Resistors (R) in AC
Resistors (R) in AC
Resistors are one of the fundamental components in both direct current (DC) and alternating current (AC) circuits. In AC circuits, resistors behave differently compared to inductors and capacitors, which have reactance that depends on frequency. The behavior of resistors in AC circuits is governed by Ohm's law, just as in DC circuits, but with consideration for the time-varying nature of AC.
Ohm's Law in AC Circuits
Ohm's law states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. In AC circuits, this relationship still holds, but voltage and current are time-dependent and can be represented as sinusoidal functions or phasors.
The formula for Ohm's law in AC circuits is:
[ V(t) = I(t) \cdot R ]
where:
- ( V(t) ) is the instantaneous voltage across the resistor
- ( I(t) ) is the instantaneous current through the resistor
- ( R ) is the resistance
In phasor form, Ohm's law is expressed as:
[ \tilde{V} = \tilde{I} \cdot R ]
where:
- ( \tilde{V} ) is the phasor representation of voltage
- ( \tilde{I} ) is the phasor representation of current
Power in Resistors
The power consumed by a resistor in an AC circuit is given by:
[ P(t) = V(t) \cdot I(t) ]
For a purely resistive load, the voltage and current are in phase, meaning their waveforms peak at the same time. Therefore, the power is always positive, indicating that the resistor is continuously dissipating energy as heat.
The average power over a cycle is given by:
[ P_{avg} = I_{rms}^2 \cdot R ]
where:
- ( I_{rms} ) is the root mean square (RMS) value of the current
Impedance and Phase Angle
In a purely resistive AC circuit, the impedance (Z) is equal to the resistance (R), and there is no phase difference between the voltage and current. The impedance is given by:
[ Z = R ]
The phase angle (( \phi )) in a purely resistive circuit is:
[ \phi = 0^\circ ]
This means that the voltage and current waveforms are aligned in time.
Differences Between Resistors in DC and AC
Here is a table summarizing the differences and important points regarding resistors in DC and AC circuits:
| Aspect | DC Circuit | AC Circuit |
|---|---|---|
| Voltage & Current | Constant and unidirectional | Time-varying and sinusoidal |
| Ohm's Law | ( V = I \cdot R ) | ( V(t) = I(t) \cdot R ) or ( \tilde{V} = \tilde{I} \cdot R ) |
| Power | ( P = I^2 \cdot R ) | ( P(t) = V(t) \cdot I(t) ), ( P_{avg} = I_{rms}^2 \cdot R ) |
| Impedance | ( Z = R ) (resistance only) | ( Z = R ) (still resistance only) |
| Phase Angle | Not applicable | ( \phi = 0^\circ ) (voltage and current in phase) |
| Frequency Dependence | Not applicable | No dependence (resistance is constant) |
Examples
Example 1: Ohm's Law in AC Circuit
Consider a resistor with a resistance of 10 ohms connected to an AC source with a peak voltage of 100V at 50Hz. The peak current can be calculated using Ohm's law:
[ I_{peak} = \frac{V_{peak}}{R} = \frac{100V}{10\Omega} = 10A ]
The RMS values of the voltage and current are:
[ V_{rms} = \frac{V_{peak}}{\sqrt{2}} = \frac{100V}{\sqrt{2}} \approx 70.7V ] [ I_{rms} = \frac{I_{peak}}{\sqrt{2}} = \frac{10A}{\sqrt{2}} \approx 7.07A ]
Example 2: Power Dissipation in AC Circuit
Using the resistor and RMS values from Example 1, the average power dissipated by the resistor is:
[ P_{avg} = I_{rms}^2 \cdot R = (7.07A)^2 \cdot 10\Omega \approx 500W ]
This power is dissipated as heat, and there is no reactive power since it is a purely resistive load.
In summary, resistors in AC circuits behave similarly to those in DC circuits with respect to Ohm's law and power dissipation. However, the time-varying nature of AC requires the use of RMS values for practical calculations, and the concept of phase angles becomes relevant even though it is zero for pure resistors.