time characteristic of sinusoidal alternating voltages and currents can be represented not only by means of the line diagrams we have seen so far, but also by vector diagrams which can prove more appropriate in certain cases. The illustrations below show the relationship between the line and vector diagrams of a sinusoidal alternating voltage u with a peak value u0 and frequency f. The vector can be seen as a line of length u0 rotating anti-clockwise at a frequency f or angular frequency w = 2·pi·f about the coordinate origin.

Calculations with Alternating Current Quantities

The zero point in the line diagram at time t = 0 where the sinusoidal curve begins corresponds to the vector's horizontal initial position where the direction arrow is pointing to the right. The vector diagram also shows a second vector at a phase angle of w·t = 60°. The perpendicular line (dashed blue) from the tip of this vector to the horizontal axis represents the instantaneous value u of the voltage at that phase angle according to the equation.

Calculations with Alternating Current Quantities

The following animation illustrates the relationship between vector and line diagrams.

Calculations with Alternating Current Quantities

If a voltage u1 with a peak value u10 exhibits an oscillation that is ahead of a voltage u2 (peak value u20) by a phase angle j, the corresponding diagram shows two vectors, u2 displaced with respect to u1 by an angle of j (see the illustration below). 

Calculations with Alternating Current Quantities

Vectors in a vector diagram are drawn at an initial position representing the phase angle they would assume at time t = 0, i.e. a sort of snapshot during the continuous rotation of the vector. The main advantage of vector diagrams over line diagrams is that they can be used very easily to represent sinusoidal alternating quantities. Vector diagrams prove especially practical if several phase-displaced alternating quantities need to be displayed at the same time as in the example above. Instead of peak values, vector diagrams can also represent rms values U and I, both differing simply by a factor of √2.

Sinusoidal alternating quantities of identical frequencies can be visualised in vector diagrams where the vector length indicates the voltage or current value, while the angles between the vectors indicate the phase shift between the respective alternating quantities.

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