@@cougar2013Not really, it's more profound than that; if we just wanted a root of -1 you can work in ℝ[X]/(X²+1) and there's your root We use complex numbers so that a single complex exponential Ae^(jωt) can represent a sinusoidal wave and encode its frequency, phase and amplitude at once. And all that is to be able to work with ideal sine waves and then apply the superposition principle (corresponding to the Fourier transform) with which you deduce what happens to your circuit with any kind of signal
I'm a mathematician/physicist doing a career switch over to electrical engineering, and this has been the clearest introduction to Impedance I've seen. Thanks a lot
@@Anonynomymon-fh8wyYou know electrical engineers can do that too when specified in the field, right? That's the deal of electrical engineering, many majors. You can specify in things like robotics and programming. You pick one or two and specialize. I'm a student yet but I started learning coding in lessons. Then I got curious and learnt some languages. I wasn't interested in AI but anyone who does can do a job about it pretty sure
@@Anonynomymon-fh8wyML is an overlap of electrical engineering and computer engineering. But still, EE is the most versatile and broad field of engineering. My master's will be an overlap of math models, machine learning, and digital signal processing
The math trick of changing from sinusoids to complex numbers is based on ignoring the transient part e^kt in the total response, but in general it take few miliseconds and can be ignored, so the trick is useful.
It is when the inductors and capacitors have some stored energy. We usually analyse the circuit when they are in steady state means the transient response has died out long ago.
These circuits can be generalized to the s-domain, where the transient response is not ignored. Using purely complex exponentials is a stepping stone to doing that
This was mind blowing, this puts to rest my age old quest on how Laplace's transform converts differential into algebraic equations and the inverse Laplace does the opposite. Also had a smile on my face when I saw how Vc in an RC circuit literally maps to the total impedance of the circuit, just as in two resistors. I learned all of these 2 decades ago and it all came back to me. You don't need to do rote learning if you understand it deep at this level. A million thanks for all your efforts. Don't underestimate the value of your work, it brings a lot of smiles to a lot of people.
It was in the 1980's when I learned about Laplace transform and complex numbers for electrical engineering during my polytechnical college education. Later on, I also did some eddy-current testing, in which this theory is put into practice. It was mesmerizing to observe how all of these concepts actually work, touching the objects and manipulating the sensors to act as they are supposed to do. It's not just theory for its own sake.
I remember having a 15 minute “discussion” with my professor in my EE lesson at uni (consistently in the top 5 in the world!) where he literally claimed there were an imaginary number of electrons moving around in the circuit. After a while people really can lose sight of their abstractions.
@@nicolaignazio it’s really not that uncommon. He was perfectly competent at his research and still is world renowned in control theory. It was so second nature to him to think of current as complex that he didn’t see the need to keep the justification for it around…. 🤷♂️
@@ivanmirones9220 He definitely wasn’t. This was 1999, in one of the first lectures. I’d done a year out, so was a bit more confident, everyone else was straight from school. Eventually I gave up. I got to know him more later, he just literally did not give a shit about the physical reality of systems, or considered it a distraction - he was completely consumed by ‘pure’ control theory.
He's metaphorically right: It's called "displacement current" and correspond on the woobling of electrons inside insulators as an effect of an alternate electric field. That result in a 90° shifted alternate current that is the virtual continuation, inside the dielectric, of the alternate current flowing within a capacitor. You can - at that point- represent it as a flow of imaginary virtual particles. If math works, why not?
In electrochemistry, phasor notation becomes really useful when investigating the electrode-electrolyte interface, in a technique called electrochemical impedance spectroscopy (EIS). The resistor component is modelled by the real impedance of charge transfer (Zct) and the real impedance of solution (Zs), and the imaginary impedance modelled by the electrical double-layer's capacitance (Zdl). Zdl and Zct are in parallel, and that parallel system is in series with Zs. It's a fairly simple circuit, but as you sweep the frequency, you can get a really nice pair of plots to model at which frequencies you have maximum and minimum impedance within a range. The plot of combined impedance with respect to frequency is called the Bode plot, which shows the big picture of impedance. Then for each point of data given, you can break the combined impedance into real and imaginary components and get what is called the Nyquist plot, which is probably one of my favorite things in the world. I love it so much because Nyquist plots give you a really nice dome-like curve, the peak of which tells you the maximum capacitance of the EDL, the farthest right point of which tells the maximum impedance of charge transfer, and beyond that, you can see if charge transfer becomes less influential than mass transfer below a certain frequency. I hope you'll do a deep dive into EIS at some point. It's just so cool. Also really useful for engineering things related to energy conversion and storage, which is something I'm sure a lot of your subscribers (me included) are very passionate about.
I wish you were my professor in the early 90's when I did my EE. I remember I had a lot of confusion at first and it took a lot of hours of studying to finally get it. Your explanation is so clear it's amazing! Thank you!
It’s the coolest stuff I’ve seen in the physical world. Linear systems and controls are damn fascinating! Well, so is waveguide and antenna theory, and communication theory! This video shows how you do steady state system analysis, but laplace transforms, which aren’t too different, let you do transient analysis using just algebra too. And what is MIND BLOWINGis that since you’re really just solving ODEs, all this stuff applies to dynamic mechanical (or chemical!!!) systems too! Eg a spring is an inductor, a shock absorber is a capacitor!
Short answer : Fourier and Laplace Transform. Avoiding solving PDEs and ODEs to study system characteristics such as stability,equilibrium points, system delays and response to different signals with different frequencies.
Right off the bat this video clarified what “phase” meant at 0:40. I knew it was related to the phase of the signal (how much it’s shifted back or forth) but I didn’t realize that it was the phase of the current in relation to the voltage, and that the derivative in the capacitor equation is what introduces it into the circuit. I graduated from ece 5 years ago, learned AC circuit analysis 8 years ago. We just learned to calculate things, not get an intuition for why things work. This video is delightful for someone who spent time calculating and wants to get some insight into how things work.
@@nestorv7627If I could code, I'd do an example of RLC analysis using phasor diagrams. Voltage, current, & power, with resonance visuals if ya like seeing humps/dips and cutoffs. Put the values on sliders so you can see what them numbers do. Str8 vectors and waves using real numbers.
- Excellent. - Very well done: clear/concise/insightful. - Thx. - And, as a formally degreed engineer (Electrical), plus a self-taught math teacher, I especially appreciated your presentation. - Keep up the great content...
As a current electrical enginggering student, this was a very informative video on a topic I had issues understanding before in class, this clears up the confusion.
My interpretation is that due to Euler’s formula, one can convert to exponential notation. Since the complex exponential is the eigenfunction of linear, time invariant systems (and derivatives and integrals become algebraic multiplication due to this choice of basis), we now have an easy way of solving these. What many people do not realize is (IIR) filtering pretty much always involves a phase shift, and phase, time, and derivatives/integrals are all related
You cannot imagine how helpful this video is for me ,Everyone in my class doesn't seem to care why we are suddenly using complexe numbers on vibrations and electricity , Thank you so much !
