Resistance, inductance and capacitance: a simple introduction

In this guide we meet the three basic properties of every circuit, with simple animated diagrams and five calculators to try.

Why R, L and C matter

Every electrical circuit has three basic properties. They are resistance, inductance and capacitance. We write them as R, L and C.

Sometimes we add them on purpose, as parts called resistors, inductors and capacitors. But they also appear on their own. A long power line has all three. So does a motor, a cable and a transformer.

With direct current, or DC, only resistance limits the steady current. With alternating current, or AC, the picture changes. So why does the same circuit carry a different current with AC than with DC? The answer is inductance and capacitance. In this guide we look at each property in turn. Then we put them together.

This is the second article in our power system basics path. The ideas here come back in every later article, from reactive power to line models.

Resistance: opposition that makes heat

Resistance is the opposition a material gives to the flow of current. When current flows through a resistance, some electrical energy turns into heat. The symbol is R. The unit is the ohm, written Ω.

Ohm's law links voltage, current and resistance: V = I × R. Voltage (V) is the electrical push, in volts. Current (I) is the flow of charge, in amperes. So for the same voltage, a larger resistance gives a smaller current.

Large values use prefixes. 1 kΩ (kilohm) = 1,000 Ω. 1 MΩ (megohm) = 1,000 kΩ = 1,000,000 Ω.

The resistance of a wire depends on three things. These are the metal, the length and the cross-section area. We write it as R = ρ × L ÷ A. Here ρ (rho) is the resistivity, a property of the metal. The figure below shows the idea.

Copper has a resistivity of about 1.7 × 10⁻⁸ Ω·m at 20 °C. The aluminium used in overhead lines is about 2.8 × 10⁻⁸ Ω·m. So 1 km of aluminium with a 400 mm² cross-section has R = 2.8 × 10⁻⁸ × 1000 ÷ 0.0004 ≈ 0.07 Ω. Resistance also rises as the metal gets hotter.

A resistor is a part made to have a known resistance. We use resistors to limit current, divide voltage, set the gain of amplifiers, act as a load, and protect other parts. A resistor has three main ratings:

To measure a resistor, we use a multimeter on its ohms range. The resistor should be removed from the circuit, or at least disconnected from any supply. Otherwise other parts change the reading.

Small resistors show their value with coloured bands. This is the resistor colour code, set out in the standard IEC 60062. The figure below shows how to read a common four-band resistor.

The table gives the full code. The same colour means a digit in the first bands, a multiplier in the next band, and a tolerance in the last band. Gold and silver are used only as multipliers or tolerances.

Resistors come with 3, 4, 5 or 6 bands. With 3 or 4 bands, there are two digits. With 5 or 6 bands, there are three digits, for more precise values. The 6th band gives the temperature coefficient, how much the resistance changes with temperature, in ppm/K (parts per million per kelvin). Try the calculator below.

Inductance: opposition to a change in current

Current in a wire always makes a magnetic field around it. If we wind the wire into a coil, the fields of all the turns add together. The magnetic flux, written Φ, is the amount of this field that passes through the coil. It is measured in webers (Wb).

Inductance is the property that opposes any change in current. When the current changes, the flux changes too. The changing flux induces a voltage in the coil itself. This voltage pushes against the change. The symbol is L. The unit is the henry, written H.

Smaller units are common. 1 H = 1,000 mH (millihenry). 1 mH = 1,000 µH (microhenry).

The figure below shows what happens when we switch a coil on to a DC supply. Look at the red arrow. It is strong at first, and then it fades away.

So inductance does not stop a steady DC current. Only the resistance of the wire does that. Inductance acts only while the current is changing. The induced voltage is V = L × ΔI ÷ Δt, where ΔI ÷ Δt is how fast the current changes. When the current falls, the coil pushes the other way and tries to keep it flowing. This is why switching off a coil can make a spark.

Inductance depends on the coil. More turns, a larger coil area and an iron core all raise it. For a coil, the inductance grows with the square of the number of turns. We can also find it from the flux: L = N × ΔΦ ÷ ΔI. Here N is the number of turns, ΔΦ is the change in flux, and ΔI is the change in current that caused it.

An inductor is a part made to have a known inductance. It is usually a coil of wire, often with a core of iron or ferrite.

With AC, the current changes all the time. So the inductor opposes it all the time. This opposition is called inductive reactance, written XL. It is measured in ohms, like resistance.

