Answer to Question #115215 in Electric Circuits for Vidurjah Perananthan

Question #115215
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?
1
Expert's answer
2020-05-12T09:38:04-0400

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 = vpeak2\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=V2Rt=i2RtVit = \frac{V^2}{R}t = i^2Rt

Calculations


1). Peak value (peak/maximum voltage) of the given waveform = Umax = 25V (@ sin100ω\omegat = 1)

` Therefore,

Effective value=25V2=17.678V\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

=Vrms2Rt=(252V)210Ω×60s=1875j\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}


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