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0:00
- This is the scariest chart in electrical engineering. - 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 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
0:25
we have today. That's because this chart solves one of the most paradoxical problems in electrical 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? (both laughing) There've been some great YouTube videos explaining how to use it, like one from my friend Zach Star, but I always felt like they were missing something essential. You also, you wanna know why this chart looks the way it does.
0:50
- Yes. But then I need to know the physics. I would wanna know what the circles are and what I'm looking at at first. - Because even once you know how to use this chart, 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 100 years to the origins of this chart. In 1928, Philip H. Smith, fresh out of electrical engineering school, got his first job at Bell Labs.
1:17
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 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.
1:45
With only one antenna, the signal radiates across the whole sky, and only a 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 about 10 degrees across, and the power in that direction is 400 times stronger. So to have any chance of their signal
2:09
reaching across the globe, Smith's team connected more than 20 smaller antennas into a massive directional array linked by over two kilometers of transmission line. Smith's job was to test this 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 planet,
2:34
he needed to make sure that all the power reached the emitting antennas. So he needed to find a way to minimize the reflections. (light music) 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 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,
2:58
electrons have to oscillate back and forth at radio frequency. As they accelerate, they create 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. 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,
3:23
let's just look at a single sinusoidal voltage and current. Let's recreate Smith's signal by shaking a slinky up and down. (light music) You can think of my arm as the power source. (light music) 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 the frequency of the wave.
3:48
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, 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 meters per second, or c.
4:13
In the transmission line, the wave travels more slowly, but for a given line, 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. (slinky clattering) If you decrease the frequency, the wavelength gets longer, and if you increase it, the wavelength gets shorter, and this wavelength is crucial to our problem.
4:39
We'll also show you in simulation here, without the effects of friction and damping. See, if we start out with a long wavelength, 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 original signal all that much, so there's nothing to worry about.
5:02
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. (light music) After a reflection, the forward and reflected waves combine.
5:18
At some points, the two waves cancel and the voltage is very small,
5:23
and at other points they add together. This is what's called a standing wave pattern. (slinky clattering) 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 reflection, and a powerful standing wave did this to the inner conductor of that transmission line.
5:53
This is the effect of the standing wave. It's not made up. - In most electrical systems we're used to, this isn't too much of a problem. Your house supplies AC power at a frequency of 50 or 60 hertz, so that corresponds to a wavelength of about 5 to 6,000 kilometers, which is far longer than any wire that takes power from the station to your home. But Smith was generating radio waves in the megahertz range, and at 10 megahertz, the wavelength is about 30 meters.
6:20
With the transmission line he was working on more than two kilometers 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 if we could recreate what Smith was up against. - As you can see, it's very small and very compact, but we can do a lot of antenna measurements in here. We can do electromagnetic compatibility measurements here. - Here's the experiment.
6:44
We've got a power source sending a radio signal down a 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 Smith's configuration. Now, based on our input power and the distance between the antennas, we expect a signal of about negative 55 decibels.
7:07
But that's not what we see. We've lost half the power. We're picking up around negative 59 decibels. And because signal is measured on a logarithmic scale, that four decibel drop means we've lost more than half the power,
7:21
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 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 with a black box. That gives us a very simple toy model with just three parts.
7:46
And we can represent these parts with slinkys. 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. Since both the antenna and the transmission line have their own electrical properties, we can model Smith's whole system by tying two slinkys together. One for the transmission line, and the other for the antenna. First, we're gonna try two very different slinkys.
8:10
One with a high mass per unit length, and one with a low mass per unit length. Let's see what happens. (slinky clattering) Well, when we send a pulse down the line, it reaches the boundary between the two Slinkies. Part of the energy continues through. (light music) 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.
8:40
But if the two slinkys have the same mass per unit length, the wave passes straight through. 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 is stretched, and this mass per unit length. But since in this setup both slinkys are tied together,
9:04
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 again.