I just wanna point out that a lot of professors may remove points, when using the method shown around 6:30. If v = Acos(wt), then v = Ae^(jwt) is technically incorrect. To fix it, just express it as v = RE(Ae^(jwt)), where RE() indicates the real part of the complex exponetial; or express it as (A/2)*(e^(jwt) + e^(-jwt)). Even the professsors that say, "I don't care if you box in the right answer; I only grade the work shown," will probably only take 1 point off on a 10 pointer (not a big deal). I enjoyed the video Zach, but I thought I should give this PSA for anyone that has a professor named Wanda or Leonard. edit: "(A/2)*(e^(jwt) + e^(-jwt))" was incorrectly express as "(A/2)*(Ae^(jwt) + Ae^(-jwt))"
@@nestorv7627 No problem. There was actually an error in my logic caused from coping and pasting the text. I wouldn't have noticed it, if you never commented, so thank you.
@@leonhardeuler9839 I think Zach did it because loves his phasor analysis too much. There is also half as much work to show, and also half the work to grade, so professors and students may like it for that reason. The alternative is to differentiate the last expression of v = cos(wt) = RE(Ae^(jwt)) = (A/2)*(e^(jwt) + e^(-jwt)) and multiply by C, then you don't need to worry about "ignoring" the imaginary part; it will just cancel out in the end anyways. I think the real question is why convert it to complex exponentials in the first place, and the answer is all about the basics of frequency domain analysis.
@@nestorv7627 It was my dream to become a mathematician, but my DE prof pulled me aside to talk (he probably didn't get many students that score perfectly on both exams and hw); I told him that I wanted to study math, but he persuaded me to get into engineering instead. I graduate EE next sem, and the only time I felt that I learned any real engineering was the second/third week of EM waves when we were solving problems using the derivative form of maxwell's equations. Every other time, they just lump the necessary parameters, needed to actually solve the DEs directly, into empirical ones, then only expect you to use the "formula" or "procedure" instead of actually understanding the actual physical process. Am I missing something, or would you also say that, for the most part, engineers suck at understanding the actual math used in their field?
It's a great tool when dealing with single frequencies, because then going through the Laplace transform can be annoying, especially with more complicated systems.
Thank you for this great video. At first I was bothered by the extra imaginary number, thinking 'Now there's extra stuff. What is it?', but then I realized that we are actually *removing* stuff in a way. We are transforming the problem from having the input voltage specified as a arbitrarily complicated function to something more constrained. We notice that the input function 'sin(...)' can be 'calculated' by reading of numbers on a disk after rotating it. So we've reduced the problem to that of rotations. Rotations are less powerful in the general case, but they are sufficient for the problem at hand, and they are simple linear algebra and so easy to work with. The imaginary part has always just been part of the "structure" that rotates. And taking the real part is "reading the number of of the disk".
Its been a minute since i was in college, but it seems i remember the Laplace transform being very similar. They could be used to reduce a 2nd order differential equation to an algebra problem. Same thing here. A simple transform using Eulers formula does the same thing. What a great tool.
interestingly enough, further into a circuits course you end up using the laplace transform to do exactly that. The laplace domain actually takes advantage of imaginary numbers as well, the S variable ends up being an imaginary number.
Actually, the s domain has it built in from the start. Laplace’s “s” parameter is a complex number. Its real part represents a signal’s growth and decay rate, and the imaginary part represents a signal’s oscillations. Both of these encode any signal you can take a Laplace transform of. Just like a Fourier transform encodes any repeating signal through pure oscillations, the Laplace transform does the same but also encodes how the signal grows or decays
Or you can use a Laplace Transform to convert a differential equation into its algebraic equivalent. Solve. Then use Inverse Laplace Transform. This way, you are not restricted to just sinusoidal inputs.
Imaginary numbers are simply tools to represent the stored energy in the reactive components. Phase shift is the effect of this energy storage (inductors use magnetic field and caps use electric field to store energy).
Thank you! I learned this stuff 5 years ago and I was still confused every time I saw j. I knew it involved phase shifts, but I didn't know how or why. Now it all makes sense.
In the 1900’s, Charles Proteus Steinmetz introduced the concept of phasor algebra which involves the complex numbers. This tool was used to analyze the performance of electric power systems in steady-state and avoids the analysis of sinusoids which often leads to finding the solution to differential equations.
I had this in circuits, signals and systems for my engineering courses. This is basic stuff for a circuit with an ac current, an impedance, resistance and capacitance. This is what the circuit is when one uses differential equations : V(t) = d^2I/dt^2 + dC/dt + L where I is current, C is capacitance and L is impedance.
First learned AC circuit analysis in the Navy. We would just compute impedances (Z = 1/(omega*C)) and keep track of 'angles'. So if V is 120v / 0 degrees, and capacitor had impedance of 10 ohms/ 90 we simply too 120v/10ohms = 12 amps at (0 - 90= -90 degrees). And combining capacitors, inductors, and resistors in circuits was similar to resistor-only combinations, but keeping track of the +90, -90, and 0 degree angles. At the beginner level, we didn't go into the complex math, just 'remember the rules to keep track of angles. When multiplying, add the angles and when dividing, subtract"
This is basically the same way they teach in highschool before complex numbers. Note that by "keeping track of the angles" and using special formulas to add there values with different phases you are basically doing the same calculations as if you were to do it with complex numbers by hand (using trigonometry and pythagoras' theorem) While simples to understand, this method has the disadvantage of having different formulas (for example you cant just calculate the parallel resistance with XY/(X+Y), as you would have no way of accounting for the phases). Also, if there is both series and parallel connections, this can quickly spiral down into lots of trigonometry just so you can keep the phases, while with complex numbers, you are basically hiding all this nasty arithmetic by using a single 2D number to store both the amplitude and phase (and mathematically this allows you to use the EXACT SAME formulas as you would for DC, so you don't need to learn any new formulas except for the generic complex arithmetic)
The simplification of using phasor notation comes with the hidden assumption that all operations on the voltage will be linear operations. For non-linear devices (for instance when analyzing distortion products) you have to go back to sines and cosines
No. I disagree. The hidden but necessary assumption is that the analysis takes place at a constant frequency. All passive components are modeled as being linear. There is a phasor representation for the Fourier series, if ... at least... non-linearity can be modeled around harmonics of a fundamental frequency.
i majored electronic engineering ,what i had been taught is totally different to this vedio. the reason why we define "impedance" is because when we're trying to analyse a circuit, a mathematic transform would be used frequently; that is, Laplace transform. In short terms, Laplace transform is a integral transform which transform the target function defined on time into complex number. mostly we denoted the Laplace transform of function f(t) as F(s) = ℒ [f(t)] ,where s = a+ib, providing a and b is real number. Electronic circuits can be only written as math functions. which means every Electronic circuits have their own equivalent math functions. Let's say the RC circuit shown is the vedio is our target of analysis and we assume the AC source can be a random periodic function denoted as E(t). By KVL, we know E(t) = Vr + Vc which can be E(t) = RC( dVc/dt ) + Vc. the equation above is a first order ODE and we can take Laplace transform from both sides ℒ [E(t)] = RC*{sℒ [Vc(t)] - Vc(0)} + ℒ [Vc(t)] the transformed equation above would no longer have derivatives and would be a quadratic function in terms of s. and after rearranging the transformed equation, we will have the equation shown in 11:01. where s = jω. Since the simple circuit like the RC circuit we solved still takes some times to be solved. Therefore, we defined impedence and transformed the input signals into complex world to make things easier.
