Alternating Currents and Voltages

Alternating current generator

Let’s consider a signal turn coil rotating freely at a constant angular velocity symmetrically between the poles of a magnet as illustrated below:

The ends of the loop are terminated in slip rings. As the loop rotates, side A will have an emf induced first in one direction and then in the other direction. Hence, because side A is permanently connected to a slip ring A, this ring will be alternatively positive and negative. The same process applies to ring B. The generated output is therefore alternating.

The Single-phase ac waveform

Figure 1(b) shows across section through a single-loop generator. In the vertical position AB, the loop sides are cutting no flux, and therefore no emf is induced, however as the loop rotates, flux is cut increasingly and hence more and more emf is being induced, up to a maximum in the horizontal position. Further rotation of the loop causes the emf to fall to zero again. This rise and fall of the emf can be traced graphically.

Since the current will flow in the same direction as induced emf, it too will rise and fall in time with the emf i.e. the current and emf are said to be in phase with one another.

Figure 2 below shows a single conductor rotating in a magnetic field.

After each 30° of revolution, the conductor is at positions, 1, 2, 3, 4, etc. The horizontal axis of figure 2(A) is the circular path taken by the conductor opened out to form a straight line, each 30° linear space corresponding to each 30° angle of movement. The vertical axis represents the magnitude of the induced emf. Since the induced emf depends on the amount of flux being cut, which itself depends on the position of the conductor, therefore, the magnitude of the emf can be represented by the conductor position.

The resulting graph indicates the emf induced in one complete revolution of the conductor. This waveform is referred to as a sine wave and any quantity that has a wave of this nature is called sinusoidal quantity.

Figure 2(B) shows the variation of the emf during one revolution of the conductor called one cycle. This is the principle of operation of the AC generator (alternator).

The time taken for an alternating quantity to complete one cycle is called the period or the periodic time, T of the waveform.

The number of cycles completed in one second is called the frequency f, of the supply and is measured in hertz (HZ). The standard frequency of supply is 60 Hz in some countries and 50 Hz in others.

AC Values

Instantaneous values are the values of alternating quantities at any instant time and they are represented by small letters, i, v, e, etc. The largest value reached in a half cycle is called the peak value or the maximum value or the amplitude of the waveform, such values are represented by V_M, I_M, etc.

Peak to peak value of emf is the difference between the maximum and minimum values in cycle.

The average or mean value of symmetrical alternating quantity such as sine wave is the average value measured over a half cycle (the average value is zero, over a complete cycle).

The area under the curve can be calculated using approximate methods such as the trapezoidal rule, mid-ordinate rule or Simpson’s rule. The average values are represented by V_AV, I_AV.

For a sine wave, the average value = 0.637 x maximum value i.e. 2/ℼ x maximum value for example I_AV = 0.637 x I_MAX

The effective value of alternating current is that current which will produce the same heating effect as equivalent direct current. The effective value is called the root mean square (rms) value and whenever an alternating quantity is given, it is presumed to be the rms value. The symbols of rms values are I, V, E, etc.

For a non-sinusoidal waveform like the one in figure 3(b) the rms value is given by:

Where n is the number of intervals used.

For a sine wave, rms value = 0.707 x maximum value

For example I_rms = 0.707 x I_max

The values of form and peak factors give an indication of the shape of waveforms.

Related: Power Measurements in DC Circuits

The Equation of a sinusoidal waveform

When linear circuits are excited by alternating current signals of a given frequency the current through and voltage across every element in the circuit are AC signals of the same frequency.

A sinusoidal AC voltage v(t) as shown in figure 3(c) can be expressed mathematically as:

v(t) = V_msin (wt + Φ)

Where V_m is the signal amplitude, w is the radian frequency measured in radians per second and Φ is the phase angle relative to the reference sinusoid V_msin (wt) measured in radians. The phase angle is related to the time shift (Δt) between the signal and reference.

A positive phase angle Φ implies a leading waveform, (i.e. it occurs earlier on the time axis) and a negative angle implies a lagging waveform (i.e. it occurs later in the time axis).

Given the general sinusoidal voltage

v = V_msin (wt ± Φ), then

1. Amplitude or maximum value = V_m
2. Peak to peak value = 2V_m
3. Angular velocity = w rad/s
4. Periodic time, T = 2ℼ/w seconds
5. Frequency f = w/2ℼ Hz
6. Φ = angle of lag or lead (compared with a reference sinusoid wave)

Worked Example

Given AC voltage v(t) = 10.00 sin (t-1) V, find the signal amplitude, frequency and phase angle.

Solution

Signal amplitude V_m = 10.00 V

The signal radian frequency is w = 1.00 rad/sec

The phase angle is Φ = – 1 rad = -57.3°

The negative phase angle indicates that the signal lags, i.e. it occurs later in time relative to reference sin (t).

Reasons why AC Power is used

Alternating current power is employed in many applications where direct current power is not practically possible. Some of the reasons why AC power is used include:

• AC power is more efficient to transmit over long distances because it is easily transformed to a high-voltage, low-current form, minimizing power losses during transmission; it is then transformed back to required levels in residential areas for consumers.
• AC power is easy to generate with rotating machinery e.g. an electric generator.
• AC power is easy to use to drive rotating machinery e.g. an electrical motor.

You can also read: Power Measurement in AC Circuits (Single-phase & Polyphase Systems)