9:13
So you might think Smith could do the same thing, except instead of matching slinkys, 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 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
9:38
by the current. So if a voltage wave hits the boundary between the 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 slinkys. Indeed, in real life when we checked, the resistances didn't match. So the antenna array is at 12.5 ohms, and the transmission line is at 50. So, what we're gonna do is we're gonna add this 37.5 ohm or 40 ohm resistor
10:03
to try to bring it up so that they match, and we'll see what happens to the power.
10:07
Okay, what are we looking at? So now it's at... Yeah, that's, it hasn't helped us at all. It's actually, yeah, we're 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. And that means by putting it in our system here, we're gonna lose energy no matter what. But there's another problem with using resistors, and that's that real-life systems don't just have resistances.
10:30
Take this disposable camera. When I turn on the flash, it takes a moment to charge. You can hear it. And then when I take a picture (camera clicking) releases all that energy at once. 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, a 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 current, so let's see what happens when we have a capacitor with an alternating current signal. Okay, so we're gonna make a chart, and on the horizontal axis, I'm gonna draw time. (marker squeaking) And on the vertical axis,
11:26
we'll just have magnitude of our wave. Okay, so we'll start with our voltage wave, which we'll just draw like this. (marker squeaking) Now we'll draw our current wave in red. (marker squeaking)
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 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.
12:07
In other words, the voltage lags the current by 90 degrees.
12:12
But there is another effect we need to take into account. That's because whenever a current flows through a wire, it creates a magnetic field,
12:19
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 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 faster you try to change that current,
12:43
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. 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.
13:09
The voltage leads the current by 90 degrees. So that's 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 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.
13:37
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 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 gonna refer to imaginary numbers with a j.
13:58
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. So on this plane, pure resistance lies along the horizontal axis, where phase is zero. Since we care about positive resistances, we're only gonna look at the right side of the plane.
14:22
Now, compare that to pure capacitors and inductors. These components do shift the phase of our waves relative to each other. Since an inductor causes voltage to lead current by 90 degrees, we rotate it up plus 90 degrees on the vertical axis. A capacitor causes voltage to lag current by 90 degrees, so we rotate it down to minus 90 degrees.
14:41
Put them together and this vertical axis is called reactance. 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.
15:00
Now, impedance is still defined as 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 how the phase has shifted. It's a pretty great real life application of "jmaginary" 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_0.
15:27
This is just a fixed property of the line. It's kind of like mass per unit length with the Slinkies.
15:32
Now, in most radio frequency or RF systems, this is designed to be 50 ohms of 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 we measure the impedance at the antenna and it's 10-j30 This is a pretty bad match. It's like having two Slinkies that are quite different. We have to cancel out that -j30, and we can do that by adding a series inductor of +30, and the imaginary part drops to zero.
16:00
So, inductive 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 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,
16:26
the line now carries two waves at once, the forward one and the reflected one.
16:30
At any point, the voltage you'd measure is now the two voltage waves added together.
16:35
But the reflected current wave is flipped, so wherever the voltage is reinforced, the current's canceled. That means that the impedance, the voltage over the current, isn't just one number anymore. It changes as you move along the line. And if the impedance is changing anyway, well, maybe there's a point on the line where our 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.
16:59
And we can do that with a lossless capacitor 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 equations. But actually using them to eliminate reflections was difficult. Calculating the right values meant working through long expressions with a slide rule.
17:25
(upbeat music) In the 1930s, this was no longer 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 powerful nations began to take notice. In the Soviet Union, engineer Amiel Volpert began working on the same problem as Smith.
17:53
And in Japan, engineer Tōsaku Mizuhashi did as well. They all wanted the same thing, a simple graphical system that would make matching impedances quick and reliable.
18:04
See, when Philip 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 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.
18:26
I was 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 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.
18:46
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, and 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've 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 work weeks. Now, brokers are one thing.
19:11
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 takes it down for you. If you wanna 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
19:35
that you'll find for your personal data. The junk stops bouncing back. So head to incogni.com/veritasium
19:41
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 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 different match.
20:08
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, Z_0. This normalized the values so that everything was now dimensionless. And now a value of one meant a perfect match, whatever the line's actual characteristic impedance.
20:28
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 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, a geometry problem.
20:57
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 numbers to address the problem.
