Average Value (DC Component)

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

RMS (Root Mean Square) Value

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

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

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: \(x_{rms} = A\)

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

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

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

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

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

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

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

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

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

Common Waveforms - Quick Reference

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