AC Resistance and Impedance

Impedance, measured in Ohms, is effective resistance to current flow around an AC circuit containing resistors and reactances.

In previous lessons, we found that in an AC circuit containing sinusoidal waveforms, voltage and current phasers, together with complex numbers, can be used to represent a complex amount.

We also found that sinusoidal waveforms and functions previously drawn in time-space transformation can be converted into spatial or phaser-space so that phaser diagrams can be created to find this phaser voltage-current relationship.

Now that we know how to represent a voltage or current as a phaser, we can look at this relationship when connected to a single-phase AC source when applied to basic passive circuit elements such as AC Resistance.

Any ideal basic circuit element, such as a resistance, can be mathematically identifiable in terms of voltage and current, and in the tutorial on resistors, we found that the voltage on a pure ohmic resistance is linearly proportional to the current flowing through it. As defined by the Ohm Act. Consider the circuit below.

AC Resistance with a Sinusoidal Feed

ac resistance

When the switch is turned off, an AC voltage, V will be applied to resistance R. This voltage will cause a current to flow as the applied voltage rises and sinusoidally decreases. Since the load is a resistance, the current and voltage will reach both maximum and peak values and fall to zero exactly at the same time, that is, they rise and fall at the same time, and therefore they are said to be "intra-phase".

Then the electric current flowing from an AC resistance changes sinusoidally over time and is represented by the expression I(t) = Im x sin(wt + ε), where Im is the maximum amplitude of the current and is the phase angle. In addition, we can say that for any given current, the maximum or peak voltage along the R terminals flowing through the resistance will be given by the Ohm Act:

ac resistance

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

ac resistance

Therefore, for a fully resistant circuit, the alternating current flowing from the resistance changes in proportion to the voltage applied, following the same sinusoidal model.Since the feed frequency is common for both voltage and current, its phasers will also be common, which causes the current to be "phased" by voltage, ( ε = 0 ).

In other words, there is no phase difference between current and voltage when an AC resistance is used, since when the voltage reaches maximum, minimum and zero values, as shown below, the current will reach maximum, minimum and zero values.

Sinusoidal WaveForms for AC Resistance

ac resistance

This "in-phase" effect can also be demonstrated by a phaser diagram.In the complex area, resistance is a real number, which means that it is not just " j " or imaginary component.Therefore, since the voltage and current are in the same phase, there will be no phase difference ( ε = 0 ) between them, so the vectors of each quantity are drawn over each other along the same reference axis.It is given as a transformation from sinusoidal time area to phaser area.

Phaser Diagram for AC Resistance

ac resistance

Because a phaser represents RMS values of voltage and current quantities, unlike a vector representing peak or maximum values, it is given as the corresponding voltage-current phaser relationship by dividing the peak value of the above time zone expressions by √2.

RMS Relationship

ac resistance

Phase Relationship

ac resistance

This indicates that a pure resistance within an AC circuit produces a relationship between voltage and current phasers in exactly the same way as the same resistance voltage and current relationship within a DC circuit. However, in a DC circuit this relationship is often called Resistance, as defined by the Ohm Act, but in a sinusoidal AC circuit this voltage-current relationship is now called Impedance. In other words, electrical resistance in an AC circuit is called "Impedance".

In both cases, this voltage-current (V-I) relationship is always linear in pure resistance. Therefore, when resistors are used in AC circuits, the term Impedance, symbol Z is often used to express its resistance. Therefore, we can accurately say that it is DC resistance = AC impedance or R = Z for a resistance.

The impedance vector is represented by the letter (Z) for an AC resistance value with the same Ohm ( Ω) units as dc. Then impedance (or AC resistance) can be defined as follows:

AC Impedance

ac resistance

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

Since the phase angle between voltage and current is zero in a fully resistant AC circuit, the power factor must also be zero and is given as follows: cos 0 o = 1.0 , Then the instantaneous power consumed in the resistance is given as follows:

ac resistance

However, since the average power in a resistant or reactive circuit depends on the phase angle and is equal to this ε = 0 in a fully resistant circuit, the power factor is equal to one, so that the average power consumed by an AC resistance can simply be defined using the Ohm Act:

ac resistance

these are the same Ohm Act equations as for DC circuits. The effective power consumed by an AC resistance is then equal to the power consumed by the same resistance in a DC circuit.

Many AC circuits, such as heating elements and lamps, consist only of pure omic resistance and have negligible inductance or capacitance values containing impedance.

In such circuits, as in DC circuit analysis, we can use both the Ohm Act, the Kirchoff Law and simple circuit rules to calculate and find voltage, current, impedance and power.When working with such rules, it is usual to use only RMS values.

AC Resistance Question Sample 1

An electric heating element with 60 Ohm AC resistance is connected to the 240V AC single phase source.Calculate the current drawn from the feed and the power consumed by the heating element.Also draw the corresponding phaser diagram showing the phase relationship between current and voltage.

1. Feed current:

ac resistance

2. The active power consumed by AC resistance is calculated as follows:

ac resistance

3. Since there is no phase difference in a resistant component ( ε = 0 ), the corresponding phaser diagram is given as follows:

ac resistance

AC Resistance Question Sample 2

A sinusoidal voltage source defined as V(t) = 100 x cos(ωt + 30 o) is connected to a pure resistance of 50 Ohms.Determine the impedance and peak value of the current that passes through.Draw the corresponding phaser diagram.

The sinusoidal voltage throughout the resistance will be the same as the feed in a fully resistant circuit.Converting this voltage from a time-spaced expression to a phaser-field expression gives us this:

ac resistance

Enforcing the Ohm Act allows us to:

ac resistance

The corresponding phaser diagram will therefore be:

ac resistance

Impedance Summary

In a pure omic AC Resistance, the current and voltage are both "in-phase" because there is no phase difference between them. The current flowing from the resistance is directly proportional to this linear relationship in an AC circuit called impedance.

Impedance, which is given the letter Z in a pure omic resistance, is a complex number consisting only of a real part (R) and zero imaginary parts (j0) with a real AC resistance value. Therefore, the Ohm Law can be used in circuits containing an AC resistance to calculate these voltages and currents.

In the next tutorial on AC Induction, we will look at the voltage-current relationship of an inductor when a fixed state sinusoidal AC waveform is applied, together with the representation of a phaser diagram for both pure and non-pure inductaces.