[00:00] This is the scariest chart in electrical  engineering. It puts the fear of God into all undergraduate students. First time that you  ever laid eyes on this chart. What was your first impression? Holy. It's like terrifying. It  looks almost like a wormhole in some sci-fi
[00:14] movie. I mean, on many versions, it literally  says black magic. But for all the fear, it's everywhere. Millions of copies have been printed,  and it's built into the most advanced software we have today. That's because this chart solves one  of the most paradoxical problems in electrical
[00:30] engineering. So, we put it to the test. This is  crazy. We're getting way more power transfer, even though we've cut the line. How does that  make any sense? There have been some great YouTube videos explaining how to use it, like  one from my friend Zach Star. But I always felt
[00:45] like they were missing something essential. You  also, you want to know why this chart looks the way it does? Yes. But then I need to know the  physics. I would want to know what the circles are and what I'm looking at at first because  even once you know how to use this chart,
[00:59] you're still left wondering how it pulls off its  strangest bit of witchcraft, trapping infinity inside a finite circle. Well, to understand that,  we have to go back a hundred years to the origins of this chart. In 1928, Phillip H. Smith,  fresh out of electrical engineering school,
[01:14] got his first job at Bell Labs. The telephone  industry was booming. Americans were placing more than 65 million calls a day. But because  telephone calls had to go through copper cables, every call had a limit, the coast. At the time, no  telephone cables crossed the Atlantic. So the only
[01:32] way to call across continents was by using radio  waves. Smith began work on a mission to send radio signals from New Jersey to receiving stations  thousands of kilometers away. One in England and one in Argentina. With only one antenna, the  signal radiates across the whole sky, and only a
[01:48] small portion of the power reaches the receiver.  But with two antennas, the waves combine. In some directions, they cancel each other out, and in  others, they reinforce. So, right here in the middle, the power doubles. If you add even more  antennas, it narrows the beam further. Focus it to
[02:03] about 10° across, and the power in that direction  is 400 times stronger. So, to have any chance of their signal reaching across the globe, Smith's  team connected more than 20 smaller antennas into a massive directional array linked by over 2 km  of transmission line. Smith's job was to test this
[02:19] massive array. But when he tried sending a signal  from the source down the line to an antenna, well, he noticed something strange. Part of it was  bouncing back. And that reflection meant that a large portion of the power never reached the  antenna. If he wanted to send a signal across the
[02:33] planet, he needed to make sure that all the power  reached the emitting antennas. So, he needed to find a way to minimize the reflections. Now, the  reason Smith was getting these reflections is that he wasn't working with the kind of electricity  that we might be used to, like with a battery and
[02:48] a light bulb. Here the battery provides a constant  voltage which drives a steady current through the wire which lights up the bulb. This is direct  current or DC. But to generate radio signals, electrons have to oscillate back and forth at  radio frequency. As they accelerate, they create
[03:03] changing electromagnetic fields that propagate  out as radio waves. So the voltage has to rise and fall and reverse direction and the current  has to as well. That's alternating current or AC.
[03:15] In the simplest case, both the voltage and the  current are smooth sine waves. Now, in reality, it does get a bit more complicated than that, but  for now, let's just look at a single sinusoidal voltage and current. Let's recreate Smith's signal  by shaking a slinky up and down. You can think of
[03:31] my arm as the power source. So, waves travel down  the line to the other end. And you can see that they have some key characteristics. The distance  from one peak to another or from one valley to another is consistent. This is the wavelength  and how many peaks pass a point per second is
[03:46] the frequency of the wave. So if you know the  number of peaks that go by and you know the distance between those peaks, well by multiplying  the two together, you get the speed of the wave. The formula is that the speed is equal to the  wavelength times the frequency. In Smith's case,
[04:00] he needed to emit radio waves and they sit on the  electromagnetic spectrum alongside microwaves, infrared, and visible light. And light waves all  travel at the same speed in a vacuum, about 300 million m/s or c. In a transmission line, the  wave travels more slowly, but for a given line,
[04:16] that speed is still fixed. And if the speed is  fixed, well, that means that frequency must be inversely proportional to wavelength. Let  me show you. If you decrease the frequency,
[04:29] the wavelength gets longer. And if you  increase it, the wavelength gets shorter. And this wavelength is crucial to our problem.  We'll also show you in simulation here without the effects of friction and damping. See,  if we start out with a long wavelength,
[04:44] the signal barely changes during the time it  takes to travel down the slinky. Once the wave reaches the other fixed end, any reflection that  comes back looks almost identical to the original wave. The resulting interference of these two  waves doesn't change the overall shape of the
[04:58] original signal all that much. So, there's nothing  to worry about. But when I increase the frequency to the point where the wave becomes shorter than  the slinky itself. Well, now when the wave reaches the fixed end, the reflection back interferes with  the incoming wave. After a reflection, the forward
[05:15] and reflected waves combine. At some points,  the two waves cancel and the voltage is very small. And at other points, they add together.  This is what's called a standing wave pattern.
