How transmission lines deliver (established?) power from generator

[arestes]

TL;DR Summary

Power is assumed to be given (a constant) when justifying why we need high voltage to minimize losses through the lines. However, I think power depends on load.

Hello:

I'm confused about transmission lines. According to Faraday's Law, what's induced is an emf that depends on how fast the coils spin, or whatever equivalent to a simple model seen on Physics textbooks.

Power, however, is then assumed constant when talking about the power loss due to heating ($P_{loss}=I^2R$I2R). Current is obtained from the power P generated at the station:
$$ IV=P_{gen} \rightarrow I=\frac{P_{gen}}{V}$$
Therefore:
$$ P_{loss} =\frac{P^2_{gen}}{V^2}R$$

My question is: According to a simple schematics of a transmission line (taken from wikipedia):

The impedance of the load is part of the circuit, it makes sense that if load changes (which is what users do) then power delivered is changed. Therefore, what we extract from the station depends upon the load, which makes sense.
But this contradicts the argument that we only need to maximize V on the denominator in the expression of $P_{loss}$
In real life, how is this taken into account? Is the load impedance assumed constant? (I think there are conditions for maximum power delivery of impedance matching). Variations of impedance are negligible?

Thanks for any clarifications on this topic. By the way, I need this to have a crystal clear explanation of why we need to have high voltages in transmission lines from a Physics perspective but this seems to touch on electrical engineering matters.
V=P→I=PV

 

[russ_watters]

Yes, it would change with different power delivered. But power delivered is an input value. It's a design requirement of the system. What you're trying to minimize is loss at design/maximum delivered power.

 

[anorlunda]

The source voltage is variable. More important, in AC power, is is the phase angle of the source voltage, not its magnitude that most strongly influences power flow to the load. So, when the load impedance changes, the phase angle difference between load and source changes.

There are also secondary ways to adjust power flows and voltages. But it is best to learn the primary methods first.

See this PF Insights article: https://www.physicsforums.com/insights/ac-power-analysis-part-2-network-analysis/

Of course, some loads do have approximately constant power. The computer you use to type PF posts, is closer to constant power than it is constant impedance. When power engineers simulate power systems, each load can be represented as a combination of constant power, constant current, and constant impedance, and perhaps factors that make load a function of frequency as well as voltage.

It is not necessary to discuss losses at all to understand these points. Indeed, we could use superconducting wires to send power from the source to the load with zero losses. But the primary methods would remain unchanged. So, please leave losses out of it.

 

[Rive]

Power, however, is then assumed constant when talking about the power loss due to heating (I2R).

If I take it right in that expression of P_loss the 'R' is not the impedance of the load, but the impedance of the transmission line.

I think there are conditions for maximum power delivery of impedance matching

There is no impedance matching in that sense for transmission (I mean, if we are talking about power distribution through the grid). Please notice that in case of maximal power impedance matching would waste half of the generated power on the generators and the transmission.

 

[arestes]

If I take it right in that expression of P_loss the 'R' is not the impedance of the load, but the impedance of the transmission line.There is no impedance matching in that sense for transmission (I mean, if we are talking about power distribution through the grid). Please notice that in case of maximal power impedance matching would waste half of the generated power on the generators and the transmission.

Yes, the R is due to the transmission line.
Oh, thanks for clarifying that impedance matching is not desirable in this case for it would waste half of power.

I still have trouble understanding how we can talk about "generated power" if the transmitted power depends upon the load.

 

[Baluncore]

I still have trouble understanding how we can talk about "generated power" if the transmitted power depends upon the load.

The line is short and mismatched. There will be a reflected wave that largely cancels the transmitted wave. The generator only has to make up the difference, which is the energy consumed by the load and the line losses.

 

[anorlunda]

I still have trouble understanding how we can talk about "generated power" if the transmitted power depends upon the load.

There can be a difference between generated power and transmitted power. The difference changes the speed of the generator, thus storing (or using up) electric energy in the form of spinning kinetic energy. The power plant's speed governor senses the change in speed (frequency) and adjusts the steam flow or water flow or fuel flow to adjust the generated power appropriately.

That is also described in the article linked above.

In the end, energy is conserved, which means that every last joule of energy generated goes someplace. It might go to speed, or to losses, or to the load, or perhaps other places difficult to describe, but somewhere.

This whole scheme of a network of generators connected by multiple transmission lines to multiple loads, and controlled by speed governors was first conceived and implemented by Thomas Edison. Edison did much more than invent the light bulb, he invented a whole power company and the power grid to supply it.

 

[Tom.G]

@arestes, part of the confusion is that the phrase "Transmission Line" is used in two different ways.

1) The images from the Wikipedia article you posted refer to Radio Frequency transmission lines. Those are used to connect an antenna to a transmitter or receiver (and a few other uses that would just confuse this conversation!)

2) For home electrical power from the electric company, the wiring from the power company (which may be hundreds of miles away) is also called a Transmission Line. Those steel towers you see out in the country, with usually six wires, are the power company transmission lines.

The operational needs for the two are completely different. I will skip the Radio Frequency (RF) lines here because you seem to be asking about the power company ones. (also the RF ones are harder to understand!)

In your Summary,

Power is assumed to be given (a constant) when justifying why we need high voltage to minimize losses through the lines. However, I think power depends on load.

I agree with the second sentence. The electricity supplied to your home is essentially at a constant voltage. To keep that voltage constant with wide load variations, the source impedance must be low. That means there should be very little resistance between you and the generators in the power plant.

Remember Ohms law, V=I²R, the voltage drop across a resistance is proportional to the square of the current flowing thru it.

Also the power used by you is W=V×I. Since voltage is fixed, when you use more power the current increases.

Now we finally get to the losses in the High Voltage transmission lines.

For simplicity, let's use some arbitrary numbers;
Distance to power plant: 10 miles, or 52 800 feet
Load for your neighborhood of one large city block:
240 000W at 120V = 2000A thru the lines to the power plant

Now if the transmission voltage is stepped up from 120V to 50 000V;
The current is W/V=I (240 000)/(50 000) = 4.8A

The end result is that the reduction in current allows a much smaller wire to the power plant. Or for a given larger wire size, many more city blocks can be supplied.

Well, this ended up a bit longer than expected. I hope it helps clear things up for you!

Cheers,
Tom

p.s. The numbers used above are arbitrary. There are others here than can likely refine them closer to what is actually used.

 

[sophiecentaur]

Oh, thanks for clarifying that impedance matching is not desirable in this case for it would waste half of power.

A common misapprehension and well worth pointing out. When power actually 'costs you' then you would generally choose a very low source impedance (aka 'voltage source). This wiki article discusses it. While you will always lose some power on any transmission line, systems are designed to minimise transmission losses. High transmitted power levels will usually involve high voltage sources - National Grid uses hundreds of kV (despite the safety issues with those voltages). At the other extreme, a motor car uses just 12V with short runs of thick wire (very low series resistance)

I still have trouble understanding how we can talk about "generated power" if the transmitted power depends upon the load.

That's just down to The Conservation of Energy. Same with money: the buyer pays for the item PLUS commission and taxes.
OR are you asking about the stated "Transmitter Power"? That refers to maximum power available from the generating equipment - often a lot more than the actual power demanded by the total load.