Voltage and Current Values of a Sine Wave

As the magnitude of the waveform is not constant, the waveform can be measured in different ways.

These are:

• Instantaneous value
• Peak value
• Peak-to-peak value
• Root mean square (RMS) value
• Average value

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Instantaneous Value

Consider the sine wave shown in Fig. 2.7.

At any given time, it has some instantaneous value. This value is different at different points along the waveform.

During the positive cycle, the instantaneous values are positive and during the negative cycle, the instantaneous values are negative.

For example:

• At time $1 \text{ ms}$, the value is $4.2 \text{ V}$
• At time $2.5 \text{ ms}$, the value is $10 \text{ V}$
• At time $6 \text{ ms}$, the value is $-2 \text{ V}$
• At time $7.5 \text{ ms}$, the value is $-10 \text{ V}$

The instantaneous value of a sine wave is given by:

$$v(t) = V_m \sin(\omega t)$$

where:

• $v(t)$ = instantaneous value
• $V_m$ = maximum value
• $\omega$ = angular frequency
• $t$ = time

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Peak Value

The peak value of a sine wave is the maximum value of the wave during the positive half-cycle or the maximum value during the negative half-cycle.

Since these two values are equal in magnitude, a sine wave is characterised by a single peak value.

In Fig. 2.7, the peak value of the sine wave is:

$$V_m = 10 \text{ V}$$

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Peak-to-Peak Value

The peak-to-peak value of a sine wave is the value from the positive peak to the negative peak.

In Fig. 2.7, the peak-to-peak value is:

$$V_{pp} = 20 \text{ V}$$

In general,

$$V_{pp} = 2V_m$$

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Average Value

In general, the average value of any function $v(t)$ with period $T$ is given by:

$$V_{avg} = \frac{1}{T} \int_0^T v(t)\,dt$$

That means the average value of a curve in the X-Y plane is the total area under the complete curve divided by the distance of the curve.

The average value of a sine wave over one complete cycle is always zero.

Therefore, the average value of a sine wave is defined over a half-cycle and not over a full-cycle period.

The average value of the sine wave is the total area under the half-cycle curve divided by the distance of the curve.

The average value of the sine wave is given by:

$$V_{avg} = \frac{1}{\pi} \int_0^{\pi} V_m \sin \theta \, d\theta$$

Taking $V_m$ outside the integral,

$$V_{avg} = \frac{V_m}{\pi} \int_0^{\pi} \sin \theta \, d\theta$$

Integrating,

$$V_{avg} = \frac{V_m}{\pi} \left[ -\cos \theta \right]_0^{\pi}$$

Substituting the limits,

$$V_{avg} = \frac{V_m}{\pi} \left[ -\cos \pi + \cos 0 \right]$$

Since,

$$\cos \pi = -1$$

and

$$\cos 0 = 1$$

Therefore,

$$V_{avg} = \frac{V_m}{\pi} \left[ 1 + 1 \right]$$ $$V_{avg} = \frac{2V_m}{\pi}$$

or

$$V_{avg} = 0.637V_m$$

The average value of a sine wave is shown by the dotted line in Fig. 2.8.

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Root Mean Square Value or Effective Value

The root mean square (RMS) value of a sine wave is a measure of the heating effect of the wave.

When a resistor is connected across a dc voltage source, a certain amount of heat is produced in the resistor in a given time.

A similar resistor is connected across an ac voltage source for the same time.

The value of the ac voltage is adjusted such that the same amount of heat is produced in the resistor as in the case of the dc source.

This value is called the RMS value.

Thus, the RMS value of a sine wave is equal to the dc voltage that produces the same heating effect.

In general, the RMS value of any function with period $T$ is given by:

$$V_{rms} = \sqrt{ \frac{1}{T} \int_0^T v^2(t)\,dt }$$

For a sinusoidal waveform,

$$v(t) = V_m \sin \omega t$$

Substituting into the RMS equation,

$$V_{rms} = \sqrt{ \frac{1}{T} \int_0^T V_m^2 \sin^2 \omega t \,dt }$$

Taking $V_m^2$ outside the integral,

$$V_{rms} = V_m \sqrt{ \frac{1}{T} \int_0^T \sin^2 \omega t \,dt }$$

Using the trigonometric identity,

$$\sin^2 \omega t = \frac{1 - \cos 2\omega t}{2}$$

Therefore,

$$V_{rms} = V_m \sqrt{ \frac{1}{T} \int_0^T \frac{1 - \cos 2\omega t}{2} \,dt }$$ $$V_{rms} = V_m \sqrt{ \frac{1}{2T} \int_0^T (1 - \cos 2\omega t) \,dt }$$

Integrating,

$$V_{rms} = V_m \sqrt{ \frac{1}{2T} \left[ t - \frac{\sin 2\omega t}{2\omega} \right]_0^T }$$

Since the sine term becomes zero over a complete cycle,

$$V_{rms} = V_m \sqrt{ \frac{1}{2T} [T] }$$ $$V_{rms} = V_m \sqrt{ \frac{1}{2} }$$ $$V_{rms} = \frac{V_m}{\sqrt{2}}$$

or

$$V_{rms} = 0.707V_m$$

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Peak Factor

The peak factor of any waveform is defined as the ratio of the peak value of the wave to the RMS value of the wave.

$$\text{Peak Factor} = \frac{V_m}{V_{rms}}$$

For a sinusoidal waveform,

$$\text{Peak Factor} = \frac{V_m}{0.707V_m} = 1.414$$

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Form Factor

Form factor of a waveform is defined as the ratio of RMS value to the average value of the wave.

$$\text{Form Factor} = \frac{V_{rms}}{V_{avg}}$$

For a sinusoidal waveform,

$$\text{Form Factor} = \frac{0.707V_m}{0.637V_m} = 1.11$$

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Summary

• Instantaneous value changes continuously with time.
• Peak value is the maximum value of the waveform.
• Peak-to-peak value is twice the peak value.
• Average value of a sine wave is defined over a half-cycle.
• RMS value represents the effective heating value.
• Peak factor is the ratio of peak value to RMS value.
• Form factor is the ratio of RMS value to average value.