Ohm's Law in AC circuits with pure resistance maintains the same simple linear relationship as in DC circuits: voltage equals current times resistance (V = IR). The key insight is that pure resistors do not introduce any phase shift between voltage and current — they remain perfectly in phase at all frequencies. This means both voltage and current reach their maximum and minimum values simultaneously, and both cross zero at the same instants. The resistance value R remains constant regardless of AC frequency, making resistive circuits the simplest case in AC analysis. This fundamental relationship applies whether using RMS values for power calculations or instantaneous values for time-domain analysis.
Physically, pure resistance in AC circuits represents the direct conversion of electrical energy into heat without any energy storage. Unlike reactive components (inductors and capacitors), resistors respond instantaneously to voltage changes. The electrons flowing through the resistive material experience friction-like collisions that convert kinetic energy to thermal energy, regardless of whether the current is DC or AC. This instantaneous response means no phase shift occurs. The frequency independence arises because the microscopic resistance mechanism (electron-lattice interactions) operates much faster than typical AC frequencies.
Ohm's Law for an AC circuit containing only a pure resistor describes a simple, linear relationship between voltage, current, and resistance, identical to its DC counterpart. The key physical property is that the resistor does not introduce any time delay, or phase shift, between the voltage across it and the current flowing through it; they remain perfectly synchronized.
| Property | Details |
|---|---|
| Nature | Voltage (V), Current (I), and Resistance (R) are treated as scalar quantities for magnitude calculations (e.g., RMS or peak values). |
| SI Units | Voltage: Volt (V), Current: Ampere (A), Resistance: Ohm (Ω) |
| Magnitude Relationship | The RMS voltage is directly proportional to the RMS current: V_rms = I_rms * R. The same holds for peak values: V_peak = I_peak * R. |
| Phase Relationship | In a purely resistive AC circuit, the voltage and current are always in phase. The phase angle difference between them is 0 degrees. |
| Conservation Law | Represents conservation of energy, where electrical energy is dissipated as heat in the resistor. The average power dissipated is P = (I_rms)^2 * R. |
| Dimensional Formula | The dimensional formula for Resistance (R) is [M L^2 T^-3 I^-2]. |
| Symbol | Quantity | SI Unit | Description |
|---|---|---|---|
| \(u(t), v(t)\) | Instantaneous Voltage | Volt (V) | The voltage at a specific moment in time. |
| \(i(t)\) | Instantaneous Current | Ampere (A) | The current at a specific moment in time. |
| \(U_{\text{rms}}, V_{\text{rms}}\) | RMS Voltage | Volt (V) | The Root Mean Square or effective value of the AC voltage. |
| \(I_{\text{rms}}\) | RMS Current | Ampere (A) | The Root Mean Square or effective value of the AC current. |
| \(U_0, V_0\) | Peak Voltage | Volt (V) | The maximum amplitude of the AC voltage waveform. |
| \(I_0\) | Peak Current | Ampere (A) | The maximum amplitude of the AC current waveform. |
| \(R\) | Resistance | Ohm (Ω) | The opposition to current flow, independent of frequency. |
| \(P_{\text{avg}}, P\) | Average Power | Watt (W) | The average rate of energy conversion to heat over a full cycle. |
| \(p(t)\) | Instantaneous Power | Watt (W) | The power at a specific moment in time; always positive for a resistor. |
| \(\phi\) | Phase Angle | Radian (rad) or Degree (°) | The phase difference between voltage and current. For a pure resistor, \(\phi = 0\). |
| \(\cos(\phi)\) | Power Factor | Dimensionless | The ratio of real power to apparent power. For a pure resistor, \(\cos(\phi) = 1\). |
| \(Z\) | Impedance | Ohm (Ω) | The total opposition to AC current. For a pure resistor, \(Z = R\). |
We can prove that the voltage and current are in phase in a purely resistive AC circuit by starting with the general form of an AC voltage and applying Ohm's law.
1. Assume a sinusoidal voltage source is applied across a resistor R. The instantaneous voltage \(u(t)\) can be described by:
where \(U_0\) is the peak voltage, \(\omega\) is the angular frequency, and \(\phi_u\) is the initial phase angle of the voltage.
2. According to Ohm's law for instantaneous values, the current \(i(t)\) is directly proportional to the voltage \(u(t)\) at every instant:
3. We can define the peak current \(I_0\) as \(U_0 / R\). Substituting this into the equation gives the expression for instantaneous current:
4. The general form for the current is \(i(t) = I_0 \cos(\omega t + \phi_i)\). By comparing this with our derived expression, we see that the phase angle of the current, \(\phi_i\), is identical to the phase angle of the voltage, \(\phi_u\).
5. The phase difference, \(\phi\), between voltage and current is defined as \(\phi = \phi_u - \phi_i\). Therefore:
This proves that for a purely resistive AC circuit, the voltage and current waveforms are perfectly in phase.
