# What are Resistors in AC Circuits : Working & Its Examples

The flow of current throughout a circuit can be done in two methods. If the current flows in a single direction then it is called DC whereas the current flows alternatively then it is known as alternating current or AC. AC circuit or alternating current is a simple circuit that can be powered through an alternating source, either voltage or current. There are different AC circuits like AC circuits with resistance, capacitance, inductance, and the combination of RC, RL, LC, and RLC. This article discusses an overview of resistors in ac circuits.

## What are Resistors in AC Circuits?

Resistors are two-terminal passive electrical components that implement electrical resistance as a circuit element. These components restrict the flow of current and also decrease the voltage levels within circuits. In both AC & DC circuits, the resistor’s resistance value is similar irrespective of the frequency of the AC voltage supply. In AC supply, the change in current flow direction does not affect the behavior of resistors. So the current within the resistor will increase & drop based on the voltage because it increases & drops.

In AC circuits, the resistance mainly depends on the phase angle and supply frequency or phase difference φ. Thus, the term ‘Impedance’ is used within AC circuits to indicate the resistance because it possesses both phase & magnitude as compared to the resistance within DC circuits where it has only magnitude.

### The Resistor in AC Circuit

Consider the following AC circuit that includes a pure resistance ‘R’ that is simply connected across a voltage source.

The alternating voltage can be given through the following equation

V = Vm sinωt……….(1)

Because of this voltage, an AC (alternating current) in the circuit ‘i’ will supply. The whole voltage applied can appear across the resistance simply. So,

V = iR

From the above equation i = v/R = Vm sinωt/R = Imsinwt ……(2)

Where, Im=Vm/R

From the above two equations, we can observe that the flow of current throughout a pure resistor is within phase through voltage across the resistor.

### Power Dissipated by a Resistor in an AC Circuit

Let us consider the following AC circuit with a resistor. The voltage supply like V (t) = VMax sin ωt is connected to a resistor R. The instant voltage across the resistor ‘R’ will be VR. The ‘IR’ in the circuit is an instantaneous current that is flowing throughout the resistor.

When the above AC circuit is pure resistive within nature, then Ohm’s principles can be applied.

The voltage supply across the ‘R” at an instant can be given as

VR = VMax sin ωt.

Likewise, the flow of current throughout the ‘R” can be determined through Ohm’s law as

IR = VR / R

We know that VR = VMax sin ωt.

Substitute this VR equation in the above IR formula, then we can get.

IR = (VMax * sin ωt) / R

Here, the VMax/R value is the maximum current within the circuit specified by Imax. So the above equation will become;

So IR = IMax sin ωt.

Therefore IR = IMax sin ωt.

In a purely resistive AC circuit, the whole voltage of the circuit is equivalent to the individual resistors’ sum of voltages as all these individual voltages are in-phase. Now let’s see

#### How to Calculate Power Dissipated by a Resistor in an AC Circuit?

To measure the power in an AC circuit, the power factor (PF) plays a significant role, so it can be defined as the phase angle’s cosine in between voltage & current. Here, the phase angle is indicated by the ‘φ’ symbol.

In a circuit, if the real power is ‘P’ which is measured in Watts, and the apparent power ‘S’ is measured in Volt-Amps. So the main relation between these two powers can be given by the following equation.

P = S Cos φ.

In pure resistive AC circuits, the phase angle between voltage & current is 0 degrees. Thus φ = 0 degrees. Therefore the power factor (PF) Cos φ is Cos 0 = 1. Thus, the real power (P) is equivalent to the apparent power (S) which is the result of voltage & current.

In these AC circuits, the power can be found by calculating the result of voltage and current at that moment. The power consumed by the AC circuit can be measured through the following equation.

P = VRMS x IRMS x Cos φ.

As φ = 0 degrees in this case, the power is

P = VRMS x IRMS

The power in pure resistive AC circuits can be consumed through the circuit is the product of voltage & the current because there is no phase angle among voltage & current.

When the voltage and current increases then the power increases and reaches its maximum value once both voltage &current reach their maximum value. After that, it drops to zero because voltage & current will drop to zero. Once the voltage & current polarity varies then the power value will be increased again & reaches maximum values when voltage &current achieve their negative peak. Once the current & voltage drop to zero then the power value drops to zero.

In a purely resistive circuit including an AC RMS power supply, the dissipation of power is the same as compared to a resistor connected to a DC power supply.

P = VRMS * IRMS = I^2RMS * R = V^2RMS/R..

In the above equation, both the VRMS and IRMS are RMS values of current & voltage respectively.
‘P’ is power in Watts and ‘R’ is resistance in Ohms (Ω).

#### Resistance in AC Circuit Formula

The current & voltage in pure resistance of an AC circuit will oscillate within phase, specifically, once the flow of current is at maximum then the voltage will also be at its maximum, thus there is no phase shift. Additionally, the value of resistance is not affected by the height of frequency.

But, whenever there are capacitors or inductors within the circuit then the phase difference will be there in between the voltage & the current because they will oscillate 90 degrees out of phase.

To monitor both the magnitude and the phase, complex numbers are used where the resistance can be used in combination with an imaginary unit to signify reactance. So, AC resistance can be simply determined through an LCR meter.

So, the formula is; Z = R + i X => R + iωL + 1/iωC

Angular frequency ω = 2 π f

#### What is the Phase Difference between Voltage and Current in an AC Circuit with a Pure Resistor?

The current & voltage within a pure resistor of an AC circuit is in-phase because there is no phase difference among them. The flow of current throughout the resistance can be directly proportional to the voltage across it which is known as impedance.

#### What is Instantaneous Value in AC Circuit?

The instantaneous value within an AC circuit is “the value of ac current or ac voltage or ac power at a particular moment of time within the cycle.

### Resistors in AC Circuits Examples

The example problems of resistors in AC circuits are solved below.

Example-1

A heating element with 500 Watt is simply connected to a 300v AC voltage supply, then calculates the AC resistance or impedance of the heating element once it is hot & the sum of the current supply used from the supply.

We know that current I = P/V

I = 500W/250V = 2 amps

R = V/I = 250/2 = 125 Ohms

Example-2

Determine the power being consumed through a 50Ω resistor which is connected across a 120v supply.

We know that I = V/R = 120/50 = 2.4A

Formula for power consumed is P = I^2*R = (2.4)^2 x 100 = 576 Watts.

Example-3

In the below circuit, if a heating element like ‘R’ is connected to a 120V AC supply then the power utilized by the heating element is 1.5 K Watts. What is the resistance value?

The flow of current throughout the heating element ‘R’ can be calculated as

I = P/V

P = 1.5kW = 1500 Watts and V = 120V

Thus, I = 1500/120 = 12.5 Amps

The resistance value for the heating element can be measured with Ohm’s law as

R = V / I = 120/12.5 = 9.6 Ohms

Example-4

In the following circuit, if 20 ohms resistor is connected to 80V voltage supply then calculate the values for the flow of current throughout the resistor & the power consumed through the resistor can be measured as;