GATE EE

Signals: RMS and Average Values GATE EE Notes

Lecture Notes

SEC 01

Definitions and Concepts

1Average Value of Periodic Signals

1Average 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

1RMS Value of Periodic Signals

1RMS (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: \(P = \dfrac{x_{rms}^2}{R}\)

SEC 02

Standard Waveforms

1Sinusoidal Signals

1Sine Wave: \(x(t) = A\sin(\omega t + \phi)\)

Average Value: \(x_{avg} = 0\)
RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{2}} = 0.707A\)

1Cosine Wave: \(x(t) = A\cos(\omega t + \phi)\)

Average Value: \(x_{avg} = 0\)
RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{2}} = 0.707A\)

1GATE Tip

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

1Square Wave

1Square Wave: Amplitude \(\pm A\), Period \(T\)

Average Value: \(x_{avg} = 0\) (symmetric)
RMS Value: \(x_{rms} = A\)

1Asymmetric 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: \(x_{rms} = A\)

1Triangular Wave

1Symmetric Triangular Wave: Peak \(\pm A\), Period \(T\)

Average Value: \(x_{avg} = 0\)
RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{3}} = 0.577A\)

1Sawtooth Wave: \(0\) to \(A\), Period \(T\)

Average Value: \(x_{avg} = \dfrac{A}{2}\)
RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{3}}\)

1Rectified Signals

1Half-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: \(x_{rms} = \dfrac{A}{2} = 0.5A\)

1Full-Wave Rectified Sine

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

Average Value: \(x_{avg} = \dfrac{2A}{\pi} = 0.637A\)
RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{2}} = 0.707A\)

SEC 03

Important Properties

1Key Properties

1Relationship 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

1Power 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

1Superposition Properties

1Linear 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)

1DC + AC Components

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

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

SEC 04

Calculation Methods

1Step-by-Step Calculation

1General Approach

Identify the period \(T\) of the signal
Set up the integral over one complete period
For Average: \(\int_0^T x(t)\,dt\) and divide by \(T\)
For RMS: \(\int_0^T x^2(t)\,dt\), divide by \(T\), then take square root
Simplify using symmetry properties when possible

1GATE Strategy

Use symmetry to reduce calculation complexity
Remember standard values for common waveforms
Check units and reasonableness of answer

1Symmetry Shortcuts

1Even Symmetry: \(x(t) = x(-t)\)

Average of odd part = 0
Can integrate over half period and multiply by 2

1Odd Symmetry: \(x(t) = -x(-t)\)

Average value = 0
RMS calculation still requires full period

1Half-Wave Symmetry: \(x(t) = -x(t + T/2)\)

Average value = 0
Contains only odd harmonics

SEC 05

GATE Examples

1Example 1: Composite Signal

1Problem

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

1Solution

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: \(x_{rms} = \sqrt{3^2 + (2\sqrt{2})^2 + (\sqrt{2})^2}\)
\(x_{rms} = \sqrt{9 + 8 + 2} = \sqrt{19}\)

1Example 2: Piecewise Function

1Problem

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

1Solution

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

SEC 06

Quick Reference

1Formula Summary

1Common 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.	\(\dfrac{2A}{\pi}\)	\(\dfrac{A}{\sqrt{2}}\)	\(\dfrac{\pi}{2\sqrt{2}}\)

1Key Takeaways for GATE

1Important 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

1Common 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