I would prefer the much easier and more natural representation of complex numbers as a vector with two components. Also I'd like to see a diagram with a voltage axis and a current axis.
i think the ideal representation would be as a vector with two components in a geometric algebra, because then we get the benefit of being able to multiply by a constant to get a phase shift as well as the intuition associated with vectors
i think it would cause a bit of confusion as well, bc they don't behave as vectors all the time, for example, there is no dot product or cross product. It makes sense to think of them like vectors, but i think we should do only that, not redefine how we represent them
@@andredetoni897 There is nothing wrong in redefining how we represent something. It all depends on context of the problem and whichever convention you start with you should also finish with, just don't change midstream. It's kind of like whipping up some batter. You're not going to start to stir or whip with a fork and while still whipping it transition the fork into a whisk. That just doesn't happen. Intuition and ingenuity is what allows us to elaborate such models and representations. If people didn't redefine things throughout time then we wouldn't have the tools of calculus that we do today. Here's a prime example; how many times throughout the last 200 years has the model of an atom changed? How many times has various equations, sequences, series, summations, etc... changed due to someone redefining them in different terms? We wouldn't have the complex number system today if people didn't redefine things! Having various representations of the same thing can give us many different perspectives of their properties and behaviors.
@@skilz8098 you're right about all of that man, absolutely. I just think that in this example particularly, it's not necessary. We already treat complex numbers like vectors in a multitude of ways (like calculating power relationships) but they're not fully vectors. I think it's tough to redefine a complex number as a vector when it follows only some of the vectors property, that's all. And I understand completely your point, it would be very convenient. I'm studying to be a professor, so I have kind of strong opinions about how we should teach topics (it doesn't mean they're right tho), and for what I've seen, it's better to define clearly that complex numbers ARE NOT vectors. You make great points thoigh, and I agree with you
@@andredetoni897 As for me, I see numbers and vectors as being the same thing. A number that has a value has a magnitude. Also the sign of that number implies its direction. Take for example +3 and -3. Their magnitude is the same, a weighted or measurable value of ||3||. Yet they differ in sign. One is positive and the other is negative. They are facing or pointing in the opposite direction. If you take the dot product over the product of their magnitudes it is equal to the cos of the angle between them. In other words cos(t) = a dot b over | a | | b |. We can take the points (3,0) and (-3,0) and we end up with cos(t) = (3,0)dot(-3,0) / |3| |-3| which would give us 180 degrees or PI radians. This is the angle of a straight line. So when we look at any number on the number line even the number 1. It is unit vector with its tail at (0,0) and its head at (1,0). Which can be simplified to the unit vector or simply . So in truth all scalars are vectors where their magnitude is emphasized and their directions is their sign. Vectors and numbers are not mutually different. In fact I tend to think that they are the same exact thing because even the number 1 itself is a linear translation of 1 unit in some arbitrary direction. Did you know that the Unit Circle and the Pythagorean Theorem are embedded both in the expression y = x as well as the simplest and first arithmetic expression 1+1 = 2? I could show proof here but his is quite long as it is... If you're interested just le me know and I can show how all mathematics as well as all applied mathematics (science) are related well particularly physics...
Because of the Fourier transform ... you replace differential equation I = CdV/dt or V = LdI/dt by normal equations with complex numbers. I = jwCV or V = jwLI It requires a small math course but d/dt => multiplication by jw by the Fourier transform It's not only in electricity ... in mechanics you can do the same to solve periodic movement like a pendulum. And when things are not periodical, but still linear, you have an extension with Laplace transform where you replace jw by s, which is a full complex number with real and imaginary parts, not only imaginary like jw. (and btw, electricians use j instead of i for complex numbers because i is symbol of current)
I was so peeved when a professor first told me that you can't just multiply vectors by each other. Why not? There must be something better than dot and cross products, right? Then I took circuit analysis. Super, super satisfying to be able to just multiply (2,2) by (2,2) and get (0,4).
What's wrong with dot and cross products? They are very rational ideas for vectors, one tells you "how parallel" they are. The other tells you how perpendicular and also gives you the normal of the plane they form. Those are GREAT operations If you just want to have linearity then you can use linear algebra, a Matrix is what multiplies a vector, and they can also multiply other matrices. And you can just to Mv and get a new vector
One major reason is definitely because it turns inductors and capacitors into "resistors", and this makes it much easier to simplify a circuit; it is pretty much like when you use the Laplace Transform to turn a differential equation into an algebraic equation (not quite the same thing, but something similar to that).
12 Years ago, when I was preparing for my Electrotechnology exam at Uni, I remember having to learn how to measure an Electric cirtuit with capacitance on it. It took me the entire night decomposing lectures, until I learned how the bloody thing works with the Euler's method. If you only did the videos then, you'd save me a night of painful studying.
Don’t forget that, for certain conventions, the magnitude of signal phasors (voltage and current) in polar form is actual their time-domain RMS values, not their amplitude. This is usually the amplitude divided by sqrt(2), though you’ll have to use the integral formula for RMS values when the signal has harmonics or is not a perfect sinusoid.
4:31 The size of the exponent on the right hand side threw me off for a second. It looks like it's theta that's squared when, in fact, the 2 should be the same size as i and theta
yeah, you get the theory of it correct, but a few things: First, one has to understand that this technique only applies to steady state circuit analysis. For transient phenomena (what happens when you throw a switch) you really do need to deep dive into the differential equations. Also, once you get a feel for the technique you can start omitting the imaginary unit. One would write a phasor as A
The two types of math video: "Today we're going to use trig to discover some really useful law of nature and gain a deeper insight into the world" and "Today we're going to prove that I don't understand math but I think this result is profound because I use calculus"
By the end of first year, i learned that imaginary related to phasor form like video explained. At the second year after learned math of complex laplace fourier z transform, i realized that impedance same as 1/sC with s=jw, and can work with any signal and condition. It just so much easier if lecturer told that those just advance math that used in introduction circuit class.
Because complex numbers have parts that remain absolutely orthogonal to each other. That is what makes them so powerful and why they are generally represented on a plane. So this keeps sine and cosine orthogonal so they don't interfere with each other. Brilliant!.
1. This mixed domain (time and phasor) analysis is tricky and treacherous. 2. You need to indicate the reference polarities of the ac voltage source and the voltage drop across the resistor to write the KVL equation for the circuit.
I think the clearer reason this is useful for well-behaved non-sinusoidal waves is that its complex Fourier series, in addition to linearity, allow you to consider a voltage wave's individual harmonics' contributions to the voltage/current equations, where each harmonic is an weighted exponential as you describe. By allowing omega to vary you are simultaneously finding impedance for all of the harmonics.
