Question

An alternating voltage u (V) across a resistor with resistance 10.0  varies with time

t (s) according to equation u = 25.0.sin 100t.

a) Determine the effective value of the voltage.

b) How much energy is converted into the resistor for 1 minute?

Answer 1

Explanation

• Effective value of a sinusoidal waveform (specially with voltage & current) is considered as the RMS value which when applied to a electrical load, it produces the same amount of power.
• When the peak (maximum) value of the sinusoidal waveform (consider voltage waveform in this case)is known the RMS can be calculated by RMS = $\large\frac{v_{peak}}{\sqrt{2}}$
• When a voltage is supplied only to a resistor, supplied energy = lost/dissipated energy which is calculated by E = $Vit = \frac{V^2}{R}t = i^2Rt$

Calculations

1). Peak value (peak/maximum voltage) of the given waveform = U_max = 25V (@ sin100$\omega$t = 1)

` Therefore,

$\qquad\qquad \begin{aligned} \small \text{Effective value} &= \small \frac{25V}{\sqrt{2}}\\ &= \small \bold{17.678V} \end{aligned}$

2). Energy converted into the resistor

$\qquad\qquad \begin{aligned} &= \small \frac{V^2_{rms}}{R}t\\ &= \small \frac{\big(\frac{25}{\sqrt{2}}V\big)^2}{10\Omega}\times60s\\ &= \small \bold{1875\,j} \end{aligned}$