Ohm's Law And Inductors: A Misunderstood Relationship

Ohm's law states that voltage is equal to the product of current and resistance, or V = IR. However, this law is not always applicable to inductors and capacitors. Inductors and capacitors are known as reactive components and are used in AC circuits. Inductors resist changes in current, while capacitors resist changes in voltage. The relationship between voltage and current for these components is more complex than for resistors, and it is described using the concept of impedance. Impedance is a complex number that can be used to generalise Ohm's law to include capacitors and inductors.

Ohm's Law and Impedance

Ohms Law and Impedance

Ohm's law states that voltage (V) equals current (I) multiplied by resistance (R), or V = IR. This law applies to circuits with only resistors. When storage devices like capacitors and inductors are added to the circuit, the relationship between voltage and current becomes more complex.

Capacitors and Inductors

Capacitors and inductors are storage devices that store energy. Capacitors resist changes in voltage, while inductors resist changes in current.

Impedance

Impedance (Z) provides a direct relationship between voltage and current for resistors, capacitors, and inductors when analysing circuits with phasor voltages or currents. Impedance is a complex number:

> Z = R + jX

Here, R is the real part, representing resistance from resistors, and X is the imaginary part, representing reactance, which comes from the effects of capacitors or inductors.

Phasor Diagrams

Phasor diagrams explain the differences between resistors, capacitors, and inductors, showing whether voltage and current are in phase or out of phase by 90 degrees. For resistors, voltage and current are in phase because an instantaneous change in current corresponds to an instantaneous change in voltage.

For capacitors, voltage doesn't change instantaneously, so even if the current changes instantaneously, the voltage will lag the current. For inductors, current doesn't change instantaneously, so when there's an instantaneous change in voltage, the current lags behind.

AC and DC Circuits

Ohm's law still applies in AC circuits, but the relationships become more complex as AC circuits involve complex sources and impedances that vary with time or frequency.

In DC circuits, Ohm's law is used mostly for analysis, while in AC circuits, you need to consider impedances as well.

Applications

Understanding the relationship between impedance, voltage, and current is important for designing circuits that can accept or reject specific ranges of frequencies for various applications, such as creating fancy Christmas displays with multicoloured lights flashing and dancing to the music.

Capacitors and Inductors

Capacitors resist changes in voltage and store energy in the form of an electric field. The voltage in a capacitor doesn't change instantaneously, and the capacitor functions as a short circuit for alternating current.

Inductors, on the other hand, resist changes in current and store energy in the form of a magnetic field. The current in an inductor doesn't change instantaneously, and the inductor functions as a short circuit for direct current.

The unit of capacitance is the farad, while the unit of inductance is the henry.

When it comes to Ohm's Law, capacitors and inductors behave differently. In a circuit with only resistors, Ohm's Law states that voltage equals current times resistance, or V = IR. However, when storage devices like capacitors and inductors are added to the circuit, the relationship becomes more complex.

For capacitors, the imaginary component of impedance is negative, and the capacitor's impedance decreases as the frequency increases. Capacitors act like an open circuit for a constant voltage source.

For inductors, the imaginary component of impedance is positive, and the inductor's impedance increases as the frequency increases. Inductors act like a short circuit for a constant voltage source.

AC and DC Circuits

Ohm's law states that voltage is equal to the current multiplied by the resistance, or V = IR. However, when storage devices like capacitors and inductors are added to a circuit, the relationship between voltage and current becomes more complex.

DC circuits

In direct current (DC) circuits, electricity flows in a constant direction, similar to the flow of water in pipes. DC circuits provide a constant voltage and current that flows in one direction, and are commonly used for lower-power items like computers and other devices that use batteries as their power source.

AC circuits

Alternating current (AC) circuits are a bit more complex. In these circuits, the direction of the current and voltage periodically change, switching between positive and negative. AC circuits are used for high-power and long-distance energy transmission, as they can transmit a lot of power at a lower current, reducing power loss during transmission.

Ohm's Law and AC/DC circuits

Ohm's law can be generalized and applied to AC circuits by using the concept of impedance, which is a complex number that takes into account both the resistance and the reactance of the circuit. Impedance provides a direct relationship between voltage and current for resistors, capacitors, and inductors when analyzing circuits with phasor voltages or currents.

In AC circuits, capacitors and inductors can be treated as open and closed circuits, respectively, and their impact on the circuit can be ignored. In DC circuits, capacitors and inductors behave differently and cannot be ignored. Capacitors resist changes in voltage, while inductors resist changes in current.

In summary, Ohm's law can be applied to both AC and DC circuits, but the way it is applied differs due to the different behaviours of capacitors and inductors in these circuits.

Phase Difference

When inductors are involved in an AC circuit, the current and voltage do not peak simultaneously. The phase difference, or the fraction of a period difference between the peaks expressed in degrees, is always less than or equal to 90 degrees. The phase difference is typically depicted in a phasor diagram.

In an AC circuit with an inductor, the voltage and current are "out-of-phase", meaning they do not cross zero or reach their peaks and valleys simultaneously. The current will lag the voltage by one-quarter of a cycle, or 90 degrees, in a circuit with an inductive component. This phase shift occurs because the inductive reactance changes with the changing current.

In a resistive circuit, the current and voltage are in phase. The phasor description of a resistor is given by:

> VR = IRR

In contrast, the phasor description of an inductor is:

> VL = ILjωL

Here, the inductor voltage leads the current by 90 degrees due to the Euler formula:

> VL = ILjωL

The imaginary component is positive for inductors. As the radian frequency (ω) increases, the inductor's impedance also increases.

Complex Impedance

In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit. Impedance is expressed in ohms (Ω). It is a complex number that can be represented in polar form as:

> Z = |Z|∠θ

Here, |Z| represents the ratio of the voltage difference amplitude to the current amplitude, and θ gives the phase difference between voltage and current.

The impedance of a circuit element can be defined as the ratio of the phasor voltage across the element to the phasor current through the element. This is identical to the definition from Ohm's law:

> V = IZ

The impedance of an ideal resistor is purely real and is called resistive impedance:

> ZR = R

In this case, the voltage and current waveforms are proportional and in phase.

Ideal inductors and capacitors have a purely imaginary reactive impedance:

> ZL = jωL

> ZC = 1/(jωC)

In both cases, for an applied sinusoidal voltage, the resulting current is also sinusoidal, but in quadrature, 90 degrees out of phase with the voltage. However, the phases have opposite signs: in an inductor, the current is lagging, while in a capacitor, the current is leading.

The impedance of a two-terminal circuit element is the ratio of the complex representation of the sinusoidal voltage between its terminals to the complex representation of the current flowing through it. It depends on the frequency of the sinusoidal voltage.

The handling of the impedance of an AC circuit with multiple components can be simplified by using complex exponential functions. If a complex plane is used with resistance along the real axis, then the reactances of the capacitor and inductor are treated as imaginary numbers.

The phase angle of reactance, either inductive or capacitive, is always 90 degrees out-of-phase with the resistive component. Thus, the resistive and reactive values cannot be simply added together arithmetically to give the total impedance value. Instead, we can plot the values on a two-dimensional graph with the x-axis being the resistive or "real axis", and the y-axis being the reactive or "imaginary axis".

> ZT = Z1 + Z2 + Z3 + ...

For a DC circuit, impedance reduces to resistance, and Ohm's law applies. However, in an AC circuit, impedance is the result of both the circuit's resistive (R) and reactive (X) components.

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