Q12Basic Electrical Engineering

Question

Q.2 Derive expression for RMS and average value of sinusoidal wave.

Answer 1

In advanced alternating current (AC) circuit analysis, accurately quantifying the exact mathematical magnitude of a continuously time-varying sinusoidal voltage or current waveform is strictly necessary for calculating physical power dissipation. A standard sinusoidal voltage is explicitly mathematically defined by the continuous trigonometric equation $v (t) = V_{m} sin (ω t)$ , where $V_{m}$ represents the absolute peak physical amplitude and $ω$ represents the exact angular frequency. The two most critical statistical parameters used in engineering are the mathematical Average Value and the Root Mean Square (RMS) value.

Derivation of the Average Value

The absolute average value of a pure, perfectly symmetrical sinusoidal wave over one complete $36 0^{\circ}$ geometric cycle ( $0$ to $2 π$ ) is identically zero, because the positive physical area rigorously cancels the negative physical area. Therefore, to obtain a mathematically meaningful metric, the average value is universally defined and systematically calculated explicitly over exactly one positive half-cycle ( $0$ to $π$ radians).

The fundamental mathematical integral formulation for the average continuous value over a specified angular period is strictly defined as:

V_{a v g} = \frac{1}{π} \int_{0}^{π} v (θ) d θ = \frac{1}{π} \int_{0}^{π} V_{m} sin (θ) d θ

We rigorously factor out the constant peak amplitude $V_{m}$ and analytically integrate the fundamental trigonometric sine function precisely with respect to the angular variable $θ$ :

V_{a v g} = \frac{V _{m}}{π} [- cos (θ)]_{0}^{π}

We meticulously evaluate this exact definite integral by strictly substituting the upper and lower mathematical boundaries:

V_{a v g} = \frac{V _{m}}{π} [- cos (π) - (- cos (0))] = \frac{V _{m}}{π} [- (- 1) - (- 1)] = \frac{V _{m}}{π} [1 + 1]

V_{a v g} = \frac{2 V _{m}}{π} \approx 0.637 V_{m}

Derivation of the Root Mean Square (RMS) Value

The Root Mean Square (RMS) or effective value is fundamentally much more significant in electrical engineering because it rigorously represents the exact physical direct current (DC) magnitude that systematically produces the identical thermal heating effect within a resistor. The exact definition rigorously mandates calculating the physical mean of the mathematical square of the entire function over a complete, full $2 π$ geometric cycle, and subsequently extracting the principal square root.

V_{r m s} = \frac{1}{2 π} \int_{0}^{2 π} [V_{m} sin (θ)]^{2} d θ

We must meticulously square the entire inner trigonometric function and rigorously pull the constant scalar $V_{m}^{2}$ completely outside the definite integral operation:

V_{r m s} = \frac{V _{m}^{2}}{2 π} \int_{0}^{2 π} sin^{2} (θ) d θ

To correctly analytically integrate the squared sine function, we must absolutely utilize the fundamental trigonometric double-angle geometric identity: $sin^{2} (θ) = \frac{1 - c o s ( 2 θ )}{2}$ .

V_{r m s} = \frac{V _{m}^{2}}{2 π} \int_{0}^{2 π} (\frac{1 - cos ( 2 θ )}{2}) d θ = \frac{V _{m}^{2}}{4 π} [\int_{0}^{2 π} 1 d θ - \int_{0}^{2 π} cos (2 θ) d θ]

We systematically evaluate both exact independent integrals. The fundamental integration of a pure full-cycle harmonic cosine wave is geometrically and analytically exactly zero. The remaining constant integral evaluates perfectly to $2 π$ :

V_{r m s} = \frac{V _{m}^{2}}{4 π} [[θ]_{0}^{2 π} - 0] = \frac{V _{m}^{2}}{4 π} [2 π]

We rigidly simplify the resulting algebraic fraction entirely to explicitly yield the final, globally verified mathematical formulation:

V_{r m s} = \frac{V _{m}^{2}}{2} = \frac{V _{m}}{2} \approx 0.707 V_{m}

Q12Basic Electrical Engineering

Question

Answer

Derivation of the Average Value

Derivation of the Root Mean Square (RMS) Value