[05:31] In electrical systems, the standing wave pattern  can be a big problem. That's because if the reflections in your transmission line are bad  enough, the peak voltages can reach up to twice the input voltage. And if your line isn't rated  for that, well, it burns out. There was a huge
[05:46] reflection and powerful standing wave did this  to the inner conductor of that transmission line. This is the effect of the standing wave. It's not  made up. In most electrical systems we're used to,
[06:00] this isn't too much of a problem. Your house  supplies AC power at a frequency of 50 or 60 Hz. So that corresponds to a wavelength of about  5 to 6,000 km, which is far longer than any wire that takes power from the station to your home.  But Smith was generating radio waves in the MHz
[06:16] range. And at 10 MHz, the wavelength is about  30 m. With the transmission line he was working on more than 2 km long, well, that's many times  the wavelength. So reflections were significant. That's why we went to Imperial College London's  screened radio frequency anechoic chamber to see
[06:32] if we could recreate what Smith was up against. As  you can see, it's very small. It's very compact, but we can do a lot of antenna measurements in  here. We can do electromagnetic compatibility measurements here. Here's the experiment. We've  got a power source sending a radio signal down a
[06:47] transmission line to an antenna array, and we put  it in an anechoic chamber to reduce interference. We've got a receiving antenna at the other end  which is able to pick up the signal and tell us just how much power is actually making  it to our antennas. It's a scaled model of
[06:59] Smith's configuration. Now, based on our input  power and the distance between the antennas, we expect a signal of about -55 dB. But that's  not what we see. We've lost half the power.
[07:11] We're picking up around -59 dB. And because  signal is measured on a logarithmic scale, that 4-decibel drop means we've lost more than  half the power, exactly what was happening to Smith. So let's find out what's stealing it.  To do that, let's simplify this like a true
[07:27] engineer. We can model the electrical signal as  a simple sinusoidal voltage and current source which we draw like this. Next is the transmission  line. It consists of two conductors that form a loop. At the far end is the antenna and its exact  design varies. So for now we'll just represent it
[07:42] with a black box. That gives us a very simple  toy model with just three parts. And we can represent these parts with slinkies. Now, so  far we've used just one slinky with one set of properties to show how waves travel. But the real  setup is more than just the transmission line.
[07:57] Since both the antenna and the transmission  line have their own electrical properties, we can model Smith's whole system by tying two  slinkies together. One for the transmission line and the other for the antenna. First, we're  going to try two very different slinkies. One
[08:10] with a high mass per unit length and one with a  low mass per unit length. Let's see what happens. between the two slinkies. Part  of the energy continues through
[08:29] and part of it reflects back. The severity of  the reflection is captured by a ratio called the reflection coefficient. Its magnitude is just the  size of the reflected wave divided by the forward wave. But if the two slinkies have the same mass  per unit length, the wave passes straight through.
[08:44] The reflection coefficient is then zero. So when  there's a big mismatch in mass per unit length, we get large reflections. But when the properties  match, the reflections disappear. That's because how the wave actually travels through the slinky  depends on two properties. How tightly the slinky
[08:59] is stretched and this mass per unit length.  But since in this setup, both slinkies are tied together, they're under the same tension. So  then the only difference is their mass per unit length. If we can match this property, it's like  the discontinuity is gone and it's just one slinky
[09:12] again. So, you might think Smith could do the same  thing, except instead of matching slinkies, he had to match his transmission line to his antenna.  But what electrical property could do that? Well, the mass per unit length decides how easy it  is for a wave to travel through our slinky. So,
[09:27] an analogous version for our electrical system  might just be its resistance. Because the higher the resistance, the harder it is for current  to flow. That's Ohm's law. The resistance is equal to the voltage divided by the current. So  if a voltage wave hits the boundary between the
[09:41] transmission line and the antenna and the  resistance changes across that boundary, well the whole wave can't pass through. Some  of it reflects just like what we saw with the slinkies. Indeed in real life when we checked the  resistances didn't match. So the antenna array is
[09:55] at 12 1/2 ohms and the transmission line is at  50. So what we're going to do is we're going to add this 37 1/2 ohm or 40 ohm resistor to try to  bring it up so that they match and we'll see what happens to the power. Okay, what are we looking  So now it's uh yeah, it hasn't helped us at all.