While the fundamental law V = IR remains constant, its application in AC circuits can be classified based on the type of values being used to represent the time-varying voltage and current.
| Type / Case | Description | When to Use |
|---|---|---|
| Instantaneous Form | Relates the voltage v(t) and current i(t) at any specific moment in time. For example, if i(t) = I_peak * sin(ωt), then v(t) = (I_peak * R) * sin(ωt). | Useful for analyzing waveform behavior and understanding the in-phase relationship at any point in the cycle. |
| Peak Value Form | Relates the maximum amplitude of the voltage and current waveforms: V_peak = I_peak * R. | Used in applications where the maximum voltage or current is a critical design parameter, such as determining component ratings to prevent damage. |
| RMS (Root Mean Square) Form | Relates the effective values of AC voltage and current: V_rms = I_rms * R. RMS values provide the equivalent DC values for power dissipation. | This is the most common form for general AC circuit analysis, power calculations, and when using standard AC measuring instruments (voltmeters, ammeters). |
Heating Elements: Devices like electric stoves, water heaters, and space heaters use resistive elements to efficiently convert electrical energy into thermal energy. Their behavior is almost purely resistive, with a unity power factor.
Lighting Systems: Incandescent and halogen bulbs operate by passing current through a tungsten filament, which acts as a resistor. The heat generated causes the filament to glow, producing light.
Industrial Processes: High-power resistive heating is used in electric furnaces for melting metals, glass manufacturing, and in welding applications where precise and intense heat is required.
Electronic Circuits: Resistors are fundamental components used for current limiting, voltage division, signal conditioning, and as pull-up/pull-down resistors in digital logic circuits. Their predictable, frequency-independent behavior is crucial for circuit design.
Electric Heaters
Space heaters, electric stoves, and toasters are common household appliances that function as nearly pure resistive loads. When connected to an AC outlet, they draw a current that is in phase with the voltage, and their primary purpose is to dissipate this electrical energy as heat according to the formula \(P = I^2R\).
Incandescent Light Bulbs
Though largely replaced by newer technologies, the classic incandescent bulb is a perfect example of a resistive load. A thin tungsten filament resists the flow of AC current, heating up to the point of incandescence and emitting light. The brightness can be controlled by varying the voltage with a dimmer, which directly alters the power dissipated (\(P = V^2/R\)).
Volume Control Knobs
In older audio equipment, a variable resistor (potentiometer) is used as a voltage divider to control volume. As you turn the knob, you change the resistance in the path of the audio signal (a complex AC waveform), which attenuates the voltage delivered to the amplifier, thereby adjusting the sound level.
At high frequencies (typically in the MHz to GHz range), a physical resistor begins to behave like an RLC circuit due to:
The actual impedance (Z) becomes frequency-dependent:
| Quantity | Symbol | SI Unit | Dimensional Formula |
|---|---|---|---|
| Voltage | \(V, U\) | Volt (V) | \([M L^2 T^{-3} I^{-1}]\) |
| Current | \(I\) | Ampere (A) | \([I]\) |
| Resistance | \(R\) | Ohm (Ω) | \([M L^2 T^{-3} I^{-2}]\) |
| Impedance | \(Z\) | Ohm (Ω) | \([M L^2 T^{-3} I^{-2}]\) |
| Power | \(P\) | Watt (W) | \([M L^2 T^{-3}]\) |
| Frequency | \(f\) | Hertz (Hz) | \([T^{-1}]\) |
| Angular Frequency | \(\omega\) | radians/second (rad/s) | \([T^{-1}]\) |
| Phase Angle | \(\phi\) | Radian (rad) | Dimensionless |
The formula is V = IR, which is identical in form to Ohm's Law for DC circuits. It describes the direct, linear relationship between the AC voltage (V) across a resistor and the AC current (I) flowing through it. This relationship holds for instantaneous, peak, and RMS values, as long as they are used consistently.
In the formula V = IR, 'V' represents the AC voltage in Volts (V), 'I' is the AC current in Amperes (A), and 'R' is the resistance in Ohms (Ω). For AC circuits, V and I are typically expressed as RMS (Root Mean Square) values, which represent the effective DC equivalent for power calculations.
In a purely resistive AC circuit, the voltage and current are perfectly in phase, meaning there is a 0° phase angle between them. This simplifies the analysis greatly, as we do not need to use complex numbers or phasors to account for phase shifts, which are present in circuits with inductors or capacitors. The simple scalar equation V = IR is therefore sufficient.
A frequent error is confusing peak values (V₀, I₀) with RMS values (V_rms, I_rms) when calculating average power. Students might incorrectly use P = V₀ * I₀. The correct formula for average power dissipated by the resistor is P = V_rms * I_rms, which is equivalent to (V₀ * I₀) / 2.
A classic example is an incandescent light bulb or an electric heating element in a toaster or space heater. These devices work by passing an AC current through a high-resistance filament, converting electrical energy into heat and light. Their behavior is dominated by resistance, making V = IR a very accurate model for their operation.
Impedance (Z) is the total opposition to current flow in an AC circuit, encompassing resistance and reactance. For a purely resistive circuit, the reactance from inductors and capacitors is zero. Therefore, the impedance Z is equal to the resistance R (Z = R), which is why the simple Ohm's Law formula V = IR can be used instead of the more general V = IZ.