Imaginary Numbers & Euler's Formula: Circuit Analysis Explained

this video is sponsored by brilliant because of phase and Euler's formula these two things are pretty much the why

and how electrical engineers use imaginary numbers in circuit analysis with just resistors we don't need

imaginary numbers because the ratio of voltage and current for resistor is always constant thus they are in phase

and the math is much easier even if the input is not sinusoidal voltage and current will still move

together with a capacitor this is not the case if you have a sinusoidal input and plot

voltage versus current across a capacitor you'll find they are out of phase by 90 degrees and that's found

with the capacitor equation so if I set C to 1 just for Simplicity and the voltage across the capacitor is

sine T then current is cosine T which can also be written as sine of t plus 90 degrees

so when it comes to sinusoidal inputs capacitors cause this shift along with likely an amplitude change but the shift

is what we care about because the math becomes more difficult when you have sinusoids with different phases even

just adding you can't just combine like terms here you can use a trig Identity or something

cooler which is Euler's formula and imaginary numbers now the first reason this formula is

amazing is that it allows us to write any complex number in exponential form I'm not going to go into the visuals of

this but as a numerical example if you take the complex number one plus one I you can write that as root 2 cosine of

45 degrees plus I times root 2 sine of 45 degrees and this can be Rewritten in exponential

form according to Euler's formula and all that matters here is the fact that there's always a way to write a

complex number in this format which can then be Rewritten in exponential form

even if that complex number looks like this it can still be simplified to sum a plus bi

then with just some basic trig we find in this case the real component 1 equals root 3.25 cosine of 56.3 degrees while

the imaginary component equals root 3.25 sine of 56.3 degrees and now that we have the correct format

same coefficient and same argument we can write this in exponential form it's full because when given a sinusoid

there's no real constant you can multiply it by to change the phase a here only changes the amplitude

but if you have e to the i t which can be written out with sinusoids and then multiply it by some complex constant

let's say 2E to the I times 30 degrees which is just this however I'm going to keep it in exponential form because all

we have to do is use simple rules of exponents to get our answer then from Euler's formula we can write

it out as such and I know there are two sinusoids here but just look at the individual

components like the real part started out as cosine t after multiplying by just a complex

number the real part of the output is still just a cosine function but with a different amplitude and phase which is

exactly what we observe with voltage versus current in a capacitor or inductor

so there's a peek at how useful complex numbers are they help when amplitude and phase shifts occur

now another thing we can do with Euler's formula is essentially solve problems that involve sinusoids by replacing them

with this exponential going through with the math then just looking at the real component

of the output not a great explanation but here's what I mean you can find the double angle

formula for cosine as well as many other trig identities by just using Euler's formula the first step is to just plug

two Theta into the argument because now the real part of this is what we're trying to find a formula for

and now you can use a simple rule of exponents to rewrite the left side as e to the I Theta all squared which is just

Euler's formula squared if we foil this out using the fact that I squared is negative one we'll get this

here which again is equal to what we started with e to the I times 2 Theta

so setting the real Parts equal to one another we get cosine of 2 theta equals cosine

squared minus sine squared which hey is the double angle formula for cosine but that's not all as the double angle

formula for sine can be found by setting the imaginary components equal to one another

so that's the idea when given a problem with cosine of something or sign sometimes we can instead make it more

complicated by adding I sine of that thing and setting it equal to our exponential

because now we can use properties of exponents to simplify this and at the end the individual components in a way

reveal our answer by the way that's also how you can solve this problem I put on the screen earlier

just put them in exponential form simplify then look at the real component so now let's apply this to the capacitor

equation and because I is used for current to avoid confusion from here on the imaginary unit will be represented

with a j instead of I which is actually what electrical engineers do now if the input is a generic sinusoid

some a cosine of Omega t plus Theta well again we can find current by just taking a derivative

so we get current is this here but let's instead say voltage is a e to the J times Omega t plus Theta which is

weird and makes no sense physically but the real part of the exponential is our actual voltage

so at the end we'll just see what the real component looks like okay so we'll go through with the

derivative and J root negative one is just a constant by the way so we get current equals this here

but a e to the J Omega t plus Theta is just our voltage so we can actually say current equals J

Omega c times V and voltage as we've defined it is this here with the real and imaginary

components so multiplying that by J Omega C we get this and if we just look at the real

component we see it's the same as what we got before when we simply took a derivative of a cosine function

now Euler's formula did not make any of this any easier but notice how by using Euler's formula we turned an equation

that involves a derivative into one that doesn't that's the key detail that makes this

all useful for real circuits because now we can rearrange this equation and make a ratio of voltage and

current this is the impedance of a capacitor which is kind of like resistance the 1

over Omega C part is the ratio in amplitude a versus Omega c a

but the J the imaginary unit tells us there will also be a phase shift of 90 degrees cosine versus negative sine

so now let's put it all together with a circuit that has a resistor a capacitor and a sinusoidal input

Financial equation that would be used to solve for the voltage across the capacitor

but if the input is sinusoidal then the capacitor voltage will be as well so we know the solution is going to have

the form a cosine of Omega t plus Theta but instead of plugging that in for V Sub C and solving which would not be fun

we can instead say V Sub C is this exponential because as we just saw the time

derivative of this is J Omega times itself VC so plugging that in for DV DT the

differential equation becomes an algebraic one and after factoring out a v Sub C then

rearranging the capacitor voltage we're looking for comes out to 1 over 1 plus J Omega RC times the input voltage

and this is just a complex number which can be Rewritten using Euler's formula so I need to make some room here but if

Omega R and C are all let's say one then the coefficient becomes one over one plus J which can be Rewritten as 0.5

minus 0.5 J or root 2 over 2 times e to the J times minus 45 degrees so if the input voltage is something

like 3 cosine of t plus 10 degrees we can replace that with Euler's formula or put it in phasor form

knowing the actual voltage is the real component so if we plug that in for the input

voltage then we find the output voltage across the capacitor is this here but of course we only care about the

real component which is 3 root 2 over 2 cosine of T minus 35 degrees that's how imaginary numbers help us

instead of having to solve a differential equation we can solve an algebraic one with imaginary numbers

once we have this equation the only math we have to do is figure out what that complex coefficient is in exponential

form because that exponential tells us in this case to multiply the input voltage

amplitude by root 2 over 2 and shift the phase by negative 45 degrees and when you have a circuit with just

resistors to find the output voltage across R2 you use this equation a simple voltage divider

when R2 is replaced by a capacitor we get the equation we saw earlier but by multiplying the top and bottom by 1 over

J Omega C we get this which hey looks very similar to the equation above just instead of R2 we have 1 over J Omega C

which remember is the impedance of the capacitor so we can treat impedance just like

resistance in that we can add the values when components are in series we do the

reciprocals when they're in parallel but the only difference is you're going to get complex numbers because there will

be a phase shift that happens as well as an amplitude change which we can find algebraically using Euler's formula

and while this is all useful for sinusoidal inputs when you have anything else

the math actually still works for the most part because all functions can be made up of sinusoids

it's beyond this video but if you apply the Fourier transform to both sides of this equation or the capacitor slash

inductor equations you get the same impedances and voltage dividers as before

you would just have to do inverse transforms at the end to get an actual solution

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