The problem basically gave me the parameters of a specific transmission line and asked for the electromagnetic values at a constant height of 1.8 meters from ground (approximately the height of a person's head), considering that the person could move on the X axis of a transverse section of the power line.
These were the given characteristics of the transmission tower:
I found the problem to be intriguing but I thought I could make it a lot more interesting if I could take it up to the next level, by using a variable height instead of a constant one and present the results as a color graph, almost like a temperature image, all using just Matlab.
I know this isn't exactly a project, more like a paper than anything else, but since there isn't much information about it and the results turned out to be interesting, I thought it would be worth sharing it.
The electromagnetic field, as the name says, is made of 2 components, the electric field and the magnetic field. This work will analyze them separately in different time instants, both in normal conditions and hypothetical abnormal or accidental situations.
The transmission line being analyzed is a 3-phased, 50 Hz alternating current power line considering a voltage of 400kV and a power of 1200MVA. Each phase is composed of 2 conductors ("bundled" 2 ways) each one having a diameter of 31.8 mm. There are also 2 overhead earth wires or shield wires on the top of the tower, each one having a diameter of 23.45 mm. All the important distances are represented on the previous image (in meters). The section being studied is an area perpendicular to the length of the conductors, considering 40 m for each side from the center of the tower (on the X axis) and a height of 50 m from the ground (on the Y axis).
First some physics. For the magnetic field we have Ampère's Law, which is an electromagnetism law that relates the magnetic field in a closed loop or surface with the electric current circulating through that same loop:
Applying this law to an infinite conductor and a circular surface we'll get:
If the current is equally distributed on the conductor we can consider the current density:
And replacing it back on Ampères Law:
The magnetic induction will be given by this:
For the electric field we have Gauss's Law, yet another electromagnetism law that establishes the relation between the electric field flowing through a closed surface, the Gaussian surface, and the sum of the electric charges inside a volume, limited by the that same surface:
Replacing it in Gauss's Law:
We'll have the electric field:
In terms of electric potential it'll be:
And finally, considering a null reference in infinite, it'll result in:
To determine the electromagnetic field of this transmission line, we're also gonna need to use the method of images, also known as method of mirror charges. This method is used as a mathematical tool to solve electromagnetic problems by adding a mirror image of the conductor, with the opposite charge, in relation to a common surface, in our case, the ground. To apply this, we just replicate the conductors underneath the soil, like the ground was a mirror, and consider the current on them to be the same but with opposite direction. The cross inside the conductors represents the back side of a vector and the point represents the front side, establishing the sense of direction.
Now we must analyze the transmission tower characteristics that were given and calculate relative positioning of all the conductors:
I'm considering the center of the image as the center of the referential for X and Y. We can now take the coordinates of all the conductors. I'm also representing them as imaginary numbers to make mathematical calculations easier. These are the coordinates for the real and imaginary conductors:
We need the distances between live conductors to be able to calculate the matrix of the coefficients of potential. Vertical and horizontal distances are directly obtained by analyzing the previous image but the diagonal distances need to be calculated. Having established all the coordinates, is now easier to get them. They are represented here:
To get the diagonal distance you can just calculate the modulus or absolute value of the difference between 2 points. The distances are:
We are also gonna need the distances between live conductors and the shield wires:
Which can be calculated the same way:
Now we know that the line to line voltage is 400kV, so the instantaneous line to neutral voltage is given by:
Where omega is the angular speed for a frequency of 50Hz. Each phase is composed of 2 conductors, therefore the voltage in each conductor is approximately the same. Also the shield wires are considered to have 0V.
We know this line has a power of 1200MVA, so we can calculate the current:
For this particular scenario we are considering a power factor of 1, even though this is obviously not what happens in reality. Although this is not a real situation, this consideration shall not change the values of the electric and magnetic fields, but just change the time instant when they occur. What this means is that the values of the final results are still correct, but their timings might not be the ones indicated.
These are the instantaneous current values for each phase:
With all these considerations done, we can finally start calculating the magnetic field. We have reached the equation for the magnetic induction:
Now we must attend to some factors:
We have seen that the direction of the current on the phase conductors is the opposite of the direction in the "image" or mirrored conductors. This must be represented in the equation with a mathematical operator, and since the coordinates are complex numbers, that operator can be just 'j' (I'm using 'j' as the imaginary unit instead of 'i' to avoid confusions with the currents).
Other factor we must pay attention happens when 'r' is larger than the radius 'R', because 'r' is the distance between the point you are analyzing and the conductor, which is given by a complex number, and the fact that is being used on the denominator of the equation means that the direction of the vector is gonna change and it'll give you a wrong result. To correct this, in that situation we are going to use the conjugate of 'r'.
Last factor to considerate is which current to use. If you use the complex expression, it's gonna result in always showing the worst case scenario, never really changing it's value. To avoid this, you need to make sure you are using the instantaneous values, mathematically speaking, this means that you must use just the real part or just the imaginary part of the current for each wire, and not both components of the current.I chose to use just the imaginary part.
Taking this into consideration, the new equation for the magnetic induction for the real conductors is now given by:
Where 'n' is the phase and 'a,b' is the conductor of that phase.
For the mirrored conductors the equation is similar but with a different direction:
With 'n' being the mirrored phase and 'a,b' the conductor.
The distance 'r' is always the distance between the point you are calculating the magnetic induction for and the coordinate of the respective conductor that is generating that induction.
To calculate the magnetic induction on the point 'Px' this equation must be applied for every conductor there is. Then the final value for the induction is given by their sum:
For presenting results, all that matters is the absolute value.