[10:11] It's actually Yeah, we're at half the power.  Interesting. But why is that? Resistors have a problem. They work by taking power and dissipating  it through heat. So they're fundamentally lossy. That means by putting it in our system here,  we're going to lose energy no matter what. But
[10:26] have resistances. Take this disposable camera.  When I turn on the flash, it takes a moment to charge. You can hear it. Then when I take a  picture, it releases all that energy at once.
[10:43] That's because inside are two conductive plates  separated by a gap. When I turn on the flash, I apply a voltage across them. And because they're  not connected, charge builds up on each side, storing energy. This forms a capacitor, and its  ability to store charge is called its capacitance.
[10:58] When I take the picture, the circuit closes,  and that stored energy is released all at once, which produces the flash. But this isn't  just in cameras. A transmission line has two conductors held a distance apart. So it has  a capacitance as well, but it uses alternating
[11:12] current. So let's see what happens when we have  a capacitor with an alternating current signal. Okay. So we're going to make a chart and on  the horizontal axis I'm going to draw time. And on the vertical axis we'll  just have magnitude of our wave.
[11:29] Okay. So we'll start with our voltage  wave which we'll just draw like this. Now we'll draw a current wave in red.
[11:42] As the voltage rises, charge builds up. At  the peak voltage, that buildup stops. And since current is the flow of charge, when the  charges stop moving, at that point the current is zero. Then as the voltage falls, the charge  flows back. And that's when the current is the
[11:55] largest. So unlike resistance, voltage and current  don't peak at the same time. Since the horizontal axis is time, this shows the current peaking  before the voltage, always by the same amount, a quarter of a cycle. In other words, the voltage  lags the current by 90°. But there is another
[12:13] effect we need to take into account. That's  because whenever current flows through a wire, it creates a magnetic field. And if you coil that  wire many times over, the magnetic field from each loop adds together, which makes the effect much  stronger. That's why in circuits, you might often
[12:26] see components that look like this. But even  on a straight wire like our transmission line, these fields still exist and they affect the  way that our current and voltage waves behave. A changing magnetic field induces a voltage  that opposes any change in the current. And the
[12:41] faster you try to change that current, the more  voltage pushes back. When the current is at zero, the slope of the curve is at its steepest. So  the current is changing the fastest. Therefore, the voltage is at its peak. Further along  the cycle, the current reaches its maximum.
[12:56] For just an instant, it has stopped changing. So  the slope is flat, and that's when voltage drops to zero. Past the peak, the current begins to  fall and the voltage flips direction. So again, the voltage and current peaks don't line up.  The voltage leads the current by 90°. So that's
[13:13] why matching resistances alone wasn't enough in  our demo. With resistance, we can only describe the relative size of our voltage and current  waves or the magnitude. But it says nothing about their timing. Capacitance and inductance  shift the current wave relative to the voltage
[13:28] wave. That is they shift the phase. So a real  match needs two things to line up not just one. We have to match the magnitude and the phase. Now  one common way to represent something with both a magnitude and a phase is to use complex numbers.  They consist of two parts. A real part represented
[13:44] on the x-axis and an imaginary part represented  on the y-axis. This is the complex plane. Now, because electrical engineers use I to refer to  current, we're going to refer to imaginary numbers
[13:56] with a J. And just to shake things up even more,  most electrical engineering textbooks swap the order. So instead of saying something like 3J, we  have J3. Now, a purely resistive component changes the ratio of voltage and current waves relative  to each other, but it doesn't shift their phase.
[14:12] So on this plane, pure resistance lies along  the horizontal axis where phase is zero. Since we care about positive resistances, we're only  going to look at the right side of the plane. Now compare that to pure capacitors and  inductors. These components do shift the
[14:26] phase of our waves relative to each other. Since  an inductor causes voltage to lead current by 90°, we rotate it up to +90° on the vertical axis. A  capacitor causes voltage to lag current by 90°. So we rotate it down to - 90°. Put them together  and this vertical axis is called reactance.
