Signals: RMS and Average Values GATE EE Notes

Definitions and Concepts

Average Value of Periodic Signals

Average Value (DC Component)

For a periodic signal \(x(t)\) with period \(T\):

\[\boxed{x_{avg} = \dfrac{1}{T}\int_0^T x(t)\,dt}\]

  • Also called DC component or mean value

  • Represents the constant component of the signal

  • For symmetric signals about zero: \(x_{avg} = 0\)

  • Units: Same as the original signal

RMS Value of Periodic Signals

RMS (Root Mean Square) Value

For a periodic signal \(x(t)\) with period \(T\):

\[\boxed{x_{rms} = \sqrt{\dfrac{1}{T}\int_0^T x^2(t)\,dt}}\]

  • Also called effective value or quadratic mean

  • Represents the equivalent DC value that delivers same power

  • Always positive and non-zero for non-zero signals

  • Units: Same as the original signal

  • Power relation: \(P = \dfrac{x_{rms}^2}{R}\)

Standard Waveforms

Sinusoidal Signals

Sine Wave: \(x(t) = A\sin(\omega t + \phi)\)

  • Average Value: \(x_{avg} = 0\)

  • RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{2}} = 0.707A\)

Cosine Wave: \(x(t) = A\cos(\omega t + \phi)\)

  • Average Value: \(x_{avg} = 0\)

  • RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{2}} = 0.707A\)

GATE Tip

For any sinusoidal signal: \(x_{rms} = \dfrac{\text{Peak Value}}{\sqrt{2}}\)

Square Wave

Square Wave: Amplitude \(\pm A\), Period \(T\)

  • Average Value: \(x_{avg} = 0\) (symmetric)

  • RMS Value: \(x_{rms} = A\)

Asymmetric Square Wave

For positive duration \(t_1\) and negative duration \(t_2 = T - t_1\):

  • Average Value: \(x_{avg} = A\dfrac{2t_1 - T}{T}\)

  • RMS Value: \(x_{rms} = A\)

Triangular Wave

Symmetric Triangular Wave: Peak \(\pm A\), Period \(T\)

  • Average Value: \(x_{avg} = 0\)

  • RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{3}} = 0.577A\)

Sawtooth Wave: \(0\) to \(A\), Period \(T\)

  • Average Value: \(x_{avg} = \dfrac{A}{2}\)

  • RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{3}}\)

Rectified Signals

Half-Wave Rectified Sine

\(x(t) = \begin{cases} A\sin(\omega t) & 0 \leq t \leq \dfrac{T}{2} \\ 0 & \dfrac{T}{2} < t \leq T \end{cases}\)

  • Average Value: \(x_{avg} = \dfrac{A}{\pi} = 0.318A\)

  • RMS Value: \(x_{rms} = \dfrac{A}{2} = 0.5A\)

Full-Wave Rectified Sine

\(x(t) = A|\sin(\omega t)|\)

  • Average Value: \(x_{avg} = \dfrac{2A}{\pi} = 0.637A\)

  • RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{2}} = 0.707A\)

Important Properties

Key Properties

Relationship between RMS and Average

  • Form Factor: \(k_f = \dfrac{x_{rms}}{x_{avg}}\)

  • Crest Factor: \(k_c = \dfrac{x_{peak}}{x_{rms}}\)

  • Peak Factor = Crest Factor

Power Relations

  • Average Power: \(P_{avg} = \dfrac{x_{rms}^2}{R}\)

  • For AC signals: \(P_{avg} = \dfrac{V_{rms} \cdot I_{rms}}{R}\)

  • Total Power = DC Power + AC Power

Superposition Properties

Linear Combination

If \(x(t) = x_1(t) + x_2(t)\):

  • Average: \(x_{avg} = x_{1,avg} + x_{2,avg}\)

  • RMS: \(x_{rms}^2 = x_{1,rms}^2 + x_{2,rms}^2\) (if orthogonal)