Real world, there is trace resistance in inductors, equivalent series resistance in capacitors, and inductance across sets of conductors. Those traits make real world circuits more complex. Also, reactive circuits have very real current flow in what theoretical math solves with imaginary units (the reason to designate reactance with j, not i), and those can cause very real power in conductors and in reactive parts, due to actual or equivalent series resistance. So, in doing this kind of engineering math, one has to keep an awareness of both approximations for convenience, and when a math theory process represents what very real physics aspect of real circuits. We now move on to AM directional antenna broadcast arrays, where the Earth is an electrical element subject to weather, and performance is optimal when complex series of reactive elements have a flat mostly resistive bandwidth.... But again, this is about math tools to handle complex vector sums, where thinking in terms of space and electrical waveform traits as vector elements allows understanding the math theory to physical reality relationship.
EE Here. I did this in college 35 years ago. I always thought of it in my head as just knowing, through repetition, the solution to the differential for the Ls and Cs and plugging it in.
These have a lot of applications, but do keep in mind, what you are seeing here are called "Phasors", they are a way to express periodic behavior (So waves, like sinusoids) using complex nombers. In such a way that you take full advantage of the fact that the derivative of e^x is itself But in the context of differential equations complex numbers DO show up in quite a few places, namely the Laplace Transform (And to a leader extent the Fourier transform) here the ideas that the complex numbers represent are quite different (But just as useful tho). So keep that in mind if you find some complex numbers in the wild while studying about differential equations
Also mechanical engineers use them, since vibration and acoustic analysis is made of identical equations! We even use electro-mechanical analogies to write (simplified) mechanical systems with circuits!
Mechanical, acoustic and electromagnetic vibrations are simple harmonic movements and are expressed with the same differential equation, which is why the phasor diagram can be made in all of them.
it is actually very natural to analyze is just a change of basis form vector space to espectrum or frecuency space. instead of representing a sinusoidal as a fucnion you represent it only by its amplitude and is just a vertical line. then is usefull for analizing signals that have a bunch of frecuencys. you can analyze music with it, but you can analize mechanical vibration also
what is important is that V(t) i(i) are rotating vectors. Linear algebra allows us to removed calculus complexity. In Linear algebra ±π/2 allows product to be 1. This also applies to Fourier and Laplace - Euler was just the beginning. Linear Algebra rules EE my brain flips when I see ( sin and cos ). EE and ME 's need to be more mathematically oriented in direction of Linear Algebra. It makes thinking about and analysis of complex analog circuits simple.
As a mechanical engineering student, I wish I watched this video when I was studying impedance (yes in my university, MechEs are required to learn some electrical stuff) thanks man.
Zack needs to be a math/physics professor. Best teacher on math science topics I've ever seen, and that's counting the profs I've had at the undergraduate and graduate level. I've been following his vids since his "math that engineering majors take" video.
At 11:43, you may have meant to reference the "Laplace Transform" instead of the "Fourier Transform". While these two transforms share similarities, they also have nuanced differences, particularly concerning integration limits, which can be crucial in various applications. Your video offers fantastic visual and audio explanations, effectively simplifying complex numbers for a clearer understanding. Thank you.
The most useful class I took in 16 years of school was numerical analysis. Virtually overnight I threw out years of acquired mathematical tricks, transforms, and approximations. I didn't have to solve another DE. All I ever wanted was a number, anyway, and numerical methods could give me that number to any precision needed.
We often used 'j' instead of 'i' in electrical engineering to avoid confusing it with the "i" used for current. Otherwise a good and clear explanation.
because we invent our own rules when convenient but in real life you just use the simulator and keep modyfing values until you get the desired output...
Short answer: Complex numbers are seemingly tailor made to handle oscillations, by interpreting them as a projection of a rotation. (I haven't seen the video yet, but I assume that that's the main point.)
This concept is analogous to quantum operators, for example the quantum operator for momentum ip/h has an i for the same reason that iwC does. The i captures the phase shift due to taking the time derivative of a wave. In the quantum operator, p represents a time derivative, and likewise in power equations w (frequency) also takes the place of d/dt. The convenience of these algebra operators in power system theory is only possibly because power equations use a single frequency (50 or 60 hz) that allows w to replace d/dt. Is d/dt is used instead, power flow equations become hellish 2nd order differential equations that nobody wants to spend time solving, but are required for solving for transient conditions with changing frequency.
short answer: because by using imaginary numbers we don't have to solve horrifying differential equations for every single circuit
We have nice imagination I'd say
the horror... the horror....
Actually, it’s just so that we can have a squared quantity come out negative.
you misspelled beautiful
@@cougar2013Not really, it's more profound than that; if we just wanted a root of -1 you can work in ℝ[X]/(X²+1) and there's your root
We use complex numbers so that a single complex exponential Ae^(jωt) can represent a sinusoidal wave and encode its frequency, phase and amplitude at once.
And all that is to be able to work with ideal sine waves and then apply the superposition principle (corresponding to the Fourier transform) with which you deduce what happens to your circuit with any kind of signal
I'm a mathematician/physicist doing a career switch over to electrical engineering, and this has been the clearest introduction to Impedance I've seen. Thanks a lot
Why electrical and not something more trendy like Artificial Intelligence?
@@Anonynomymon-fh8wyYou know electrical engineers can do that too when specified in the field, right? That's the deal of electrical engineering, many majors. You can specify in things like robotics and programming. You pick one or two and specialize. I'm a student yet but I started learning coding in lessons. Then I got curious and learnt some languages. I wasn't interested in AI but anyone who does can do a job about it pretty sure
@@Anonynomymon-fh8wy Artificial intelligence is considered too concentrated and specialized, EE's can also do Ai jobs and do a masters in Ai.
May I ask why? I'm interested in Physics and the statistical component of mathematics, I'd love to hear about your experience.
@@Anonynomymon-fh8wyML is an overlap of electrical engineering and computer engineering. But still, EE is the most versatile and broad field of engineering. My master's will be an overlap of math models, machine learning, and digital signal processing
The math trick of changing from sinusoids to complex numbers is based on ignoring the transient part e^kt in the total response, but in general it take few miliseconds and can be ignored, so the trick is useful.
It is when the inductors and capacitors have some stored energy. We usually analyse the circuit when they are in steady state means the transient response has died out long ago.
Quando se adotou números complexos na Engenharia Elétrica, não tinha como se detectar um fenômeno de frações de segundo.
These circuits can be generalized to the s-domain, where the transient response is not ignored. Using purely complex exponentials is a stepping stone to doing that
This was mind blowing, this puts to rest my age old quest on how Laplace's transform converts differential into algebraic equations and the inverse Laplace does the opposite. Also had a smile on my face when I saw how Vc in an RC circuit literally maps to the total impedance of the circuit, just as in two resistors. I learned all of these 2 decades ago and it all came back to me. You don't need to do rote learning if you understand it deep at this level. A million thanks for all your efforts. Don't underestimate the value of your work, it brings a lot of smiles to a lot of people.
I got chills the moment my mind made the link between the Laplace transform and the imaginary numbers.
It was in the 1980's when I learned about Laplace transform and complex numbers for electrical engineering during my polytechnical college education. Later on, I also did some eddy-current testing, in which this theory is put into practice. It was mesmerizing to observe how all of these concepts actually work, touching the objects and manipulating the sensors to act as they are supposed to do. It's not just theory for its own sake.