[14:45] We can now describe any combination of resistance,  capacitance, and inductance pretty easily on this two-dimensional plane as just a combination of a  resistance value and a reactance. This combined quantity we represent with Z. It's called the  impedance. Now, impedance is still defined as
[15:01] the ratio of voltage to current. It's Ohm's law  for AC circuits. Z is equal to V over I. But now, its magnitude tells you the relative size of your  current and voltage waves, and its angle tells you
[15:13] how the phase has shifted. It's a pretty great  real life application of imaginary numbers. So perhaps this is what Smith needed to match. See,  in every transmission line, there's a built-in impedance called its characteristic impedance  or Z-naught. This is just a fixed property of
[15:28] the line. It's kind of like mass per unit length  with the slinkies. Now, in most radio frequency or RF systems, this is designed to be 50 ohms,  a pure resistance. This is the value we need to match. Do that and it's like we've made the two  slinkies identical. No reflections. Let's suppose
[15:45] we measure the impedance at the antenna and it's  10 minus J30. This is a pretty bad match. It's like having two slinkies that are quite different.  We have to cancel out that negative J30. And we can do that by adding a series inductor of plus 30  and the imaginary part drops to zero. So inductive
[16:00] and capacitive parts cancel each other out. That  leaves the real part 10 ohms and we need 50. So you might think just add 40 ohms of resistance,  but a resistor loses power through heat. That's the very thing we're trying to stop. And  what if the resistance were too high? You can't
[16:16] add a negative resistance in a passive system. So,  how do you match a resistance without a resistor? Well, the answer actually already lies on the  transmission line. When part of a wave reflects, the line now carries two waves at once, the  forward one and the reflected one. At any point,
[16:31] the voltage you'd measure is now the two voltage  waves added together. But the reflected current wave is flipped. So, wherever the voltages  reinforce, the currents cancel. That means that the impedance, the voltage over the current, isn't  just one number anymore. It changes as you move
[16:47] along the line. And if the impedance is changing  anyway, well, maybe there's a point on the line where a resistance is matched. All that we have  to do then is find that point and then deal with the remaining imaginary part of the impedance.  And we can do that with a lossless capacitor
[17:01] or inductor. No resistor needed. At that point,  the characteristic impedance of the line will be matched. But how do you find that point? Well,  more than half a century before Smith, Oliver Heaviside had described how voltage and current  waves behave on transmission lines with a set of
[17:16] equations. But actually using them to eliminate  reflections was difficult. Calculating the right values meant working through long expressions with  the slide rule. In the 1930s, this was no longer
[17:28] just an abstract problem. As political tensions  rose across the globe, the ability to send clear signals across oceans became strategically  important. It meant you could coordinate ships, aircraft, entire supply networks, activities  crucial to waging war. So scientists in other
[17:45] powerful nations began to take notice. In the  Soviet Union, engineer Amiel Volpert began working on the same problem as Smith. And in Japan,  engineer Tosaku Mizuhashi did as well. They all
[17:57] wanted the same thing, a simple graphical system  that would make matching impedances quick and reliable. See, when Phillip Smith tried sending  a signal down his transmission line, some of the energy didn't reach the other end. It reflected  and it came bouncing straight back at him. And the
[18:12] same thing happens with your personal data. You  put it online, meaning for it to go to one place, but it ends up coming right back at you through  scam calls, phishing texts, and identity theft. I know I once got a text from a person who I thought  was Derek, asking me for some personal info. I was
[18:26] in a rush, so I just sent it right over to him.  And of course, that wasn't actually Derek. They just gathered info about me and then used it to  trick me into responding. There's a whole business around this. People who grab your data and then  sell it to the highest bidder. But there's no hope
[18:39] for an impedance match to solve this problem.  Fixing it means tracking down hundreds of data brokers and then forcing each one to delete  your info. But with today's sponsor, Incogni, you don't have to. Incogni finds those brokers for  you. They figure out who's holding your data. Then
[18:53] they send removal requests to each one. You just  make an account, tell them whose data to scrub, and you let them do the work. So far, they filed  120 requests for me, and 115 have already been completed. That's roughly 86 hours that I didn't  have to spend arguing with data brokers. Two full
[19:08] work weeks. Now, brokers are one thing. What  about that one specific site that you know has your info with your address sitting right there?  Well, with Incogni's custom removals feature, you point them at any site where your information  is exposed, and then one of their privacy agents
[19:23] takes it down for you. If you want to try it, go  to incogni.com/veritasium. Click the link in the description or scan this QR code on screen and  use code veritasium for 60% off. It's the closest thing to an impedance match that you'll find for  your personal data. The junk stops bouncing back.
