Hey yall!
So I'm an electrical engineering student and I wanted to try and put together a little blog post explaining how Alternating Current (AC) circuits work. This is mostly to help me study and understand it all myself, and like I said I am a student so there may be some errors. If any colleagues find an error in here please feel free to correct me!!
Okay so first off there are two ways to power a circuit: Direct Current (DC) and Alternating Current (AC). There was a whole 1800s/early 1900s twitter drama about which one was better and safer, but idc about all that tbh. All that you really need to know is that Tesla and Edison realized that their little spat was stupid and that AC and DC each had their own uses and then they kiss. THE END. Most house hold appliances use DC where as the outlets in most houses are AC. This is because AC is better at transmitting power over long distance where as DC is more efficient at shorter distance, so many house hold appliances need to convert the AC wall outlet voltage to a DC voltage. The most common and easily recognizable form of AC adapter is found in laptop chargers (and PC power supplies).
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That little black box is a AC Adapter :D (it says it right there :0). The actual reason why AC is better at long distances has something to do with the inherent stability of a wave opposed to a constant signal. A ball that spins will fly straighter and farther than a ball that was lobbed.
DC is relatively simple compared to AC. 70% of DC is just Ohm's law which I'm not gonna get into rn, but if you are interested here's the wikipedia article, and then the other 30% is kirchhoff's voltage/current laws and nodal analysis.
AC, on the other hand, is fairly complicated because it involves everyone's favorite highschool math subject, drum roll please:
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Trigonometry!!!! O_o
Some of you may be confused. "What do triangles have to do with electricity?". Well trig isn't just triangles rly, it's also waves. The sine function can be graphed as an infinitely repeating, well, graph that looks like this:
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The cosine function is the same as the sine function but shifted ahead by a quarter of a wave length, or +90° (the wave starts at point B).
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This gif from wikipedia helps to show how the angle θ is used as a variable for both functions. it does not show the relationship between it and complex numbers, but i don't wanna spend all day on explaining the basics of trig so I'm gonna move on.
Also side tangent here: I think that they should just be called complex numbers and not imaginary numbers when teaching them to students. The term "imaginary number" was coined by a critic of complex numbers and he called them "imaginary" cause he hated them and the name stuck, but then they were found to be extremely useful for explaining cyclic motion in mathematics.
ANYWAYS
Right circuit components. I'm going to assume that everyone mostly understands what a resistor is, if not its smth that resists the flow of electricity and its measured using Ω (ohms) and the circuit diagram symbol for it is a jagged saw wave.
Capacitors are mini batteries (basically) which impede the flow of electricity while the capacitor is charging and once it is fully charged it will act similar to an empty wire and will cause minimal obstruction of electricity. When power is disconnect from the circuit a capacitor will discharge its voltage into the circuit which will cause the circuit to stay powered for a time after being unplugged. A capacitor has a value of capacitance which is measured in F (farads). They also come in both polar (cares about orientation) and non-polar (doesn't care about orientation) forms. The circuit diagram symbol for a capacitor is two parallel bars of equal length for the non-polar ones, and for the polar ones its either one bar and a curved bar to denote the negative side, or its two bars that are different lengths with the smaller side denoting the negative side.
(ignore variable)
Inductors (physically I understand them the least but mathematically I understand them for the most part) are coils of wires which when electricity flows through them they create a magnetic field which can influence how electricity flows through it (descriptive ik ;-;). The inductance of an inductor is measured in H (Henry (omg hai henry :D)), and the circuit diagram symbol for an inductor is a little coil.
AC Voltage
So AC voltage has a few measurements, unlike DC its not just a single value that's either on or off (like bits in a computer :0). AC has peak to peak voltage (V_p-p), frequency (measured in Hertz, Hz), and voltage offset (V_offset). Typically a voltage signal would look like this:
Where the voltage varies from positive to negative voltage as time progresses. Physically this can be shown as the current (flow of voltage) in the circuit repeatedly changing directions.
The V_p-p measures the total variance of the voltage source (e.g. if a AC voltage source has a maximum voltage of +12V and a minimum voltage of -12V then the V_p-p would be (+12)-(-12)V = 24V). The frequency measures how quickly the voltage source alternates between its max and min, if it alternates once every second it is 1 Hz and if it alternates 1000 times every second then it is 1kHz. Finally the V_offset is how much the wave is shifted up/down from zero.