The formula is XL = 2π × f × L. Here f is the frequency in hertz (Hz), and L is the inductance in henries. So XL grows in step with the frequency. At 0 Hz, which is DC, XL is zero.

For example, a 100 mH inductor at 50 Hz has XL = 2π × 50 × 0.1 ≈ 31.4 Ω. At 60 Hz the same inductor has about 37.7 Ω.

Capacitance: storing charge between two plates

A capacitor is two conductors, usually metal plates, with an insulator between them. The insulator is called the dielectric. The plates do not touch, so no current can flow straight through.

When we connect a voltage, charge moves onto one plate. The same amount of charge leaves the other plate. One plate becomes positive and the other negative. An electric field builds up in the dielectric, and it stores energy. The figure below shows this.

Capacitance is how much charge a capacitor stores for each volt across it. We write it as C = Q ÷ V. Here Q is the charge in coulombs (C) and V is the voltage. The unit of capacitance is the farad, written F.

One farad is very large. So we usually see smaller units. 1 F = 1,000,000 µF (microfarad). 1 µF = 1,000 nF (nanofarad) = 1,000,000 pF (picofarad).

Capacitance depends on the shape of the capacitor, not on the voltage. For flat plates, C = ε × A ÷ d. A larger plate area A gives more capacitance. A smaller gap d gives more capacitance too. The dielectric material sets ε, its permittivity. When the supply is removed, a capacitor can keep its charge for a long time. It gives the charge back when a path is connected.

With DC, a capacitor charges up and then the current stops. So a capacitor blocks steady DC. With AC, the voltage keeps reversing. So the capacitor keeps charging and discharging, and current flows in the circuit all the time.

The opposition to this AC current is called capacitive reactance, written XC. The formula is XC = 1 ÷ (2π × f × C), with C in farads. So XC falls as frequency rises. It also falls as capacitance rises.

For example, a 10 µF capacitor at 50 Hz has XC = 1 ÷ (2π × 50 × 0.00001) ≈ 318 Ω. At 5 kHz, the same capacitor has only about 3.2 Ω.

How R, L and C behave with AC

All three properties limit an AC current. But they do it in different ways. The key difference is timing. With AC, voltage and current are sine waves. The question is whether they peak at the same moment.

This energy that moves back and forth is the basis of reactive power, the next topic in this path. The two kinds of reactance also change with frequency in opposite ways. The graph below shows this.

To find the current through one component on an AC supply, we follow four steps. The cards below use a capacitor on a 230 V, 50 Hz supply.

Worked example: R, L and C in a power line

Now we use these ideas on a real power system. We take a 50 km overhead line at 50 Hz. Each phase has one aluminium conductor with a 400 mm² cross-section. Typical values for this kind of line are about 0.07 Ω per km of resistance, 1.3 mH per km of inductance and 9 nF per km of capacitance.

The figure below shows these values in a simple line model. The capacitance is split into two halves, one at each end. This is a common way to model a short or medium line.

Look at the numbers. The reactance XL is about six times the resistance R. This is normal for overhead lines. So the inductance, not the resistance, mostly limits how much power the line can carry. It also causes most of the voltage drop.

The capacitance is in parallel with the load, between the line and ground. Its very high reactance means only a small current flows through it. But on long lines and underground cables this charging current becomes important.

Where we use resistors, inductors and capacitors

Inductors and capacitors often work together. Their reactances change with frequency in opposite directions. So a pair can pick out one frequency and block others. This is the basis of tuned filters, for example harmonic filters in industrial plants.

Key takeaways

• Resistance (R, ohms) opposes current and turns energy into heat. For a wire, R = ρ × L ÷ A.
• A resistor's colour bands give two or three digits, a multiplier and a tolerance. Gold and silver are only multipliers or tolerances.
• Inductance (L, henries) opposes changes in current and stores energy in a magnetic field. With AC, XL = 2πfL.
• Capacitance (C, farads) is C = Q ÷ V. It stores energy in an electric field. With AC, XC = 1 ÷ (2πfC).
• In an inductor the current lags the voltage by 90°. In a capacitor it leads by 90°. Reactance stores energy but does not turn it into heat.
• In overhead lines, XL is usually several times R, so inductance mostly limits power flow and causes voltage drop.

Next in this path we look at reactive power, the energy that inductors and capacitors trade back and forth with the grid. The guides below go further.