[19:39] So head to incogni.com/veritasium and use code  veritasium for 60% off. I'd like to thank incogni for sponsoring this video. Now Smith, of course,  didn't have a shortcut. To stop his reflections, he had to build the solution himself. So Smith  started designing a chart. He started with the
[19:55] complex plane but with a small change. Most  transmission lines today have a characteristic impedance of 50 ohms but that value can change.  The 75 ohm line for example needs a completely
[20:07] different match. So if Smith had just plotted raw  values well then every system would need its own chart with a different goal. Instead he divided  everything by the characteristic impedance of the line Z0. This normalized the values so that  everything was now dimensionless. And now a
[20:23] value of one meant a perfect match, whatever  the line's actual characteristic impedance. But already there was a problem. See, the values  of real circuits can fall anywhere between two extremes. A short circuit where current flows  freely with almost no voltage, so impedance is
[20:39] zero. And an open circuit where there's a voltage  but no current, so impedance is infinite. A real impedance can take on any value between these  two extremes. So to show everything, Smith's chart would also have to be infinite. Not so  practical. Smith then had a new problem to solve,
[20:55] a geometry problem. How to represent infinity  on a finite chart. Smith realized he was a bit out of his depth. So he called in the help of two  of his colleagues who specialized in mathematics, Ferrell and McRae. And together they realized  they could use a powerful property of complex
[21:11] numbers to address the problem. To see how this  works, we'll follow an approach 3Blue1Brown laid out in an excellent video that I highly recommend  you checking out. See, if we multiply any complex number Z by some constant like three, everything  scales by a factor of three. So, multiplying by
[21:28] real numbers lets us scale our value. What about  multiplying by something imaginary like J? Well, if we start with one, multiply by J, then you move  to J, a 90° rotation. Multiply by J again and by
[21:42] definition you get -1. Again and you get -J. One  more time and you're back at one. So multiplying by imaginary numbers lets us rotate our value. In  general, multiplying by any complex constant does
[21:56] two things. It scales and it rotates. Now suppose  instead of multiplying by a constant, we try a more complicated function like 1 / z and we apply  that transformation to the entire complex plane.
[22:11] Now the lines curve and the whole plane warps  and at first glance it seems like all geometric structure is lost. But zoom in and you'll see  that at a small enough scale every tiny patch still looks square. This transformation preserves  shapes and angles. It's called a conformal map.
[22:26] And you'll notice that all the values in this  map that used to stretch off to infinity, well, they now fit in the center of this chart. So it  represents infinity in a finite space. And 1 / Z is just one example. There are other conformal  maps that warp the plane this way. And that's
[22:41] what Smith realized. If he could find the right  map for his problem, then he could bend the impedance plane into something far more useful.  See, up till now, he'd been working in impedance, but it had two problems. It stretched off to  infinity, so it wouldn't fit in one chart. And it
[22:55] changed as you moved along the line. So there was  no single value for engineers to match. But think back to those two waves. And instead of dividing  voltage by current, try dividing the forward wave by the reflected one. That gives you the  reflection coefficient. Now on a lossless line,
[23:10] the amplitude of our forward wave stays the same  size everywhere. So does the reflected wave. So the magnitude of our reflection coefficient is  constant along the line. The only thing that changes as you move is the phase. So the phase  angle of the reflection coefficient picks out
[23:24] an impedance on the line. The two quantities then  carry the same information, but unlike impedance, it's a lot more tame. That's because the reflected  wave can at most be the same size as the forward wave. So the reflection coefficient can never grow  past one. So you no longer have to deal with this
[23:40] problem of infinity. Let's plot it. We start by  plotting one vertical line on the complex plane. This is a line of constant resistance. Now apply  the transformation and that line becomes a circle.
