home site map printer-friendly November 26, 2020
Transmission Lines Photo courtesy AEM.

All modern countries are crisscrossed with high-voltage transmission lines, which transport electrical power from generators at power plants to substations and ultimately consumers. Why are high voltages used? What are the advantages of alternating current (AC) versus direct current (DC)? How much energy is lost in transmitting electrical power over long distances? The main physics principle this topic addresses is electrical resistance.

## Electrical Resistance

Electrical current, the flow of charge, has a sort of friction associated with it, which is called resistance. Good conductors, like most metals, allow current to flow without much loss. Poor conductors, like most non-metals, impede the flow of current to a great extent. Superconductors like very cold niobium-tin, are special substances that allow current to flow with essentially zero loss; semiconductors, like silicon, are either good or poor conductors depending on certain conditions.

You get current to flow through a conductor by applying a voltage across it. The amount of current that flows is measured in amperes, or amps, named after a 19th-century French physicist and abbreviated A. An ampere is a fairly large amount of current: 0.1 A flowing between your hands across your heart will kill you. (Fortunately, your body has fairly high resistance so it takes a substantial voltage to drive that much current.) Voltage, or electric potential, is measured in volts, named after a physicist named Volta, and abbreviated V. Most small batteries (size AAA, AA, C, D) are 1.5 V; there is the familiar box-like 'transistor' 9 V battery, and car batteries are 12 V. In contrast, high-voltage lines have many thousands of volts between them.

Resistance quantifies how much current you get across something per volt applied. Namely, if you apply a voltage V across a wire and measure current I, the resistance R is defined by

R = V/I

Resistance therefore has units of V/A, which get another name, ohms, represented by the Greek letter .

### Electrical Power

We are all aware that electric current can transport energy from one place to another: the energy given off by a 100 Watt light bulb in your bedroom originated by burning coal or slowing down falling water or releasing nuclear energy at a power plant, for example. The expression for electric power comes from the definitions of electric potential (volts) and electric current (amps).

The MKS unit of energy is the joule (J) and the MKS unit of electric charge is the coulomb (C), which is the amount of charge that flows by in one second if the current is one amp. The volt is therefore defined by saying that if a charge of 1 C moves across a potential drop of 1 V it picks up energy 1 J:

1 V = 1 J/C

In general, then, a charge Q picks up energy

U = QV

when it moves across a potential drop V.

Electric power is the rate at which energy is transported. Since current is the rate of transport of charge, electric power is given by the above expression, but using current I instead of charge Q:

P = IV

This is a very handy formula. For example, you may see written on your hair dryer that it draws 10 A current on the hot setting from a standard US 110 V outlet. This means that the power drawn by the hair dryer is 10x110=1100 W, or 1.1 kW. That's about as high a power as home appliances go, and this is not too far from tripping a 15 A circuit breaker, standard in modern US houses. For very high power appliances, like a clothes washer or dryer, you may need a special outlet and dedicated circuit breaker. (Note: even though house current is alternating, or AC, at 60 cycles/sec (50 in Europe), this formula works because the average or RMS current and voltage are quoted and you therefore get the average power.)

Another handy version of the power formula replaces voltage V with resistance and current: V=IR:

P = I²R

### High-Voltage Transmission Lines

So we now finally come to the topic of this page: the transport of large amounts of electrical power over long distances. This is done with high-voltage transmission lines, and the question is: why high voltage? It certainly has a negative safety aspect, since a low voltage line wouldn't be harmful (you can put your hands on a 12 V car battery, for example, you won't even feel it; but make sure you don't put metal across the terminals, you'll get a huge current and a nasty spark!). Electric energy is transported across the countryside with high-voltage lines because the line losses are much smaller than with low-voltage lines.

All wires currently used have some resistance (the development of high-temperature superconductors will probably change this some day). Let's call the total resistance of the transmission line leading from a power station to your local substation R. Let's also say the local community demands a power P=IV from that substation. This means the current drawn by the substation is I=P/V and the higher the transmission line voltage, the smaller the current. The line loss is given by Ploss=I²R, or, substituting for I,

Ploss = P²R/V²

Since P is fixed by community demand, and R is as small as you can make it (using big fat copper cable, for example), line loss decreases strongly with increasing voltage. The reason is simply that you want the smallest amount of current that you can use to deliver the power P. Another important note: the loss fraction

Ploss/P = PR/V²

increases with increasing load P: power transmission is less efficient at times of higher demand. Again, this is because power is proportional to current but line loss is proportional to current squared. Line loss can be quite large over long distances, up to 30% or so. By the way, line loss power goes into heating the transmission line cable which, per meter length, isn't very much heat.

### Alternating (AC) vs. Direct Current (DC)

Given that we want to reduce line loss by using high voltage, the choice between AC and DC becomes straightforward. It is quite difficult to reduce a DC high voltage to low voltage without additional loss; it is easy to reduce an AC high voltage to low voltage using a step-down transformer. You see lots of these when you walk by a substation. An ideal transformer reduces V and increases I so that the power IV is constant. A neighborhood substation typically reduces the voltage to a reasonable value for street lines, say 330 V, and then a small transformer outside and/or inside your house reduces it to 110 V (220 in Europe). Since the current and voltage are alternating with sine waves, the power delivered to, say, a toaster also oscillates. The current or voltage oscillation frequency is 60 cycles/sec (60 Hz) in the US and 50 Hz in Europe. The figure below shows how the current, voltage and power look as a function of time along with the average (RMS) values for a load drawing 10 A in the US. Voltage, current and power for a resistive appliance that draws 10 amps (like a toaster). The average (RMS) values are shown with dotted lines. This appliance draws 1100 watts RMS.

## Equations

• electric resistance: R = V/I
• electric power: P = IV = I²R

## Summary

• Resistance quantifies the amount of current that will flow in a wire per volt.
• Power loss due to wire resistance increases as the square of the current and therefore decreases as the square of the voltage at fixed total power. The loss fraction in a transmission line increases with demand.