When power plants supply power to buildings and households, they send them over long distances in the form of direct current (DC). But households appliances and electronics generally rely on alternating current (AC).

Converting between the two forms can show you how the resistances for the forms of electricity differ from one another and how they're used in practical applications. You can come up with DC and AC equations to describe the differences in DC and AC resistance.

While DC power flows in a single direction in an electric circuit, the current from AC power sources alternates between forward and reverse directions at regular intervals. This modulation describes how AC changes and takes the form of a sine wave.

This difference also means that you can describe AC power with a dimension of time that you can transform into a spatial dimension to show you how the voltage varies across different areas of the circuit itself. Using the basic circuit elements with an AC power source, you can describe the resistance mathematically.

## DC vs. AC Resistance

For AC circuits, treat the power source using the sine wave alongside **Ohm's Law**,

for voltage *V*, current *I* and resistance *R*, but use **impedance Z** instead of

*R.*

You can determine resistance of an AC circuit the same way you do for a DC circuit: by dividing the voltage by current. In the case of an AC circuit, resistance is called impedance and can take other forms for the various circuit elements such as inductive resistance and capacitive resistance, measuring resistance of inductors and capacitors, respectively. Inductors produce magnetic fields to store energy in response to current while capacitors store charge in circuits.

You can represent the electrical current across through an AC resistance

for maximum value of current *Im*, as the phase difference *θ*, angular frequency of the circuit *ω* and time *t*. The phase difference is the measurement of the angle of the sine wave itself that shows how current is out of phase with voltage. If current and voltage are in phase with one another, then the phase angle would be 0°.

**Frequency** is a function of how many sine waves have passed over a single point after one second. Angular frequency is this frequency multiplied by 2π to account for the radial nature of the power source. Multiply this equation for current by resistance to obtain voltage. Voltage takes a similar form

for the maximum voltage V. This means you can calculate AC impedance as the result of dividing voltage by current, which should be

AC impedance with other circuit elements such inductors and capacitors take use the equations

for the inductive resistance *X _{L}*, capacitive resistance

*X*

_{C}to find AC impedance Z. This lets you measure the impedance across the inductors and capacitors in AC circuits. You can also use the equations

*X* and

_{L}= 2πfL*X* to compare these resistance values to the inductance

_{C}= 1/2πfC*L* and capacitance

*C* for inductance in Henries and capacitance in Farads.

## DC vs. AC Circuit Equations

Though the equations for AC and DC circuits take different forms, they both depend on the same principles. A DC vs. AC circuits tutorial can demonstrate this. DC circuits have zero frequency because, if you were to observe the power source for a DC circuit would not show any sort of waveform or angle at which you can measure how many waves would pass a given point. AC circuits show these waves with crests, troughs and amplitudes that let you use frequency to describe them.

A DC vs. circuit equations comparison may show different expressions for voltage, current and resistance, but the underlying theories that govern these equations are the same. The differences in DC vs. AC circuit equations come about by the nature of the circuit elements themselves.

You use Ohm's Law *V = IR* in both cases, and you sum up current, voltage and resistance across different types of circuits the same way for both DC and AC circuits. This means summing up the voltage drops around a closed loop as equal to zero, and calculating the current that enters each node or point on an electric circuit as equal to the current that leaves, but, for AC circuits, you use vectors.

## DC vs. AC Circuits Tutorial

If you had a parallel RLC circuit, that is, an AC circuit with a resistor, inductor (L) and capacitor arranged in parallel with one another and in parallel with the power source, you would calculate current, voltage and resistance (or, in this case, impedance) the same way you would for a DC circuit.

The total current from the power source should equal the **vector** sum of the current flowing through each of the three branches. The vector sum means squaring the value of each current and summing them to get

for supply current *I _{S}*, resistor current

*I*, inductor current

_{R}*I* and capacitor current

_{L}*I*. This contrasts the DC circuit version of the situation which would be

_{C}Because voltage drops across branches remains constant in parallel circuits, we can calculate the voltages across each branch in the parallel RLC circuit as *R = V/I _{R}*,

*X* and

_{L}= V/I_{L}*X*. This means, you can sum up these values using one of the original equations

_{C}= V/I_{C}*Z = √ (R* to get

^{2}+ (X_{L}– X_{C})^{2}This value *1/Z* is also called admittance for an AC circuit. In contrast, the voltage drops across the branches for the corresponding circuit with a DC power source would be equal to the voltage source of the power supply *V*.

For a series RLC circuit, an AC circuit with a resistor, inductor and capacitor arranged in series, you can use the same methods. You can calculate the voltage, current and resistance using the same principles of setting current entering and leaving nodes and points as equal to one another while summing up the voltage drops across closed loops as equal to zero.

The current through the circuit would be equal across all elements and given by the current for an AC source *I= I _{m} x sin(ωt)*. The voltage, on the other hand, can be summed around the loop as

*V* = 0 for

_{s}- V_{R}- V_{L }- V_{C}*V* for supply voltage

_{R}*V*, resistor voltage

_{S}*V*, inductor voltage

_{R}*V* and capacitor voltage

_{L}*V*.

_{C}For the corresponding DC circuit, current would simply be *V/R* as given by Ohm's Law, and the voltage would also be *V _{s} - V_{R} - V_{L }- V_{C}* = 0 for each component in series. The difference between the DC and AC scenarios is that while, for DC you can measure resistor voltage as

*IR*, inductor voltage as

*LdI/dt* and capacitor voltage as

*QC* (for charge

*C* and capacitance

*Q)*, the voltages for an AC circuit would be

*V*

_{R}= IR, VL = IX_{L}*sin(ωt + 90*°

*)* and

*VC =*

*IX*

_{C}*sin(ωt - 90*°

*).* This shows how AC RLC circuits have an inductor ahead of the voltage source by by 90° and capacitor behind by 90°.

References

- Electronics Tutorials: AC Resistance and Impedance
- Electrical Technology: What is the difference between AC and DC Resistance & How to calculate it?
- UNSW: AC circuits: alternating current electricity
- Diffen: AC vs. DC (Alternating Current vs. Direct Current)
- Indian Institute of Technology Bombay: EE101: RLC Circuits (with DC sources)

About the Author

S. Hussain Ather is a Master's student in Science Communications the University of California, Santa Cruz. After studying physics and philosophy as an undergraduate at Indiana University-Bloomington, he worked as a scientist at the National Institutes of Health for two years. He primarily performs research in and write about neuroscience and philosophy, however, his interests span ethics, policy, and other areas relevant to science.