[23:53] Let's try another constant resistance line. Again  this line becomes a circle. And we can do this for all positive constant resistance lines. What we  get is this whole family of circles. At r equals
[24:06] 0, the circle is the largest. Its diameter runs  from -1 to 1 on the reflection coefficient plane. As we increase resistance, the circles shrink and  their centers move to the right. At r equals 1, the circle passes through the center. That's  the match case where resistance equals our
[24:21] characteristic impedance. As r increases further  going off to infinity, the circles get smaller and cluster towards the right approaching the point  1 0. Now let's do the same thing but plot all the lines of constant reactance. These are horizontal  lines on the complex plane. For positive values,
[24:38] these are inductances and they map to circles  above the axis. As reactance increases, the circles shrink. For negative values, these  are capacitances and they map to circles below the axis. As reactance decreases, you get the  same pattern mirrored below. When reactance
[24:53] equals zero, the circle has infinite radius.  So, it appears as a straight line along the horizontal axis. We can get rid of the left half  of the impedance plane, and we're left with our final chart. One family of circles now tells you  resistance. The other tells you reactance. So,
[25:07] we can still easily read each point on the chart  in terms of its impedance by just seeing where these circles intersect. But because this is  the reflection coefficient plane, every point also maps to a reflection coefficient. And how far  it sits from the center tells you the reflection
[25:20] coefficient's magnitude. Now, that's really the  magic. Every point now holds two things at once, an impedance and a reflection coefficient.  And as that reflection coefficient rotates, it walks you along the line, cycling through  every impedance you'd actually measure. And just
[25:34] like our 1 / Z example, those values can now be  anything. Impedances that once ran off to infinity now sit inside this single circle. The entire  infinite range captured on one finite chart. So
[25:47] Smith found the transformation he needed for the  Smith chart. For the first time, engineers had a practical way to analyze transmission lines and it  opened up a whole new way of solving problems. Let me show you how. Here it is. We printed a massive  chart to show you how it works. We're going to
[26:03] start over here. If you walk along this horizontal  line, you're still moving along resistances. Above the line, you still have inductances. And below  the line, you still have capacitances. And the
[26:15] whole chart fits in a circle with magnitude of  one all the way around. So that's our chart. Now, let's put it to work to help us with the line we  measured earlier, the one that was losing half its power. So the measurement in the lab gave us 36  ohms of real resistance and 74 ohms of reactance.
[26:30] So we have to match that impedance. We divide by  50 ohms to normalize. That gives us a value of Z = 0.7 + 1.5J. So we start with the real part of  our impedance. These circles get bigger along the horizontal axis. So 0.7 is the circle right here.  Now let's find the imaginary part. It's positive.
[26:49] So it's above this axis. So I'll walk along  this way looking at these circles here to see where they intersect. And once we find where it  intersects with 1.5, well that's our impedance. Finding any impedance on this chart, we just go  through the same process. We just find where the
[27:03] two circles intersect. Now, we know to eliminate  reflections, we need to match the impedance of the transmission line. And that happens when the  resistance is one. So, it matches our reference impedance. And that reactance is then zero.  That's at the very center of our chart. Now,
[27:17] on a reflection coefficient plane, that's the  point 0 0, which makes sense. That's when you have no reflection. So, VR has to be zero. Okay.  So, our goal is pretty clear. We need to move from our impedance here all the way to the center of  the chart. So we draw an arrow from the center
[27:34] to our point. Because this plane represents  the reflection coefficient, the length of that arrow is then the magnitude of our reflection  coefficient. In this case, the magnitude is 0.68, which means that the reflected wave is 0.68  times the forward wave. And that's exactly
[27:49] what we saw in the demo. We also know that as we  move along our transmission line, because the same forward and reflected waves are interfering,  the magnitude of the reflection coefficient shouldn't change. So if we keep the distance  from the center fixed and we rotate around it,
[28:04] we can trace out a circle. And then every point  on that circle corresponds to a different position along the line. Because the standing wave pattern  repeats every half wavelength, we know that a 360° rotation on the chart corresponds to moving half  a wavelength along the line. And we can use this
[28:19] to make our life easier. In our example, we want  to move to the center of the chart. And for that, we need a normalized resistance of one. So, we  want our point on this circle. Lucky for us, there are already two places on our line  that match the resistance we want. Here.
[28:40] We can take advantage of that by moving to that  place on the line. To do that, just see how far it is to rotate. In this case, it's 12°. Now, since  360° is half a wavelength, then 12° here with a
[28:52] signal of 1085 MHz. Well, that's 3.1 mm. It's a  bit close to where we want to put the line. So, we're actually going to go to the next available  spot, which is over here. We measure it out on this plate and lay it down in copper tape. 28 mm  of extra line. I'm going to say it's there, which
[29:10] is approximately correct. Nice. Okay, perfect.  Now, all we have is this remaining reactance. In this case, it's on this circle, so it's roughly -  1.8. All that's left to do is cancel it out like
[29:22] what we did before. In this case, since it's  a negative capacitive 1.8, we can just add a positive inductive element in series. The equation  is pretty simple. 1.8= omega L over Z0. We
[29:35] rearrange and we get 13.2 nanohenries. So, we can  try adding something like that in the lab. Now, we decided to start out with a smaller inductor just  to test how we move. All right. Yeah. Okay. Okay. So, what you can what you can see is we've moved  along the constant resistance circle. Not quite
[29:50] enough. So, we have to actually up the size of our  inductor. Now, we add the inductance we actually calculated. And sure enough, okay, bold. There we  go. Not bad. There we go. Just like that. Yeah. No
[30:08] reflections. We did it. We did it. Let's go. Okay.  Come on. Come on, brother. Come on. That's fire. Let's go. Look at that. Okay, we're getting and  we're getting what? Like optimal power transfer down. Yeah, there we go. For sure, man. That's  the That's the highest power output we've seen.
