AC Resistance with a Sinusoidal Supply

When the switch is closed, an AC voltage, V will be applied to resistor, R. This voltage will cause a current to flow which in turn will rise and fall as the voltage rises and falls. The current and voltage will both reach their maximum or peak values and fall through zero at exactly the same time, i.e. they rise and fall simultaneously and are therefore said to be "in-phase ". Then the electrical current that flows through an AC resistance varies sinusoidally with time and is represented by the expression, I(t) = Im x cos(ωt + θ), where Im is the maximum amplitude of the current and θ is its phase angle. In addition we can also say that for any given current, i  flowing through the resistor the maximum or peak voltage across the terminals of R will be given by Ohm's Law as:

and the instantaneous value of the current, i will be:

So for a purely resistive circuit the AC current flowing through the resistor varies in proportion to the applied voltage across it following the same sinusoidal pattern. As the supply frequency is common to both the voltage and current, their phasors will also be common resulting in the current being "in-phase" with the voltage, ( θ = 0 ). In other words, there is no phase difference between the current and the voltage when using an AC resistance as the current will achieve its maximum, minimum and zero values whenever the voltage reaches its maximum, minimum and zero values as shown below.

Sinusoidal Waveforms for AC Resistance

This effect can also be represented by a phasor diagram. In the complex domain, resistance is a real number only meaning that there is no "j" or imaginary component. Therefore, as the voltage and current are both in-phase with each other, as there is no phase difference ( θ = 0 ), so the vectors of each quantity are drawn super-imposed upon one another along the same reference axis. The transformation from the sinusoidal time-domain into the phasor-domain is given as.

Phasor Diagram for AC Resistance

As a phasor represents the RMS values of the voltage and current quantities unlike a vector which represents the peak or maximum values, dividing the peak value of the time-domain expressions above by √2 the corresponding voltage-current phasor relationship is given as.

RMS Relationship

Phase Relationship

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This shows that a pure resistance within an AC circuit produces a relationship between its voltage and current phasors in exactly the same way as it would relate the same resistors voltage and current relationship within a DC circuit. However, in an sinusoidal AC circuit this voltage-current relationship is now called Impedance while it is commonly called Resistancein a DC circuit as defined by Ohm's Law. In both cases this voltage-current ( V-I ) relationship is always linear in a pure resistance. So when using resistors in AC circuits the term Impedance, symbol Z is the generally used and we can say that DC resistance = AC impedance, R = Z.

The impedance vector is represented by the letter, ( Z ) for an AC resistance value with the units of Ohm's (Ω) the same as for DC. Then Impedance (AC resistance) is defined as:

AC Impedance

Impedance can also be represented by a complex number as it depends upon the frequency of the circuit, ω when reactive components are present. But in the case of a purely resistive circuit this reactive component will always be zero and the general expression for impedance in a purely resistive circuit given as a complex number will be.

Z = R + j0 = R Ω's

Since the phase angle between the voltage and current in a purely resistive AC circuit is zero, the power factor must also be zero and is given as: cos 0^o = 1.0. Then the instantaneous power consumed in the resistor is given by

However, as the average power in a resistive or reactive circuit depends upon the phase angle and in a purely resistive circuit this is equal to θ = 0, the power factor is equal to one so the average power consumed by an AC resistance can be defined simply by using Ohm's Law as:

which are the same Ohm's Law  equations as for DC circuits. Then the effective power consumed by an AC resistance is equal to the power consumed by the same resistor in a DC circuit.

Many AC circuits such as heating elements and lamps consist of a pure ohmic resistance only and have negligible values of inductance or capacitance containing on impedance. In such circuits we can use both Ohm's Law , Kirchoff's Law  as well as simple circuit rules for calculating the voltage, current, impedance and power as in DC circuit analysis. When working with such rules it is usual to use RMS values only.