KAHIBARO
Table of Contents
Current and voltage for a pure resistor
In an AC circuit containing only a resistor, the relationship between voltage and current is especially simple. The resistor obeys Ohm's law at every instant in time. If the applied voltage changes with time, the current changes in exactly the same way.
If the voltage across a resistor is
$$
v(t) = V_0 \sin(\omega t),
$$
then the current through the resistor is
$$
i(t) = \frac{v(t)}{R} = \frac{V_0}{R}\sin(\omega t).
$$
This means the current is also sinusoidal, with amplitude
$$
I_0 = \frac{V_0}{R}.
$$
The voltage and current reach their maximum values at the same time, cross zero at the same time, and reverse direction at the same time. We say that they are in phase.
For a pure resistor in an AC circuit, voltage and current are in phase.
$$
v(t) = i(t)R
$$
If
$$
v(t) = V_0 \sin(\omega t),
$$
then
$$
i(t) = \frac{V_0}{R}\sin(\omega t)
$$
Phase relationship
Phase tells us how two oscillating quantities are shifted relative to each other in time. For a resistor, there is no phase shift between current and voltage.
So if the voltage is at a positive maximum, the current is also at a positive maximum. If the voltage is zero, the current is zero.
This is different from capacitors and inductors, where current and voltage are not in phase. Here, the resistor gives the simplest AC behavior.
Instantaneous power in a resistor
The instantaneous power delivered to the resistor is
$$
p(t) = v(t)i(t).
$$
Using
$$
v(t) = V_0\sin(\omega t), \qquad i(t) = I_0\sin(\omega t),
$$
we get
$$
p(t) = V_0 I_0 \sin^2(\omega t).
$$
Since $\sin^2(\omega t)$ is never negative, the power is always positive or zero. This means the resistor always absorbs energy, converting electrical energy into thermal energy.
Using the identity
$$
\sin^2(\omega t) = \frac{1 - \cos(2\omega t)}{2},
$$
the power can also be written as
$$
p(t) = \frac{V_0 I_0}{2}\left(1 - \cos(2\omega t)\right).
$$
So the power oscillates, but it never becomes negative.
A pure resistor does not store energy and return it later. It continuously dissipates energy as heat.
$$
p(t) = i^2(t)R = \frac{v^2(t)}{R}
$$
Average power
Because AC voltage and current change with time, the instantaneous power is not constant. In practice, we often want the average power over one complete cycle.
For a resistor,
$$
P_{\text{avg}} = \frac{1}{2}V_0 I_0.
$$
Using RMS values, this becomes
$$
P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}}.
$$
Since for a resistor $V$ and $I$ are in phase, this formula is especially direct.
If we also use Ohm's law in RMS form,
$$
V_{\text{rms}} = I_{\text{rms}}R,
$$
then the average power can be written in the familiar forms
$$
P_{\text{avg}} = I_{\text{rms}}^2 R = \frac{V_{\text{rms}}^2}{R}.
$$
For a pure resistor in AC,
$$
P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} = I_{\text{rms}}^2 R = \frac{V_{\text{rms}}^2}{R}
$$
These are the AC versions of the usual power formulas for resistors.
RMS form of Ohm's law
In AC circuits, RMS values are used because they give the equivalent DC values for heating and power calculations. For a pure resistor, Ohm's law works with RMS values exactly as expected:
$$
V_{\text{rms}} = I_{\text{rms}}R.
$$
So if you know any two of resistance, RMS voltage, and RMS current, you can find the third.
Example
Suppose a resistor of $R = 100\,\Omega$ is connected to an AC source with
$$
V_{\text{rms}} = 120\,\text{V}.
$$
Then the RMS current is
$$
I_{\text{rms}} = \frac{V_{\text{rms}}}{R} = \frac{120}{100} = 1.2\,\text{A}.
$$
The average power dissipated is
$$
P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} = 120 \times 1.2 = 144\,\text{W}.
$$
The same answer comes from
$$
P_{\text{avg}} = \frac{V_{\text{rms}}^2}{R} = \frac{120^2}{100} = 144\,\text{W}.
$$
Summary table
| Quantity | Pure resistor result |
|---|---|
| Instantaneous relation | $v(t) = i(t)R$ |
| Phase difference | $0$ |
| Current amplitude | $I_0 = V_0/R$ |
| RMS Ohm's law | $V_{\text{rms}} = I_{\text{rms}}R$ |
| Average power | $P_{\text{avg}} = V_{\text{rms}}I_{\text{rms}}$ |
| Power form | $P_{\text{avg}} = I_{\text{rms}}^2R = V_{\text{rms}}^2/R$ |
| Energy behavior | Dissipated as heat |
Waveform picture
The graph below shows voltage and current in phase for a resistor.
Physical meaning
A resistor opposes current flow, but it does not delay the current relative to the voltage. Its effect is to set the size of the current, not to shift its timing. In AC circuits, this makes the resistor the simplest circuit element to analyze.