I remember having a 15 minute “discussion” with my professor in my EE lesson at uni (consistently in the top 5 in the world!) where he literally claimed there were an imaginary number of electrons moving around in the circuit. After a while people really can lose sight of their abstractions.
How did he become a professor?
@@nicolaignazio it’s really not that uncommon. He was perfectly competent at his research and still is world renowned in control theory. It was so second nature to him to think of current as complex that he didn’t see the need to keep the justification for it around…. 🤷♂️
Have you considered the possibility that the professor was messing with you?
@@ivanmirones9220 He definitely wasn’t. This was 1999, in one of the first lectures. I’d done a year out, so was a bit more confident, everyone else was straight from school. Eventually I gave up. I got to know him more later, he just literally did not give a shit about the physical reality of systems, or considered it a distraction - he was completely consumed by ‘pure’ control theory.
He's metaphorically right: It's called "displacement current" and correspond on the woobling of electrons inside insulators as an effect of an alternate electric field. That result in a 90° shifted alternate current that is the virtual continuation, inside the dielectric, of the alternate current flowing within a capacitor. You can - at that point- represent it as a flow of imaginary virtual particles. If math works, why not?
In electrochemistry, phasor notation becomes really useful when investigating the electrode-electrolyte interface, in a technique called electrochemical impedance spectroscopy (EIS). The resistor component is modelled by the real impedance of charge transfer (Zct) and the real impedance of solution (Zs), and the imaginary impedance modelled by the electrical double-layer's capacitance (Zdl). Zdl and Zct are in parallel, and that parallel system is in series with Zs. It's a fairly simple circuit, but as you sweep the frequency, you can get a really nice pair of plots to model at which frequencies you have maximum and minimum impedance within a range.
The plot of combined impedance with respect to frequency is called the Bode plot, which shows the big picture of impedance. Then for each point of data given, you can break the combined impedance into real and imaginary components and get what is called the Nyquist plot, which is probably one of my favorite things in the world. I love it so much because Nyquist plots give you a really nice dome-like curve, the peak of which tells you the maximum capacitance of the EDL, the farthest right point of which tells the maximum impedance of charge transfer, and beyond that, you can see if charge transfer becomes less influential than mass transfer below a certain frequency.
I hope you'll do a deep dive into EIS at some point. It's just so cool. Also really useful for engineering things related to energy conversion and storage, which is something I'm sure a lot of your subscribers (me included) are very passionate about.
I would love a video on this topic
@@rennoc6478 facts
Man nerds
had us in the first half not gonna lie
The part with the double angle formulas completely blew my mind! I really liked this video and sometimes wish you'd upload more of these!
I wish you were my professor in the early 90's when I did my EE. I remember I had a lot of confusion at first and it took a lot of hours of studying to finally get it. Your explanation is so clear it's amazing! Thank you!
Going to Graduate as an Electrical Engineer in June, 2024. This video sparked inside me the excitement that I felt when I first learnt about these. 😄
Welcome to poverty world of poors ...
Welcome to hell then.
It’s the coolest stuff I’ve seen in the physical world. Linear systems and controls are damn fascinating! Well, so is waveguide and antenna theory, and communication theory! This video shows how you do steady state system analysis, but laplace transforms, which aren’t too different, let you do transient analysis using just algebra too. And what is MIND BLOWINGis that since you’re really just solving ODEs, all this stuff applies to dynamic mechanical (or chemical!!!) systems too! Eg a spring is an inductor, a shock absorber is a capacitor!
Congratulations on your degree
I chose electrical engineering because of you, and I love it! This is literally what I‘m learning right now!
When you realize free energy exist
@@lukiepoole9254 when you steal a line from your neighbours or charge your phone at restaurant/hotel.
same here!
@@mandarbamane4268 Haha no. Parametric resonance exist and electrical engineers are NOT taught about it.
Short answer : Fourier and Laplace Transform. Avoiding solving PDEs and ODEs to study system characteristics such as stability,equilibrium points, system delays and response to different signals with different frequencies.
Actually this Is something different. It's called Steinmetz Transform, look It up. The connections are incredibles.
Right off the bat this video clarified what “phase” meant at 0:40. I knew it was related to the phase of the signal (how much it’s shifted back or forth) but I didn’t realize that it was the phase of the current in relation to the voltage, and that the derivative in the capacitor equation is what introduces it into the circuit. I graduated from ece 5 years ago, learned AC circuit analysis 8 years ago. We just learned to calculate things, not get an intuition for why things work. This video is delightful for someone who spent time calculating and wants to get some insight into how things work.
Because real numbers are too easy for us!
because I sat there in class and they just spewed it at us
Because real numbers are too complicated for us*
@@nestorv7627 yes, sometimes the longer way is actually the shortcut
@@nestorv7627If I could code, I'd do an example of RLC analysis using phasor diagrams. Voltage, current, & power, with resonance visuals if ya like seeing humps/dips and cutoffs. Put the values on sliders so you can see what them numbers do.
Str8 vectors and waves using real numbers.
@@nestorv7627 Because real numbers are complex for us.
- Excellent.
- Very well done: clear/concise/insightful.
- Thx.
- And, as a formally degreed engineer (Electrical), plus a self-taught math teacher, I especially appreciated your presentation.
- Keep up the great content...
As a current electrical enginggering student, this was a very informative video on a topic I had issues understanding before in class, this clears up the confusion.
My interpretation is that due to Euler’s formula, one can convert to exponential notation. Since the complex exponential is the eigenfunction of linear, time invariant systems (and derivatives and integrals become algebraic multiplication due to this choice of basis), we now have an easy way of solving these.
What many people do not realize is (IIR) filtering pretty much always involves a phase shift, and phase, time, and derivatives/integrals are all related
You cannot imagine how helpful this video is for me ,Everyone in my class doesn't seem to care why we are suddenly using complexe numbers on vibrations and electricity , Thank you so much !
I just wanna point out that a lot of professors may remove points, when using the method shown around 6:30. If v = Acos(wt), then v = Ae^(jwt) is technically incorrect. To fix it, just express it as v = RE(Ae^(jwt)), where RE() indicates the real part of the complex exponetial; or express it as (A/2)*(e^(jwt) + e^(-jwt)).
Even the professsors that say, "I don't care if you box in the right answer; I only grade the work shown," will probably only take 1 point off on a 10 pointer (not a big deal).
I enjoyed the video Zach, but I thought I should give this PSA for anyone that has a professor named Wanda or Leonard.
edit: "(A/2)*(e^(jwt) + e^(-jwt))" was incorrectly express as "(A/2)*(Ae^(jwt) + Ae^(-jwt))"
Thank you for this. As a mathematician switching to EE for a fresh start, it scares me how engineers sometimes ignore nuances like these
Why do we ignore the imaginary part?
@@nestorv7627 No problem. There was actually an error in my logic caused from coping and pasting the text. I wouldn't have noticed it, if you never commented, so thank you.
@@leonhardeuler9839 I think Zach did it because loves his phasor analysis too much. There is also half as much work to show, and also half the work to grade, so professors and students may like it for that reason. The alternative is to differentiate the last expression of v = cos(wt) = RE(Ae^(jwt)) = (A/2)*(e^(jwt) + e^(-jwt)) and multiply by C, then you don't need to worry about "ignoring" the imaginary part; it will just cancel out in the end anyways. I think the real question is why convert it to complex exponentials in the first place, and the answer is all about the basics of frequency domain analysis.
