K323 / 2027
Electromagnetic induction overview

Topic 5 of 5

Reducing transmission losses

Transmission cables have resistance and warm when current flows. For the same delivered power, using a higher transmission voltage reduces current and therefore reduces cable heating.

Electrical power is P = VI for the stated operating conditions. For a resistance R carrying current I, the voltage drop across that resistance is V = IR.

Use the voltage across the cable

Let R be the total resistance of the outgoing and return cables. Their combined voltage drop is Vdrop = IR.

Ploss = I Vdrop = I x IR = I2RCable heating power uses cable current and the cable's own voltage drop. Use I in amperes and R in ohms to obtain watts.

The receiving-end voltage is across the equipment receiving the useful power. It is not the same as the drop along the cables. Multiplying cable current by the full receiving-end voltage gives the delivered power in this model, not the cable loss.

A grid overview and a separate cable power model

Overview: change voltage before and after transmission

This overview has no numerical voltages. The outgoing and return conductors are both shown. Brown arrows show one instant of alternating current.

Generation, step-up transformer, cable pair, step-down transformer and loadA generation block connects through two conductors to a step-up transformer. Its two secondary terminals connect through outgoing and return cables, each with resistance, to the primary terminals of a step-down transformer. The step-down secondary connects through a separate pair to the final load at a lower voltage. The high receiving-end transmission voltage is across the step-down input, not across the final load. The illustrated cable current is down the left conductor and up the right at one instant; both reverse during a.c. This is a functional block overview, not the numbered direct cable model below.GenerationStep-uptransformerLong cablesfinite resistanceCurrentreverses in a.c.Step-downtransformerLoadlower voltage

The receiving-end transmission voltage is across the step-down transformer's input. The final load receives the lower output voltage.

Direct cable model: 1000 V received

Direct cable model: 1000 V receivedThis separate idealised steady power account has a sending supply, two resistive cables and a direct receiving load, with no step-down transformer inside the model. The load receives 10.0 kilowatts at 1000 V. Current 10.0 A travels through the outgoing cable and returns through the other cable. The two cable resistances add to 2.0 ohms; neither individual resistance is specified. The total cable drop is 20.0 V, so the sending supply voltage is 1020 V. Cable heating is 200 W, and required sending power is 10200 W. The receiving load voltage and cable drop are different quantities.SupplyLoadSending voltage1020 VReceiving voltage1000 VRoutRreturnI = 10.0 AUseful receiving power10.0 kWRout + Rreturn= 2.0 ohm in totalCable drop = 20.0 VCable heating = 200 WSending power = 10200 W

This is a direct cable-and-load model, separate from the overview. Its receiving voltage is across this load. The sending supply must provide the useful 10.0 kW plus the cable heating.

Direct cable model: 10000 V received

Direct cable model: 10000 V receivedThis separate idealised steady power account has a sending supply, two resistive cables and a direct receiving load, with no step-down transformer inside the model. The load receives 10.0 kilowatts at 10000 V. Current 1.00 A travels through the outgoing cable and returns through the other cable. The two cable resistances add to 2.0 ohms; neither individual resistance is specified. The total cable drop is 2.00 V, so the sending supply voltage is 10002 V. Cable heating is 2.00 W, and required sending power is 10002 W. The receiving load voltage and cable drop are different quantities.SupplyLoadSending voltage10002 VReceiving voltage10000 VRoutRreturnI = 1.00 AUseful receiving power10.0 kWRout + Rreturn= 2.0 ohm in totalCable drop = 2.00 VCable heating = 2.00 WSending power = 10002 W

This is a direct cable-and-load model, separate from the overview. Its receiving voltage is across this load. The sending supply must provide the useful 10.0 kW plus the cable heating.

A higher voltage allows the same useful power to be carried at lower current. The comparison uses 10.0 kW delivered at the receiving end and 2.0 ohm total outgoing-and-return cable resistance in both cases.

Compare two systems delivering 10.0 kW

Use an idealised power account with receiving-end power 10.0 kW = 10 000 W and total cable resistance 2.0 ohm. The receiving equipment is arranged to take that same useful power at either stated voltage; this is not a comparison of different voltages applied to an unchanged resistor.

Case A

1000 V at the receiving end

Cable current I = Preceived/Vreceived = 10 000/1000 = 10.0 A.

Cable drop Vdrop = IR = 10.0 x 2.0 = 20.0 V.

Cable heating Ploss = I2R = 10.02 x 2.0 = 200 W.

The sending-end voltage must be 1000 + 20.0 = 1020 V. Its input power is 1020 x 10.0 = 10 200 W: 10 000 W delivered plus 200 W lost in the cables.

Case B

10 000 V at the receiving end

Cable current I = 10 000/10 000 = 1.00 A.

Cable drop Vdrop = 1.00 x 2.0 = 2.00 V.

Cable heating Ploss = 1.002 x 2.0 = 2.00 W.

The sending-end voltage must be 10 000 + 2.00 = 10 002 V. Its input power is 10 002 x 1.00 = 10 002 W: 10 000 W delivered plus 2.00 W lost in the cables.

The receiving-end voltage is ten times larger in B, so the current is one tenth as large. Because heating depends on current squared, the cable loss is one hundredth as large: 2 W instead of 200 W.

Keep the full power account

Sending-end power = receiving-end power + cable heating lossThis account treats the cables as resistive and leaves other equipment losses out of the comparison.

A step-up transformer can raise the voltage before a long transmission cable. A step-down transformer reduces it afterwards to the voltage required by the receiving equipment. The transformers change the voltage/current combination; the cable resistance still causes a finite loss.

The advantage of higher voltage follows from keeping delivered power and cable resistance fixed. Increasing voltage does not universally reduce heating: across the same unchanged resistance, a higher voltage would increase current and heating. Name the conditions before making the comparison.

Real transformers and other equipment also have losses. Include them if supplied; the calculated 200 W and 2 W are the cable losses under these stated conditions.

Optional check Two systems each deliver 10.0 kW at the receiving end through cables with total resistance 2.0 ohm. The receiving-end voltage rises from 1000 V to 10000 V. How does the cable heating loss change?
Two systems each deliver 10.0 kW at the receiving end through cables with total resistance 2.0 ohm. The receiving-end voltage rises from 1000 V to 10000 V. How does the cable heating loss change?