[30:23] I think so. I do think this is important. Like  we've verified that this crazy map switch from the complex plane to the reflection coefficient plane  leading to these insane appearing circles. They
[30:35] actually mean something. Like in real life, we've  used it to increase the power. Crazy. It is crazy. It is crazy. This is cool. This chart represented  the whole problem graphically. And in a way, radio engineers could actually use. And Smith wasn't  alone. In 1937, Smith finished his chart and
[30:52] almost simultaneously Mizuhashi in Japan arrived  at a similar representation that same year. And Volpert in the Soviet Union in 1939. Three groups  working independently all converged on the same elegant solution. And it revealed a much better  way to solve the impedance matching problem.
[31:08] One that was rarely used before because it had  been too difficult to implement. On the chart, an open circuit has infinite resistance. So, it's  here. A short circuit is all the way on the other side because it has no resistance. So, it's over  here. What you'll notice is both lie on this outer
[31:25] circle. And that makes sense. An open or a short  absorbs nothing. Every bit of the wave comes back. So, it sits on this reflection coefficient circle  of magnitude 1, the biggest reflection there is.
[31:37] And this rim works like every other circle on the  chart. Moving along the line walks you around it. One full lap every half wavelength. So start at  a short or an open, add a bit of line and look what you pass through. Every reactance there is.  What that means is that by adding a short or open
[31:52] circuit with any length of line, we can make any  reactance. So instead of adding an inductor or capacitor, we get the same effect by just using a  length of transmission line. So we need to cancel out the imaginary part of 1.8. Now the cable we're  using is coaxial. It's really two conductors,
[32:08] an inner wire and an outer shield held a precise  distance apart. Normally, it just runs to the antenna carrying the signal in a loop. But we  can splice in a side branch of the same cable, a bit of extra transmission line called a stub.  We've connected one branch to the inner cable of
[32:23] the stub, one branch to the outer. And since at  the end of our stub, the inner wire and the outer shield aren't actually connected, this means we've  created a break in our circuit. On the Smith chart that means we know that the end of the stub lies  here at the open circuit position. But the rest
[32:38] of the circuit doesn't see the far end directly.  It sees the impedance at the input of the stub, which is this connection point. And that depends  on how long the stub is. Think of the stub as a little cove branching off a river. Water runs  in, bounces off the back wall, and flows out
[32:53] again. And the length of the cove sets the timing  of that returning water. Size the cove right, and the water flowing out pushes exactly when  you want it to. And that's what the stub's length does. It sets the timing of the wave that reflects  back. We want it to equal 1.8 positive. So,
[33:09] we start at where the open circuit is  and walk all the way around our chart. And we stop at 1.8. When we measure the angle, we  see that that's about 302°. Okay. Now, 302 / 360,
[33:25] that gives us how far along we've been on the  circle. And we know that one full circle turn gives us half the wavelength. We get 77 mm.  That should be the length of our stub. Now, we could add another full lap, 92 mm, and reach  exactly the same reactance. We're going to start
[33:44] with a longer stub and then trim it down to try  and tune our match. We haven't matched it yet, cancel out. Exactly. And hopefully as we cut  this transmission line shorter and shorter,
[33:57] tell me what's happening. Okay. Just like a  little bit basically like that might be. Come on. Give me that. Ooh, that's not too bad.  Yes. All right. Here's the last little bit.
