Ohms Law: Ac Circuit Applicability Explored

Ohm's Law, formulated in 1827 by German physicist Georg Ohm, states that the electric current passing through a conductor is directly proportional to the potential difference across the conductor. The law is best demonstrated in a simple DC electric circuit, but it also applies to AC circuits. However, AC circuits involve complex sources and impedances, so the variables used in the calculations aren't always real numbers. Additionally, for AC circuits with components like capacitors or inductors, Ohm's Law does not apply.

Ohm's Law applies to AC circuits

Ohm's Law, which states that the electric current passing through a conductor is directly proportional to the potential difference across the conductor, applies to both direct current (DC) and alternating current (AC) circuits.

In an AC circuit, Ohm's Law can be expressed as:

> $V = IZ$

Where:

• $V$ is the voltage
• $I$ is the current
• $Z$ is the impedance of the circuit

The impedance $Z$ is associated with a phase angle, which is necessary to calculate the average power used by the circuit. This phase angle means that, even though $Z$ is the ratio of the voltage and current peaks, these peaks do not occur at the same time.

Ohm's Law can be used to calculate voltage, current, or resistance in an AC circuit, as long as the values of the other two variables are known. For example, if the current is 0.2 A and the resistance is 1000 ohms, then the voltage can be calculated as:

> $V = 0.2 A * 1000 \Omega = 200 V$

However, it is important to note that Ohm's Law does not apply to AC circuits that include components such as capacitors or inductors. In such cases, the mathematics becomes more complex, climbing into the domain of complex numbers.

AC circuits: Impedance and phasors

Ohm's Law, which relates current, voltage, and impedance, applies to both DC and AC circuits. However, in AC circuits, the concept of impedance becomes more complex and involves the use of phasors to represent the magnitude and phase angle of sinusoidal waveforms or time-varying quantities.

Impedance

Impedance (Z) is defined as the ratio of phasor voltage to phasor current in a circuit. It is a generalisation of the concept of resistance and is given by:

> Z = V/I

In AC circuits, impedance is typically a complex number, consisting of a real part (resistance) and an imaginary part (reactance). The impedance of a circuit depends on the frequency of the AC source and the presence of reactive components such as inductors and capacitors.

For a resistor, the impedance is purely real and is given by:

> Z = R

For an inductor, the impedance is imaginary and is given by:

> Z = jωL

For a capacitor, the impedance is also imaginary but has the opposite sign compared to an inductor:

> Z = -j/ωC

Phasors

Phasors are complex numbers that represent the magnitude and phase angle of sinusoidal waveforms or time-varying quantities. They are used to simplify calculations involving AC circuits by treating the waveforms as rotating vectors in a complex plane. The length of a phasor represents the magnitude of the waveform, while the angle represents the phase.

Phasors can be added, subtracted, multiplied, and divided using standard complex number arithmetic. This allows for the calculation of the total voltage and current in a circuit, taking into account the phase differences between different components.

Phasors are often used in conjunction with impedance to

AC circuits: Capacitors and inductors

AC circuits involve complex sources and impedances, which vary with time or frequency. Ohm's Law still applies to AC circuits, but the values of voltage, current, and resistance are not always real numbers, and can be complex expressions.

Capacitors

Capacitors do not behave the same way as resistors. Resistors allow a flow of electrons through them directly proportional to the voltage drop, while capacitors oppose changes in voltage by drawing or supplying current as they charge or discharge to the new voltage level. The flow of electrons through a capacitor is directly proportional to the rate of change of voltage across the capacitor.

The opposition to voltage change is another form of reactance, but one that is the opposite of the kind exhibited by inductors. The reactance of a capacitor is expressed in ohms and is symbolised by the letter X (or XC to be more specific).

The instantaneous current is zero whenever the instantaneous voltage is at a peak, and the instantaneous current is at its peak wherever the instantaneous voltage is at its maximum change. This results in a voltage wave that is -90° out of phase with the current wave.

Inductors

Inductors oppose changes in current. The energy stored in an inductor is given by the equation:

> \(\mathrm { E } _ { \mathrm { ind } } = \dfrac { 1 } { 2 } \mathrm { LI } ^ { 2 }\)

The rms current through an inductor L is given by a version of Ohm's Law:

> \(\mathrm { I } _ { \mathrm { rms } } = \frac { \mathrm { V } _ { \mathrm { rms } } } { \mathrm { X } _ { \mathrm { L } } }\)

XL is called the inductive reactance, and is given as:

> \(\mathrm { X } _ { \mathrm { L } } = 2 \pi \nu \mathrm { L }

Where ν is the frequency of the AC voltage source in hertz.

AC circuits: High frequencies

At high frequencies, the rules and equations used for low-frequency AC circuits or DC circuits are no longer valid. This is because the wavelength of the signal becomes much smaller and can sometimes be smaller than the wiring on PCBs.

At high frequencies, the current density in wires can no longer be assumed to be constant across the cross-section of a wire. Instead, the current is predominantly in the outer parts of the conductors, in what is known as the "skin effect". This must be taken into account when designing high-frequency circuits.

The rules and equations you are now learning also apply to simpler, lower-frequency circuits. You can use the new things to solve the simpler circuits—you just have to have more information and solve more complicated equations.

Ohm's Law: Power calculation for AC and DC circuits

Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage. This is true for many materials, over a wide range of voltages and currents, and the resistance and conductance of electronic components made from these materials remain constant.

Ohm's Law is true for circuits that contain only resistive elements (no capacitors or inductors), regardless of whether the driving voltage or current is constant (DC) or time-varying (AC). It can be used to calculate the values of circuit components, current levels, voltage supplies, and voltage drops around a circuit.

The formula for Ohm's Law is:

V = I x R

Where:

• V is voltage in Volts
• I is current in Amperes
• R is resistance in Ohms

Power is the rate at which electrical energy is transferred by an electric circuit per unit of time, typically expressed in Watts. The formula for power is:

P = V x I

Where:

• P is power in Watts
• V is voltage in Volts
• I is current in Amperes

Ohm's Law can be used to calculate power in both AC and DC circuits. However, AC circuits involve complex sources and impedances that vary with time or frequency, so the values of voltage, current, and resistance may be complex expressions.

In AC circuits, the analogue to Ohm's Law is:

V = IZ

Where:

• V and I are the rms or effective values of voltage and current
• Z is the impedance of the circuit

The impedance Z is associated with a phase angle, which is necessary to calculate the average power used by the circuit. This phase angle means that while Z is the ratio of the voltage and current peaks, these peaks do not occur at the same time.

Frequently asked questions