Phase Angle

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  Fundamental Electrical and Electronic Principles

  • C R Robertson(Author)
  • 2008(Publication Date)
  • Routledge

    (Publisher)

  same frequency. This is because the value of ω is the same for both. The angular difference, of π/6 radian, would then be described as the phase difference between them.

  We can therefore, represent an alternating quantity by means of a phasor. The length of the phasor represents the amplitude. Its angle, with respect to some reference axis, will represent its Phase Angle. Considering the two waveforms in Fig. 6.19 , the plot has been started with V1 in the horizontal position (vertical component of V1 = 0). This horizontal axis is therefore taken as being the reference axis. Thus, if these waveforms represent two voltages, v1 and v2 , then the standard expressions would be:

  The inconvenience of representing a.c. quantities in graphical form was pointed out earlier, in section 6.3. This section introduced the concept of using a standard mathematical expression for an a.c. However, a visual representation is also desirable. We now have a much simpler means of providing a visual representation. It is called a phasor diagram. Thus the two voltages we have been considering above may be represented as in Fig. 6.20 .

  Fig. 6.20

  Notice that v1 has been chosen as the reference phasor. This is because the standard expression for this voltage has a Phase Angle of zero (there is no ±φ term in the bracket). Also, since the phasors are rotating counterclockwise, and v2 is lagging v1 by π/6 radian, then v2 is shown at this angle below the reference axis.

  Notes

  1    Any a.c. quantity can be represented by a phasor, provided that it is a sinewave.

  2    Any number of a.c. voltages and/or currents may be shown on the same phasor diagram, provided that they are all of the same frequency

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  Higher Engineering Science

  • William Bolton(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  ω t, where V is the maximum value of the voltage and ω the angular frequency and equal to 2πƒ. We can imagine such a signal being produced by the vertical projection of a radial line of length V rotating with a constant angular velocity ω from some initial start position (Figure 8.4). Thus instead of specifying the variation of the voltage with time by the above equation, we can specify it by the length of the line V and whether it starts at t = 0 at some angle, termed the Phase Angle ϕ, to the reference axis which is usually taken as the horizontal axis. Such lines are termed phasors. Figure 8.4 (a) v = V sin ωt, (b) v = V sin(ωt + ϕ) A phasor can be described by drawing an arrow-headed line, the length of the line representing the amplitude and its direction, relative to a reference direction, as the Phase Angle (Figure 8.5). Because with alternating currents and voltages we are normally concerned with root-mean-square (r.m.s.) values rather than maximum value, for sinusoidal waves the maximum value is just the r.m.s. value divided by √2, generally when the term phasor is used for an arrow-headed line describing alternating currents and voltages the length of the line represents the r.m.s. value. Figure 8.5 A phasor The Phase Angle is the angle between a phasor and some reference direction. In the case of a series circuit it is customary to use the direction of the current phasor for the circuit as the reference direction, the current being the same for all the series components. For a parallel circuit it is customary to use the direction of the voltage phasor for the parallel circuit as the reference phasor, the voltage being the same for all parallel components. In textbooks the common practice to indicate that a symbol represents a phasor is to use bold print, e.g. V represents a voltage phasor. The voltage representing the length of the phasor would be given by the italic, non-bold, symbol V

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  eBook - ePub

  Engineering Science

  • W. Bolton(Author)
  • 2015(Publication Date)
  • Routledge

    (Publisher)

  Currents and voltages in the same circuit will have the same frequency and thus the phasors used to represent them will rotate with the same angular velocity and maintain the same Phase Angles between them at all times; they have zero motion relative to one another. For this reason, we do not need to bother about drawing the effects of their rotation but can draw phasor diagrams giving the relative angular positions of the phasors as though they were stationary.

  The following summarise the main points about phasors:

  1. A phasor has a length that is directly proportional to the maximum value of the sinusoidally alternating quantity or, because the maximum value is proportional to the root- mean- square value, a length proportional to the r.m.s. value.
  2. Phasors are taken to rotate anticlockwise and have an arrow- head at the end which rotates.
  3. The angle between two phasors shows the Phase Angle between their waveforms. The phasor which is at a larger anticlockwise angle is said to be leading, the one at the lesser anticlockwise angle lagging (Figure 16.4 ).
  4. The horizontal line is taken as the reference axis and one of the phasors given that direction, the others have their Phase Angles given relative to this reference axis.

  Figure 16.4 Leading and lagging

  Note that, in textbooks, it is common practice where we are concerned with just the size of a phasor to represent it using italic script, e.g. V , but where we are referring to a phasor quantity with both its size and phase we use bold non- italic text, e.g. V. Thus we might say that phasor V has size of V and a Phase Angle of φ .

  Example

  Draw the phasor diagram to represent the voltage and current in a circuit where the current is described by i = 1.5 sin ωt A and the voltage by v = 20 sin (ωt + π /2) V.

  Figure 16.5 shows the phasors with their lengths proportional to the maximum values of 1.5 A and 20 V.

  Figure 16.5 Example

  16.3 R, L, C in a.c. circuits

  In the following discussion the behaviour of resistors, inductors and capacitors are considered when each individually is in an a.c. circuit.

  16.3.1 Resistance in a.c. circuits

  Consider a sinusoidal current i = Im sin ωt passing through a pure resistance (Figure 16.6

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