@@nestorv7627 It was my dream to become a mathematician, but my DE prof pulled me aside to talk (he probably didn't get many students that score perfectly on both exams and hw); I told him that I wanted to study math, but he persuaded me to get into engineering instead. I graduate EE next sem, and the only time I felt that I learned any real engineering was the second/third week of EM waves when we were solving problems using the derivative form of maxwell's equations. Every other time, they just lump the necessary parameters, needed to actually solve the DEs directly, into empirical ones, then only expect you to use the "formula" or "procedure" instead of actually understanding the actual physical process.
Am I missing something, or would you also say that, for the most part, engineers suck at understanding the actual math used in their field?
I've been dealing with phasors, impedances for over a month now without understanding its purpose.. this made it so much clearer, thank you!
I just started a Circuit Analysis and Design class and man has this cleared up my confusion. Thanks!
That class is bread and butter EE!
It's a great tool when dealing with single frequencies, because then going through the Laplace transform can be annoying, especially with more complicated systems.
Thank you for this great video. At first I was bothered by the extra imaginary number, thinking 'Now there's extra stuff. What is it?', but then I realized that we are actually *removing* stuff in a way. We are transforming the problem from having the input voltage specified as a arbitrarily complicated function to something more constrained. We notice that the input function 'sin(...)' can be 'calculated' by reading of numbers on a disk after rotating it. So we've reduced the problem to that of rotations. Rotations are less powerful in the general case, but they are sufficient for the problem at hand, and they are simple linear algebra and so easy to work with. The imaginary part has always just been part of the "structure" that rotates. And taking the real part is "reading the number of of the disk".
Its been a minute since i was in college, but it seems i remember the Laplace transform being very similar. They could be used to reduce a 2nd order differential equation to an algebra problem. Same thing here. A simple transform using Eulers formula does the same thing. What a great tool.
interestingly enough, further into a circuits course you end up using the laplace transform to do exactly that. The laplace domain actually takes advantage of imaginary numbers as well, the S variable ends up being an imaginary number.
Actually, the s domain has it built in from the start. Laplace’s “s” parameter is a complex number. Its real part represents a signal’s growth and decay rate, and the imaginary part represents a signal’s oscillations. Both of these encode any signal you can take a Laplace transform of. Just like a Fourier transform encodes any repeating signal through pure oscillations, the Laplace transform does the same but also encodes how the signal grows or decays
That was very nice refresher. Thank you.
BEST start to a youtube video ever
Top lesson. Well done and plenty to expand on as individual work through to check the solutions.
i LOVE your vids, they're so helpful . would love to see you make vids on the Math in Machine Learning!
Or you can use a Laplace Transform to convert a differential equation into its algebraic equivalent. Solve. Then use Inverse Laplace Transform. This way, you are not restricted to just sinusoidal inputs.
MS in EE and I knew all these equations... but don't think I really understood where they came from. Awesome vid. Thank you!
Ok
Imaginary numbers are simply tools to represent the stored energy in the reactive components. Phase shift is the effect of this energy storage (inductors use magnetic field and caps use electric field to store energy).
Now put an inductor in that series and make it REALLY interesting!
I love you put the straight answer in the front. Great
Thank you! I learned this stuff 5 years ago and I was still confused every time I saw j. I knew it involved phase shifts, but I didn't know how or why. Now it all makes sense.
In the 1900’s, Charles Proteus Steinmetz introduced the concept of phasor algebra which involves the complex numbers. This tool was used to analyze the performance of electric power systems in steady-state and avoids the analysis of sinusoids which often leads to finding the solution to differential equations.
I had this in circuits, signals and systems for my engineering courses. This is basic stuff for a circuit with an ac current, an impedance, resistance and capacitance. This is what the circuit is when one uses differential equations : V(t) = d^2I/dt^2 + dC/dt + L where I is current, C is capacitance and L is impedance.
First learned AC circuit analysis in the Navy. We would just compute impedances (Z = 1/(omega*C)) and keep track of 'angles'. So if V is 120v / 0 degrees, and capacitor had impedance of 10 ohms/ 90 we simply too 120v/10ohms = 12 amps at (0 - 90= -90 degrees). And combining capacitors, inductors, and resistors in circuits was similar to resistor-only combinations, but keeping track of the +90, -90, and 0 degree angles. At the beginner level, we didn't go into the complex math, just 'remember the rules to keep track of angles. When multiplying, add the angles and when dividing, subtract"
This is basically the same way they teach in highschool before complex numbers. Note that by "keeping track of the angles" and using special formulas to add there values with different phases you are basically doing the same calculations as if you were to do it with complex numbers by hand (using trigonometry and pythagoras' theorem)
While simples to understand, this method has the disadvantage of having different formulas (for example you cant just calculate the parallel resistance with XY/(X+Y), as you would have no way of accounting for the phases).
Also, if there is both series and parallel connections, this can quickly spiral down into lots of trigonometry just so you can keep the phases, while with complex numbers, you are basically hiding all this nasty arithmetic by using a single 2D number to store both the amplitude and phase (and mathematically this allows you to use the EXACT SAME formulas as you would for DC, so you don't need to learn any new formulas except for the generic complex arithmetic)
These videos are incredible. I understand everything that my courses haven't been able to
The timing of these videos with my schooling is incredible! Thank you.
The simplification of using phasor notation comes with the hidden assumption that all operations on the voltage will be linear operations. For non-linear devices (for instance when analyzing distortion products) you have to go back to sines and cosines
I thought we EEs were sharp til I started studying fluids. I don’t EVER want to have to find any PDE BVP solution in closed form….
No. I disagree. The hidden but necessary assumption is that the analysis takes place at a constant frequency. All passive components are modeled as being linear. There is a phasor representation for the Fourier series, if ... at least... non-linearity can be modeled around harmonics of a fundamental frequency.
@@willthecat3861Yeah, but like, the second you add a transistors, or even a simple spark gap, then the math becomes a LOT harder
i majored electronic engineering ,what i had been taught is totally different to this vedio. the reason why we define "impedance" is because when we're trying to analyse a circuit, a mathematic transform would be used frequently; that is, Laplace transform. In short terms, Laplace transform is a integral transform which transform the target function defined on time into complex number. mostly we denoted the Laplace transform of function f(t) as F(s) = ℒ [f(t)] ,where s = a+ib, providing a and b is real number.
Electronic circuits can be only written as math functions. which means every Electronic circuits have their own equivalent math functions. Let's say the RC circuit shown is the vedio is our target of analysis and we assume the AC source can be a random periodic function denoted as E(t). By KVL, we know E(t) = Vr + Vc which can be E(t) = RC( dVc/dt ) + Vc.
the equation above is a first order ODE and we can take Laplace transform from both sides
ℒ [E(t)] = RC*{sℒ [Vc(t)] - Vc(0)} + ℒ [Vc(t)]
the transformed equation above would no longer have derivatives and would be a quadratic function in terms of s. and after rearranging the transformed equation, we will have the equation shown in 11:01. where s = jω.