I apologize for the poor image quality here, but all the good images I could find to explain this were on the Electrical Engineering Stack Exchange which for some reason none of those images work here ;-;. Regardless in this image we can still see that the blue and green waves are identical, but the green wave seems to be shifted up by about 2.5V. This changes the green wave's max and min values from the blue wave's +/-5V to now be +7.5V and -2.5V. This keeps the same V_p-p, but causes the current to actually change directions less often, as the green wave spends the majority of it's time above 0V.
Impedance
All of the electrical components we went over earlier impede the flow of current in some way. We can express the impedance (Z) of in two primary ways: complex numbers, and phasor form.
When writing impedance with complex numbers there are two parts the real and the imaginary numbers.
Due to the fact that "i" is already take by the symbol for current, when working with electrical circuits we use "j" to show an imaginary number. Each electrical component have their own formulas to find the impedance. Resistors are the simplest way to find the impedance as impedance is measure in Ωs!
Z_R = R
There is no imaginary part needed to find the impedance of a resistor, as a resistor works the exact same regardless of the power source. The same cannot be said of capacitors and inductors though.
Z_L = jωL
Z_C = (-1)/(jωC)
The equations for both inductors and capacitors are not only purely imaginary numbers, but are also entirely dependent on the frequency of the power source. The symbol of lowercase omega (ω) is equivalent to 2(pi)*F_s. This does also bring up the fact that impedance is only present for AC circuits, as in a DC circuit the frequency is essentially zero, so the only things that impede a DC circuit for most of its operation time are resistors!
Let's do an example :D
Lets say that we have an AC voltage source with a frequency of 1 kHz and a 10 Ω resistor and a 1.2 mH inductor in series, what is the total impedance?
The circuit would look something like the image above. To find the total impedance first lets look at the resistor:
Z_R = 10
Perfect :D!!! Wow your so smart :3
Now if we want to find the impedance of the inductor it will be a little bit more complicated, as we need to look at the frequency of the power source. So it would look something like this:
Z_L = j(2*pi*1000)(1.2*10^-3)
≈ j7.540
Now that we have both parts we can find the total impedance of this circuit:
Z_t = Z_R + Z_L = 10 + j7.540
Great!! With this number we now have 1/3 of the great and all mighty ohm triangle!
Now lets do one more example!
Using the same circuit as before the voltage is 12∠-32°V, what is the current?
Uh oh!!!
What's this voltage value we're given???
Okay so don't panic, but its just in that secret and evil second form of writing impedance: phasors (not actually evil). Now I may have caused some confusion in my previous introduction of the ways of writing impedance, but complex numbers and phasors are not exclusive to impedance, and they can be used in voltage and current (I) as well!
Both complex numbers and phasors are are just really good ways to express cyclic patterns in mathematics (like an AC wave 😱), and the reason why we use both of them is because you can't multiply/divide complex numbers but you can add/subtract them, and u cannot add/subtract phasors but you can multiply/divide them.
So now lets approach this problem with our new found knowledge of phasors:
Ohm's law gives us that I = V/R, so we just need to divide the voltage that we were given by the impedance, or resistance, we found in the previous example. In order to do this we need to convert the complex number we got for our impedance to a phasor. To do this we need to find the two components of a phasor: the magnitude (the number) and the angle (the degrees).
|Z| = √(10^2+7.540^2)
= 12.524 Ω
θ = arctan(7.540/10)
=37.016°
Z_phasor = 12.524∠37.016°Ω
So now that we have both our voltage and impedance as phasors we can divide the two in order to find our voltage:
12∠-32°V/12.524∠37.016°Ω
= 12/12.524∠-32°-37.016°
=0.958∠-69.016°A
We did it :D!!!
We were able to find that the current flowing through our circuit is 0.958∠-69.016° Amps. Now you may be asking me what exactly that means? And tbh I need to study the theory a little bit more ;-;, but thats for a later day. I missed two classes and lunch while typing this blog post and i still have an exam for another class tmr as well soooooooooooo BYE!!
I hope that you found this blog interesting and that you have a great rest of your day/afternoon/evening/night, and always remember that you are awesome :D!!!!!!!
Signing off
- Cybele74 <3
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Comments
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Joey
I’m mechanical engineering; fuck this shit lmao I like my real numbers (do not bring up those cursed damped springs)
Coils > Springs
by Cybele74; ; Report
prismeau
Using BYJUS is such a power move ngl
I was just googling images to try and explain it all lol. I would've just drawn the graphs and diagrams all myself, but finding images online is much easier
by Cybele74; ; Report
ah np, cuz it's lowk a problematic service in India lol
by prismeau; ; Report
omg i was unaware, i'll swap the image for another one
by Cybele74; ; Report