But I'm not just calculating a single point, I'm calculating an entire area, varying the values of X and Y, therefore, 'Bp' in actuality is a matrix. The area being analyzed is defined by:
Because of the processing capacity of my computer, I decided to use increments of just 0.1m which corresponds to 801 points for X and 501 points for Y, resulting in a total of 401301 points analyzed. The value of 'Bp' for each point will be saved in the induction matrix on the correspondent position:
The spacial distribution of the magnetic induction represents the magnetic field.
If I consider the initial problem, it asked for the electromagnetic values at a constant height of 1.8m but varying X. I'm going to answer it now, before I move on to the electromagnetic field for varying X and Y. It didn't specify the time instant, so I'm going to choose an instant of t=1.667ms because it's an instant where all phases have voltages and where the spacial distribution will be vertically symmetrical.
This is the magnetic induction for t=1.667ms and height=1.8m:
To make sense of this graph we should look at the voltage waveforms for this instant:
You can see that the phase S (the one in the center of the tower) has the highest absolute value and the R and T phases share the same value but lower than S, that is why the magnetic induction is higher in the middle.
Now lets move on to the electric field. This is a bit more complex than the magnetic induction. We already have the formula for calculating the electric field, however we don't have lambda or charge density, therefore we're gonna need to calculate them first. But we do have the equation for the electric potential at a given point, which we can use to calculate the charge in every conductor:
For the mirrored conductors, we don't need to calculate them, because as it happened with the current, their values will be the same but with opposite direction.
Adapting for the matrix form we'll have:
But we need the charges:
Where [A] is the matrix of the coefficients of potential and [V] is the voltages vector:
Vg1 and Vg2 are the voltages for the shield wires which are both equal to zero.
The coefficients of potential are given by:
Where 'c' is the distance between the conductor 'i' and a mirrored image 'j', both 'i' and 'j' varying from 1 to 3. And 'd' is the distance between the conductor 'i' and the conductor 'j'.
When i = j the equation is a bit different:
Here, 'e' is the distance between the conductor 'i' and the mirrored image 'i' but 'g' is the radius of the conductor 'i'.
Now it's time to calculate them. To do it, it's important to pay close attention the the previous figures of the relative positions and diagonal distances. They come:
The matrix [A] can now be filled:
And finally, the charges can be obtained by:
Going back to the electric field equation, we now have the electric charges values, so we can adapt to:
Just like before I use just the imaginary part of Q, to be able to get instant values and not just the maximum value. Here, 'rn' is the distance between the point we want to calculate the electric field for and conductor generating it, and 'rni' is the distance between that same point and the mirrored conductor generating the field.
The electric field for a specific point is given by the vectorial sum of all the values generated for all conductors:
I calculated this separately, once for the X component in which I used just the real part of 'rn' and 'rni', and then for the Y component where I used the imaginary part of 'rn' and 'rni'. Then I merge them together and get the absolute value:
By varying X and Y we'll get the values for a specific area:
The spacial distribution of this values gives us the electric field.
Again, before I move on to the electromagnetic field while varying X and Y, I'm gonna answer to the initial problem where the hight is constant at 1.8m. I chose the instant of t=1.667ms because it's an instant where all phases have voltages and where the spacial distribution will be vertically symmetrical.
This is the electric field for t=1.667ms and height=1.8m:
It's curious to observe that this graph shows that the electric field has actually a different shape than the magnetic induction for the same circumstance, basically quite the opposite form.
And this is it, my work would have been complete only if I wasn't so curious. But I wanted to know more, so by applying all these laws and equations, and varying Y as well, I got the real interesting graphs.
Since the frequency (f) is 50Hz and the period (T) is 20ms, I decided to analyze 6 time instants, from 0ms to 8.333ms, each with an interval of T/12 ms. I'm analyzing just 6 because after those, the values will start repeating, even though the voltage and current will have opposite values, in absolute values, they will be the same.
To have some kind of reference here are some common EMF values for domestic appliances:
- Fridge - 0,3μT and 90V/m;
- Stereo - 1μT and 90V/m;
- TV - 2μT and 60V/m;
- Toaster - 0,8μT and 40V/m;
Now I'll be presenting the results I've obtained for each instant, considering that everything is working normally and without any anomalies:
For t = 0 ms:
For t = 1.667 ms:
For t = 3.333 ms:
For t = 5 ms:
For t = 6.667 ms:
For t = 8.333 ms:
From this instant on, the graphs will start repeating themselves, passing through the same situations. Like this:
By analyzing the figures, it is curious to observe that sometimes, the center of the tower, at ground level, has lower EMF values than the sides of the tower, a fact that is commonly unknown by most, which proves that is safer to be right beneath than to be just next to it.
I also decided to analyze 3 different situations where anomalies have occurred to see how much they would change this electromagnetic distribution. For that I created the following situations:
- Phase T is interrupted (V3 = 0V and I3 = 0A)
- Phase S is interrupted (V2 = 0V and I2 = 0A)
- Both phase R and T are interrupted (V1 = V3 = 0V and I1 = I3 = 0A)
Abnormal situation 1) t = 1.667ms (Phase T = 0V):
Abnormal situation 2) t = 1.667 ms (Phase S = 0V):
Abnormal situation3) t = 1.667 ms (Phase R and T = 0V):
As we can observe, these abnormal situations cause more peculiar effects on the electromagnetic field than the normal working conditions. They are also generating higher EMF values at ground level, which shows that you are more affected by electromagnetic radiation when this scenarios happen.
And this is it. This was an interesting work that I've done and that a lot of people wonder about, hopefully this might bring a little light on the matter, even though this is not 100% exact, it's still a very very close approximation, and helps us understand a bit more about what's really going on around a high voltage transmission power line.
Here's a small slowed down animation representing the electric field for 40ms:
I've also used this work recently to experiment the plotly MATLAB/Octave API.
To make it compatible with Octave, you just have to change the "clearvars" function to "clear".