Since the simple circuit like the RC circuit we solved still takes some times to be solved. Therefore, we defined impedence and transformed the input signals into complex world to make things easier.
Thanks for such a clear explanation to something that looked so “complex”
I Watched this not expecting this to be by the same guy who makes leprechauns put pots of gold in a country I can't remember the name of.
I would prefer the much easier and more natural representation of complex numbers as a vector with two components. Also I'd like to see a diagram with a voltage axis and a current axis.
i think the ideal representation would be as a vector with two components in a geometric algebra, because then we get the benefit of being able to multiply by a constant to get a phase shift as well as the intuition associated with vectors
i think it would cause a bit of confusion as well, bc they don't behave as vectors all the time, for example, there is no dot product or cross product. It makes sense to think of them like vectors, but i think we should do only that, not redefine how we represent them
@@andredetoni897 There is nothing wrong in redefining how we represent something. It all depends on context of the problem and whichever convention you start with you should also finish with, just don't change midstream. It's kind of like whipping up some batter. You're not going to start to stir or whip with a fork and while still whipping it transition the fork into a whisk. That just doesn't happen. Intuition and ingenuity is what allows us to elaborate such models and representations. If people didn't redefine things throughout time then we wouldn't have the tools of calculus that we do today. Here's a prime example; how many times throughout the last 200 years has the model of an atom changed? How many times has various equations, sequences, series, summations, etc... changed due to someone redefining them in different terms? We wouldn't have the complex number system today if people didn't redefine things! Having various representations of the same thing can give us many different perspectives of their properties and behaviors.
@@skilz8098 you're right about all of that man, absolutely. I just think that in this example particularly, it's not necessary. We already treat complex numbers like vectors in a multitude of ways (like calculating power relationships) but they're not fully vectors. I think it's tough to redefine a complex number as a vector when it follows only some of the vectors property, that's all. And I understand completely your point, it would be very convenient. I'm studying to be a professor, so I have kind of strong opinions about how we should teach topics (it doesn't mean they're right tho), and for what I've seen, it's better to define clearly that complex numbers ARE NOT vectors. You make great points thoigh, and I agree with you
@@andredetoni897 As for me, I see numbers and vectors as being the same thing. A number that has a value has a magnitude. Also the sign of that number implies its direction. Take for example +3 and -3. Their magnitude is the same, a weighted or measurable value of ||3||. Yet they differ in sign. One is positive and the other is negative. They are facing or pointing in the opposite direction. If you take the dot product over the product of their magnitudes it is equal to the cos of the angle between them. In other words cos(t) = a dot b over | a | | b |. We can take the points (3,0) and (-3,0) and we end up with cos(t) = (3,0)dot(-3,0) / |3| |-3| which would give us 180 degrees or PI radians. This is the angle of a straight line. So when we look at any number on the number line even the number 1. It is unit vector with its tail at (0,0) and its head at (1,0). Which can be simplified to the unit vector or simply . So in truth all scalars are vectors where their magnitude is emphasized and their directions is their sign. Vectors and numbers are not mutually different. In fact I tend to think that they are the same exact thing because even the number 1 itself is a linear translation of 1 unit in some arbitrary direction. Did you know that the Unit Circle and the Pythagorean Theorem are embedded both in the expression y = x as well as the simplest and first arithmetic expression 1+1 = 2? I could show proof here but his is quite long as it is... If you're interested just le me know and I can show how all mathematics as well as all applied mathematics (science) are related well particularly physics...
Because of the Fourier transform ... you replace differential equation I = CdV/dt or V = LdI/dt by normal equations with complex numbers. I = jwCV or V = jwLI
It requires a small math course but d/dt => multiplication by jw by the Fourier transform
It's not only in electricity ... in mechanics you can do the same to solve periodic movement like a pendulum.
And when things are not periodical, but still linear, you have an extension with Laplace transform where you replace jw by s, which is a full complex number with real and imaginary parts, not only imaginary like jw.
(and btw, electricians use j instead of i for complex numbers because i is symbol of current)
I sometimes forget Zack how experienced Zach is in math and electrical engineering.
I was so peeved when a professor first told me that you can't just multiply vectors by each other. Why not? There must be something better than dot and cross products, right?
Then I took circuit analysis. Super, super satisfying to be able to just multiply (2,2) by (2,2) and get (0,4).
What's wrong with dot and cross products? They are very rational ideas for vectors, one tells you "how parallel" they are. The other tells you how perpendicular and also gives you the normal of the plane they form. Those are GREAT operations
If you just want to have linearity then you can use linear algebra, a Matrix is what multiplies a vector, and they can also multiply other matrices. And you can just to Mv and get a new vector
One major reason is definitely because it turns inductors and capacitors into "resistors", and this makes it much easier to simplify a circuit;
it is pretty much like when you use the Laplace Transform to turn a differential equation into an algebraic equation (not quite the same thing, but something similar to that).
12 Years ago, when I was preparing for my Electrotechnology exam at Uni, I remember having to learn how to measure an Electric cirtuit with capacitance on it.
It took me the entire night decomposing lectures, until I learned how the bloody thing works with the Euler's method.
If you only did the videos then, you'd save me a night of painful studying.
Don’t forget that, for certain conventions, the magnitude of signal phasors (voltage and current) in polar form is actual their time-domain RMS values, not their amplitude. This is usually the amplitude divided by sqrt(2), though you’ll have to use the integral formula for RMS values when the signal has harmonics or is not a perfect sinusoid.
As a student of IIT Kharagpur, I also used it in my class XII, while studying AC.
The EE right of passage is internalizing s=jw
8:13 Well technically there is a transient factor of e^-alpha t which decays over time but for most practical applications you dont need it.
Yes he should have told that it's steady state analysis
beautifully and clearly presented, thank you
4:31 The size of the exponent on the right hand side threw me off for a second. It looks like it's theta that's squared when, in fact, the 2 should be the same size as i and theta
This video can clarify only things to people that have just studied the concepts.
yeah, you get the theory of it correct, but a few things: First, one has to understand that this technique only applies to steady state circuit analysis. For transient phenomena (what happens when you throw a switch) you really do need to deep dive into the differential equations. Also, once you get a feel for the technique you can start omitting the imaginary unit. One would write a phasor as A
this was just awsome. I am a mechanical engineering student and this helped me a lot with my basics of ee course. Thank you so much
The two types of math video:
"Today we're going to use trig to discover some really useful law of nature and gain a deeper insight into the world"
and
"Today we're going to prove that I don't understand math but I think this result is profound because I use calculus"
Essentially it's because the eigenvalues for the general second-order linear equation are complex.
By the end of first year, i learned that imaginary related to phasor form like video explained. At the second year after learned math of complex laplace fourier z transform, i realized that impedance same as 1/sC with s=jw, and can work with any signal and condition. It just so much easier if lecturer told that those just advance math that used in introduction circuit class.
Because complex numbers have parts that remain absolutely orthogonal to each other. That is what makes them so powerful and why they are generally represented on a plane.
So this keeps sine and cosine orthogonal so they don't interfere with each other. Brilliant!.
