Definitions and Concepts
Average Value of Periodic Signals
Average Value (DC Component)
For a periodic signal \(x(t)\) with period \(T\):
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Also called DC component or mean value
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Represents the constant component of the signal
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For symmetric signals about zero: \(x_{avg} = 0\)
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Units: Same as the original signal
RMS Value of Periodic Signals
RMS (Root Mean Square) Value
For a periodic signal \(x(t)\) with period \(T\):
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Also called effective value or quadratic mean
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Represents the equivalent DC value that delivers same power
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Always positive and non-zero for non-zero signals
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Units: Same as the original signal
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Power relation: \(P = \dfrac{x_{rms}^2}{R}\)
Standard Waveforms
Sinusoidal Signals
Sine Wave: \(x(t) = A\sin(\omega t + \phi)\)
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Average Value: \(x_{avg} = 0\)
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RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{2}} = 0.707A\)
Cosine Wave: \(x(t) = A\cos(\omega t + \phi)\)
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Average Value: \(x_{avg} = 0\)
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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 \(T\)
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Average Value: \(x_{avg} = 0\) (symmetric)
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RMS Value: \(x_{rms} = A\)
Asymmetric Square Wave
For positive duration \(t_1\) and negative duration \(t_2 = T - t_1\):
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Average Value: \(x_{avg} = A\dfrac{2t_1 - T}{T}\)
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RMS Value: \(x_{rms} = A\)
Triangular Wave
Symmetric Triangular Wave: Peak \(\pm A\), Period \(T\)
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Average Value: \(x_{avg} = 0\)
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RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{3}} = 0.577A\)
Sawtooth Wave: \(0\) to \(A\), Period \(T\)
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Average Value: \(x_{avg} = \dfrac{A}{2}\)
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RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{3}}\)
Rectified Signals
Half-Wave Rectified Sine
\(x(t) = \begin{cases} A\sin(\omega t) & 0 \leq t \leq \dfrac{T}{2} \\ 0 & \dfrac{T}{2} < t \leq T \end{cases}\)
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Average Value: \(x_{avg} = \dfrac{A}{\pi} = 0.318A\)
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RMS Value: \(x_{rms} = \dfrac{A}{2} = 0.5A\)
Full-Wave Rectified Sine
\(x(t) = A|\sin(\omega t)|\)
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Average Value: \(x_{avg} = \dfrac{2A}{\pi} = 0.637A\)
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RMS Value: \(x_{rms} = \dfrac{A}{\sqrt{2}} = 0.707A\)
Important Properties
Key Properties
Relationship between RMS and Average
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Form Factor: \(k_f = \dfrac{x_{rms}}{x_{avg}}\)
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Crest Factor: \(k_c = \dfrac{x_{peak}}{x_{rms}}\)
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Peak Factor = Crest Factor
Power Relations
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Average Power: \(P_{avg} = \dfrac{x_{rms}^2}{R}\)
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For AC signals: \(P_{avg} = \dfrac{V_{rms} \cdot I_{rms}}{R}\)
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Total Power = DC Power + AC Power
Superposition Properties
Linear Combination
If \(x(t) = x_1(t) + x_2(t)\):
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Average: \(x_{avg} = x_{1,avg} + x_{2,avg}\)
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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:
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Average: \(x_{avg} = A\)
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RMS: \(x_{rms} = \sqrt{A^2 + x_{ac,rms}^2}\)
Calculation Methods
Step-by-Step Calculation
General Approach
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Identify the period \(T\) of the signal
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Set up the integral over one complete period
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For Average: \(\int_0^T x(t)\,dt\) and divide by \(T\)
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For RMS: \(\int_0^T x^2(t)\,dt\), divide by \(T\), then take square root
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Simplify using symmetry properties when possible
GATE Strategy
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Use symmetry to reduce calculation complexity
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Remember standard values for common waveforms
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Check units and reasonableness of answer
Symmetry Shortcuts
Even Symmetry: \(x(t) = x(-t)\)
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Average of odd part = 0
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Can integrate over half period and multiply by 2
Odd Symmetry: \(x(t) = -x(-t)\)
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Average value = 0
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RMS calculation still requires full period
Half-Wave Symmetry: \(x(t) = -x(t + T/2)\)
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Average value = 0
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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
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DC component: \(3\)
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\(\cos(t)\) component: RMS = \(\dfrac{4}{\sqrt{2}} = 2\sqrt{2}\)
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\(\sin(3t)\) component: RMS = \(\dfrac{2}{\sqrt{2}} = \sqrt{2}\)
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Total RMS: \(x_{rms} = \sqrt{3^2 + (2\sqrt{2})^2 + (\sqrt{2})^2}\)
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\(x_{rms} = \sqrt{9 + 8 + 2} = \sqrt{19}\)
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
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RMS is always \(\geq\) Average (equality only for DC)
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For power calculations, always use RMS values
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Symmetry properties can simplify calculations significantly
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Form factor and crest factor are important for waveform analysis
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Remember the \(\dfrac{1}{\sqrt{2}}\) factor for sinusoidal RMS
Common GATE Mistakes
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Forgetting to square the function for RMS calculation
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Not considering the complete period
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Mixing up average and RMS in power calculations
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Incorrect handling of negative values in average calculation