[34:17] That's not bad. That's a good match. That's  a good match. Okay. That's awesome. It's actually so weird. Literally by just changing  the length of this line here, this this stub, we've matched the impedance. Insane. Look.  Look at it. We stopped every reflection on
[34:33] this line with just a break in the circuit. A  dangling piece of cable connected to nothing trimmed to the right length. And because the  stub is just more of the same cable, the fix is made of the thing that it's fixing. Now, the  series stub we've shown gets the idea across, but
[34:48] you'd rarely build one this way because a series  stub means cutting into your line. In practice, you'd rather just solder it in parallel. But  that introduces a new problem. In series, impedances add together cleanly. In parallel,  they don't, but one over impedance does. That's
[35:01] admittance. The only catch is that admittance  curves the other way on the Smith chart. So, engineers use a flipped version of it called  the admittance Smith chart. Using this chart, matching stubs in parallel is easy. So, what  we've done today is basically what RF engineers
[35:16] do every day. They play a game trying to zigzag  their way to the middle of the Smith chart. But all these charts have one big drawback.  They only work for a single frequency. But real they spread across a range of frequencies. There  are even radio techniques that encode information
[35:34] by changing the frequency. I mean, you've probably  heard of them, frequency modulation or FM radio. In practice, then the response of a system doesn't  just map to a single point on a chart, but rather it traces out a curve across its frequency range.  So instead of matching one point on a Smith chart,
[35:48] you try to bring that curve as close to the  center as possible over the range you care about. So even then the chart is still useful.  So Phillip Smith designed a chart that could revolutionize how radio engineers worked. But at  first, no one really cared. It took about 2 years
[36:04] of slow responses and rejections from technical  magazines before Electronics finally published Smith's chart. Engineers already had their own  methods, and this one required a completely different way of thinking. At first, adoption  was slow, but World War II changed everything.
[36:21] In the Battle for the Atlantic, German U-boats  were sinking Allied ships faster than they could be replaced. At night or in bad weather, the  predators were almost impossible to find. At the MIT Radiation Laboratory, scientists  were building new microwave radar systems
[36:35] that could detect submarines on the surface, even  in complete darkness. And in the middle of a war, you don't have time for trial and error. You  need a fast, reliable way to fix these systems. So Smith's chart was used constantly. When the  war ended, those same engineers took the chart
[36:50] with them out into industry. Into the universities  they'd work in and the textbooks they'd go on to write. Meanwhile, the other versions of the chart  by Mizuhashi and Volpert were used in Japan and the Soviet Union locally, but weren't spread  as widely. That's why today we refer to this
[37:04] chart as the Smith chart. Now granted, engineers  don't need to impedance match by hand anymore. Computers can do it a lot faster. But this chart  is still taught around the world in engineering classrooms and many engineers still rely on it  to this day. That's because while a computer
[37:20] can tell you the perfect match, it doesn't tell  you what to try or why it works. The Smith chart encodes that intuition. And because of that,  it's still a popular visualization tool in modern software. And it's used in almost all radio  frequency measurement tools. So, historically,
[37:34] we would get a Smith chart and we'd use rulers and  we'd use protractors and compasses. These days, we use a Smith chart more like a back of the  envelope calculation where it's like a map. It's like if if somebody asks you, "How do you get  to the nearest tube station?" You'd say, "Well,
[37:50] um, here's an envelope, you go down  the street here, turn left there, go further down there." And in the same  way, we use it to navigate from where we are to where we want to be. Sure. So it  gives you an intuition of what direction to
[38:03] go. That's right. Okay. Before the Smith chart,  engineers were stuck using cumbersome formulas and guessing. It took a concerted effort by  Phillip Smith, Tosaku Mizuhashi, and Amiel Volpert to make this complex world of reflections  and standing waves easier to understand. For me,
[38:18] it brings to mind Mendeleev and the periodic table  or Feynman and his diagrams. The Smith chart is in that same vein. So much of scientific  progress comes not from new discoveries, but from new forms of representation. They make  hard problems easier to solve and make further
[38:34] innovation and discovery possible. The Smith chart  helped advance radio frequency engineering. It was crucial in building modern radar and sensing  systems. And the communication networks we rely on today were no small part built from this chart.  So yes, it's scary, but it's also damn useful.
[38:54] One last thing, last year we launched  the official Veritasium board game, Elements of Truth. It's a trivia game with over  800 questions about science and technology. But the twist is you don't just answer. You also  have to put down a number from 1 to 10 for
[39:07] how confident you are that you're right. So, you  can know the answer and still miss out on points by doubting yourself, which it turns out gets  people pretty fired up. Our team is competitive, so if you want to pre-order, there's a link in  the description that'll take you to the Elements
[39:23] of Truth website. Thank you for supporting this  project, and as always, thanks for watching. Before we go, I want to thank Imperial College  London for having us and especially Dr. Stepan
[39:36] Lucyszyn and Ian Rossuck for setting up the demo.  And as always, want to thank you for watching.