### High Voltage Direct Current (HVDC)

##### Page 3: Electricity Transmission

###### Resistance

Resistance is the property of a material to oppose (resist) current flow through the material, causing some of the electricity to be lost (e.g., as heat). For example, electric wires have resistance which causes some of the electricity that is transmitted through the wire to be lost.

All materials have some resistance. Conductors have less resistance than insulators, which is what makes them conductors. But even conductors have resistance, just less resistance than insulators.

For a given resistance of a material, denoted R, the amount of power lost is given by the following formula:

PR = I 2 × R

where PR denotes power that is lost to resistance, I 2 denotes the current squared (current times itself), and R denotes resistance of the material (measured in Ohms). This is known as the “I2R Loss”.

Notice that the I2R loss increases as a square of the current (an exponential relationship): doubling the current quadruples the loss, tripling the current increases the loss nine-fold, etc. Thus, to reduce loss, we need to reduce current by increasing voltage, which is possible with transformers.

Reducing current by increasing voltage is a linear relationship, not an exponential relationship, given by this formula from the previous page of this report:

P = V × I

where P denotes power, V denotes voltage, and I denotes current.

For a given initial electrical power P0 we are trying to transmit, the power that is transmitted PT becomes the initial power minus the I2R loss PR

PT = P0 – ( I 2 × R )

with P = V × I holding throughout: for the power that is transmitted, for the power we are trying to transmit, and for the I2R loss.

For a given initial electrical power P0 we are trying to transmit, say we want to reduce the I2R loss of transmitting that electrical power by doubling the voltage. In that case, the current is halved (according to the formula P = V × I), keeping the power we are trying to transmit constant. Then, plugging the new current level, which is half the current (at double the voltage), into the I2R formula shows that the power lost will be one-fourth of the power lost.

Likewise, reducing the current to one-third of the current (by tripling the voltage) reduces I2R loss to one-ninth of the loss. Quadrupling the voltage drops the I2R loss to 1/16 of the loss, etc.

Huge amounts of electricity are prevented from being lost by using transformers to step up the voltage for transmission and step it back down when needed for end uses. That is why AC transmission became popular, because transformers only work on AC (see previous page of this report).

There will always be a need for AC transmission, to be able to use transformers. The role of DC will be to transmit bulk electricity long distances. Electrical transmission systems need both AC and DC.

###### AC Skin Effect

One of the disadvantages of trying to use AC for bulk electricity transmission is that larger wires are needed because AC electricity migrates toward the edges of the conductor, not utilizing the interior of a wire to transmit electricity. This is called the skin effect.

The skin effect is caused by induced emfs opposing current flow.

Electricity flowing in a wire magnetizes the wire which creates an electric field that opposes AC reversing directions. The effect is greater in the portion of the wire that has more wire surrounding it (the center of the wire).

This does not happen with DC. Direct current transmits electricity uniformly across the cross section of a wire, using the entire wire to transmit electricity.

For AC wires, the transverse distance in from the edge of the wire, where most of the current is flowing, is called the skin depth.

The skin depth, denoted δ (small delta), is where 1/e (38.6 %) of the intensity of the wire edge current flows. For transmission line frequencies, this can be calculated with the following formula (eq. 9 in Riba, see References below):

δ = 1 ∕ k = 1 ∕ sqrt( π f μσ )

where f is the AC frequency (50 or 60 cycles per second), and μ and σ are material constants (and of course π is 3.14… which is also constant). The skin depth is constant for a given frequency (and conductor conditions). Note: The denominator is a constant, denoted k, which can be used as a scalar in subsequent calculations, not to be confused with the k of (i,j,k) vectors.

Frequency of an AC cycle is measured in cycles per second which is also called hertz (abbreviated Hz). 50 Hz has a deeper skin depth than 60 Hz, resulting in less resistive line losses. For that reason, Europe has switched to 50 Hz, and Asian countries have been adopting 50 Hz. Note: The US still uses 60 Hz.

The following map shows examples of HVDC lines and back to back stations around the world that connect two AC grids together, specifying which AC grids are 50 Hz and which are 60 Hz.