1. This mixed domain (time and phasor) analysis is tricky and treacherous. 2. You need to indicate the reference polarities of the ac voltage source and the voltage drop across the resistor to write the KVL equation for the circuit.
I think the clearer reason this is useful for well-behaved non-sinusoidal waves is that its complex Fourier series, in addition to linearity, allow you to consider a voltage wave's individual harmonics' contributions to the voltage/current equations, where each harmonic is an weighted exponential as you describe. By allowing omega to vary you are simultaneously finding impedance for all of the harmonics.
Angle theta should be expressed in radians if omega is in radian per second.
This imaginary thing can only be done, if the given input is in form of sinusoidal.
Or you can write any function into sum of infinite number of sinusoidal waves, by using Fourier transform
Real world, there is trace resistance in inductors, equivalent series resistance in capacitors, and inductance across sets of conductors. Those traits make real world circuits more complex.
Also, reactive circuits have very real current flow in what theoretical math solves with imaginary units (the reason to designate reactance with j, not i), and those can cause very real power in conductors and in reactive parts, due to actual or equivalent series resistance.
So, in doing this kind of engineering math, one has to keep an awareness of both approximations for convenience, and when a math theory process represents what very real physics aspect of real circuits.
We now move on to AM directional antenna broadcast arrays, where the Earth is an electrical element subject to weather, and performance is optimal when complex series of reactive elements have a flat mostly resistive bandwidth.... But again, this is about math tools to handle complex vector sums, where thinking in terms of space and electrical waveform traits as vector elements allows understanding the math theory to physical reality relationship.
Love your video’s man!
EE Here. I did this in college 35 years ago. I always thought of it in my head as just knowing, through repetition, the solution to the differential for the Ls and Cs and plugging it in.
Yup, but it is a little deeper that that, it is a tool that works for ALL constant periodic behavior. It removed a TON of differential equations
very good video - flow, graphics, refreshes what we knew 30 years ago. good tone
i love watching these types of videos while im still in precalc
I'm gonna have to revisit my Differential Equations class and see if I can apply this to those topics.
These have a lot of applications, but do keep in mind, what you are seeing here are called "Phasors", they are a way to express periodic behavior (So waves, like sinusoids) using complex nombers. In such a way that you take full advantage of the fact that the derivative of e^x is itself
But in the context of differential equations complex numbers DO show up in quite a few places, namely the Laplace Transform (And to a leader extent the Fourier transform) here the ideas that the complex numbers represent are quite different (But just as useful tho). So keep that in mind if you find some complex numbers in the wild while studying about differential equations
Takes me back to intro to signal processing and circuits 2... thanks for the refresher about why we do this at all.
Also mechanical engineers use them, since vibration and acoustic analysis is made of identical equations! We even use electro-mechanical analogies to write (simplified) mechanical systems with circuits!
Mechanical, acoustic and electromagnetic vibrations are simple harmonic movements and are expressed with the same differential equation, which is why the phasor diagram can be made in all of them.
Euler's formula, my beloved. Saving me from memorizing dozens of trig identities!
Really great vídeo, I love understands these formulas and methodology that in the university teach like a mechanical method. Big hug from Colombia.
it is actually very natural to analyze is just a change of basis form vector space to espectrum or frecuency space. instead of representing a sinusoidal as a fucnion you represent it only by its amplitude and is just a vertical line. then is usefull for analizing signals that have a bunch of frecuencys. you can analyze music with it, but you can analize mechanical vibration also
what is important is that V(t) i(i) are rotating vectors. Linear algebra allows us to removed calculus complexity. In Linear algebra ±π/2 allows product to be 1. This also applies to Fourier and Laplace - Euler was just the beginning. Linear Algebra rules EE my brain flips when I see ( sin and cos ). EE and ME 's need to be more mathematically oriented in direction of Linear Algebra. It makes thinking about and analysis of complex analog circuits simple.
You can also use Laplace's transform instead of Fourier's and exchenge 'jw' by 's'
As a mechanical engineering student, I wish I watched this video when I was studying impedance (yes in my university, MechEs are required to learn some electrical stuff) thanks man.
I think what’s important is to emphasize that it’s more of a tool than anything else.
Zack needs to be a math/physics professor. Best teacher on math science topics I've ever seen, and that's counting the profs I've had at the undergraduate and graduate level. I've been following his vids since his "math that engineering majors take" video.
A great summation. Thanks.
It remembers me to my days in electrical engineering..., thanks!
This video should absolutely mention that this transformation is only valid AFTER the transient response !
At 11:43, you may have meant to reference the "Laplace Transform" instead of the "Fourier Transform". While these two transforms share similarities, they also have nuanced differences, particularly concerning integration limits, which can be crucial in various applications. Your video offers fantastic visual and audio explanations, effectively simplifying complex numbers for a clearer understanding. Thank you.
Omg I'm legit learning this right now, and was thinking to my self "I wish there was some ez to understand videos on this" thanks math dude!
Saying good bye to the transient period! But very accurate for frequency domain analysis 😅
Six seconds in and he's answered the question. Now that's effeciency!
The most useful class I took in 16 years of school was numerical analysis. Virtually overnight I threw out years of acquired mathematical tricks, transforms, and approximations. I didn't have to solve another DE. All I ever wanted was a number, anyway, and numerical methods could give me that number to any precision needed.
I wanna learn that too.
We often used 'j' instead of 'i' in electrical engineering to avoid confusing it with the "i" used for current. Otherwise a good and clear explanation.
watch the video before commenting next time 😂😂😂😂😂😂
because we invent our own rules when convenient but in real life you just use the simulator and keep modyfing values until you get the desired output...
Great video! Only thing I couldn’t follow is why you were using degrees instead of radians :)
Engineers prefer to use degrees
Short answer: Complex numbers are seemingly tailor made to handle oscillations, by interpreting them as a projection of a rotation. (I haven't seen the video yet, but I assume that that's the main point.)
10 seconds in and this guy already answered the title question
A great video. Should have at least mentioned the Laplace transform at the end, to make the explanation complete.
He probably meant to say "Laplace Transform" when he said "Fourier Transform"
No you can *NOT* write 1+i as sqrt2 e^(45i). The angle MUST be in radians in exponential form.
They had the degrees sine after 45. So if we just treat the degrees sign as a scaler than maps degrees to radians, then the maths is true
@@gregoryfenn1462 Except that's not what it is.
1) Because they can
2) Because all oscillatory phenomena are easier to calculate when you have exp() instead of a bunch od sin() and cos() functions.
finally I understood! thanks!
This concept is analogous to quantum operators, for example the quantum operator for momentum ip/h has an i for the same reason that iwC does. The i captures the phase shift due to taking the time derivative of a wave. In the quantum operator, p represents a time derivative, and likewise in power equations w (frequency) also takes the place of d/dt. The convenience of these algebra operators in power system theory is only possibly because power equations use a single frequency (50 or 60 hz) that allows w to replace d/dt. Is d/dt is used instead, power flow equations become hellish 2nd order differential equations that nobody wants to spend time solving, but are required for solving for transient conditions with changing frequency.
kindly make more of the electrical engineering content😌. Where possible even tutor🙏