Question 3: A sinusoidal alternating voltage is of angular frequency ω is connected across a resistor R. Find mathematical expressions for instantaneous voltage, instantaneous current and average power dissipated per cycle of the applied voltage.
Consider a circuit in which a resistor R is connected across a source of sinusoidal voltage.
The instantaneous voltage across R is given by
Here Vm = peak or maximum voltage, ω = angular velocity
Similarly, the accompanied sinusoidal current is
Equations (i) and (ii) are expressions for instantaneous voltage and instantaneous current, respectively.
Both equations indicate the applied voltage and current are in phase with one another.
The average power dissipated in resistor R is
Put the values from equations (i) and (ii)
Since Im and Vm are constants, therefore,
Now the equation can be written as
Here the rms values of voltage and current are substituted.
Power curve of a pure resistive circuit is shown below.
It shows the power is always positive except for the point when it touches the x-axis. At these points both current and voltage are also zero.