DC + AC Components

If \(x(t) = A + x_{ac}(t)\) where \(x_{ac}\) has zero mean:

  • Average: \(x_{avg} = A\)

  • RMS: \(x_{rms} = \sqrt{A^2 + x_{ac,rms}^2}\)

Calculation Methods

Step-by-Step Calculation

General Approach

  1. Identify the period \(T\) of the signal

  2. Set up the integral over one complete period

  3. For Average: \(\int_0^T x(t)\,dt\) and divide by \(T\)

  4. For RMS: \(\int_0^T x^2(t)\,dt\), divide by \(T\), then take square root

  5. Simplify using symmetry properties when possible

GATE Strategy

  • Use symmetry to reduce calculation complexity

  • Remember standard values for common waveforms

  • Check units and reasonableness of answer

Symmetry Shortcuts

Even Symmetry: \(x(t) = x(-t)\)

  • Average of odd part = 0

  • Can integrate over half period and multiply by 2

Odd Symmetry: \(x(t) = -x(-t)\)

  • Average value = 0

  • RMS calculation still requires full period

Half-Wave Symmetry: \(x(t) = -x(t + T/2)\)

  • Average value = 0

  • Contains only odd harmonics

GATE Examples

Example 1: Composite Signal

Problem

Find RMS value of \(x(t) = 3 + 4\cos(t) + 2\sin(3t)\)

Solution

  • DC component: \(3\)

  • \(\cos(t)\) component: RMS = \(\dfrac{4}{\sqrt{2}} = 2\sqrt{2}\)

  • \(\sin(3t)\) component: RMS = \(\dfrac{2}{\sqrt{2}} = \sqrt{2}\)

  • Total RMS: \(x_{rms} = \sqrt{3^2 + (2\sqrt{2})^2 + (\sqrt{2})^2}\)

  • \(x_{rms} = \sqrt{9 + 8 + 2} = \sqrt{19}\)

Example 2: Piecewise Function

Problem

Square wave: \(+5V\) for \(0 \leq t < 2\), \(-2V\) for \(2 \leq t < 5\), \(T = 5\)

Solution

Average: \(x_{avg} = \dfrac{1}{5}[5 \times 2 + (-2) \times 3] = \dfrac{10-6}{5} = 0.8V\)

RMS: \(x_{rms}^2 = \dfrac{1}{5}[5^2 \times 2 + (-2)^2 \times 3] = \dfrac{50+12}{5} = 12.4\)

Therefore: \(x_{rms} = \sqrt{12.4} = 3.52V\)

Quick Reference

Formula Summary

Common Waveforms - Quick Reference

Waveform Average RMS Form Factor
Sine/Cosine 0 \(\dfrac{A}{\sqrt{2}}\) \(\infty\)
Square 0 \(A\) \(\infty\)
Triangle 0 \(\dfrac{A}{\sqrt{3}}\) \(\infty\)
Sawtooth \(\dfrac{A}{2}\) \(\dfrac{A}{\sqrt{3}}\) \(\dfrac{2}{\sqrt{3}}\)
Half-wave rect. \(\dfrac{A}{\pi}\) \(\dfrac{A}{2}\) \(\dfrac{\pi}{2}\)
Full-wave rect. \(\dfrac{2A}{\pi}\) \(\dfrac{A}{\sqrt{2}}\) \(\dfrac{\pi}{2\sqrt{2}}\)

Key Takeaways for GATE

Important Points

  • RMS is always \(\geq\) Average (equality only for DC)

  • For power calculations, always use RMS values

  • Symmetry properties can simplify calculations significantly

  • Form factor and crest factor are important for waveform analysis

  • Remember the \(\dfrac{1}{\sqrt{2}}\) factor for sinusoidal RMS

Common GATE Mistakes

  • Forgetting to square the function for RMS calculation

  • Not considering the complete period

  • Mixing up average and RMS in power calculations

  • Incorrect handling of negative values in average calculation