21:14
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 real numbers lets us scale our value. What about multiplying by something imaginary, like j? Well, if we start with one, multiply it by j, then you move to j, a 90-degree rotation. Multiply by j again,
21:42
and by definition you get negative one.
21:45
Again, and you get negative j. One more time and you're back at one. So, multiplying by imaginary numbers lets us rotate our value.
21:53
In general, multiplying by any complex constant it scales and it rotates.
22:00
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. (light music) 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.
22:27
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 one over Z is just one example. There are other conformal maps that warp the plane this way, and that's what Smith realized. If you could find the right map for his problem, then he could bend the impedance plane into something far more useful. See, up 'til now, he'd been working in impedance,
22:50
but it had two problems.
22:52
It stretched off to infinity, so it wouldn't fit in one chart, and it 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, the amplitude of our forward wave stays the same size everywhere. So, does the reflected wave.
23:15
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 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.
23:38
So you no longer have to deal with this 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.
23:56
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=0, the circle is the largest. Its diameter runs from negative one to one on the reflection coefficient plane. As we increase resistance, the circles shrink and their centers move to the right. At r=1, the circle passes through the center. That's the match case, where our resistance equals our characteristic impedance.
24:23
As r increases further, going off to infinity, the circles get smaller and cluster towards the right, approaching the point one, zero. 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, 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,
24:50
you get the same pattern mirrored below. When reactance 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.
25:03
One family of circles now tells you resistance, the other tells you reactance. So 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 coefficient's magnitude. Now, that's really the magic. Every point now holds two things at once,
25:26
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 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, Smith found the transformation he needed for the Smith Chart. For the first time,
25:54
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've printed a massive chart to show you how it works. We're gonna 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 whole chart fits in a circle with magnitude of one all the way around.
26:19
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, so we have to match that impedance. We divide by 50 ohms to normalize, and that gives us a value of z equals 0.7 plus 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.
26:46
Now let's find the imaginary part. It's positive, 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 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,
27:11
so it matches our reference impedance, and that reactance is then zero. That's at the very center of our chart.
27:17
Now, on a reflection coefficient plane, that's the point zero, zero, which makes sense. That's when you have no reflection, so V_r 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 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,
27:44
which means that the reflected wave is 0.68 times the forward wave, and that's exactly what we saw in the demo.
27:51
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, 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-degree rotation on the chart
28:16
corresponds to moving half a wavelength along the line. And we can use this to make our life easier. In our example, we wanna 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. (light music) And here. 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.
28:46
In this case, it's 12 degrees. Now, since 360 degrees is half a wavelength, then 12 degrees here with a signal of 1085 megahertz, well, that's 3.1 millimeters. It's a bit close to where we want to put the line, so we're actually gonna 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 millimeters of extra line.
29:08
- I'm gonna say it's there, which is approximately correct. (light music) 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 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's pretty simple. 1.8 equals omega L over Z_0. We rearrange and we get 13.2 nanohenries.
29:37
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, so what you can, what you can see is we've moved along the constant resistance circle. Not quite enough, so we have to actually up the size of our inductor. (machine buzzing) Now we add the inductance we actually calculated.
29:59
And sure enough.
30:02
Okay. - Lo and behold. - There we go. - Not bad. - There we go, just like that. Yeah, no reflections. - We did it. - We did it. - Let's go. Okay. Come on, come on, brother. Come on. That's fine. Let's go. Look at that. Okay, and we're getting, and we're getting what? Like, optimal power transfer now. - Yeah. - There we go. - For sure. - Man, that's the highest power output we've seen. - I think so. - I do think this is important. Like, we've verified that this crazy map switch
30:29
from the complex plane to the reflection coefficient plane leading to these insane appearing circles, they 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 almost simultaneously Mizuhashi in Japan arrived at a similar representation that same year,
30:58
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, 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.
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A short circuit is all the way on the other side because it has no resistance, so it's over here.
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What you'll notice is both lie on this outer 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 one, the biggest reflection there is. 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.
31:50
What that means is that by adding a short or open 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 negative 1.8. Now, the cable we're using is coaxial. It's really two conductors, 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,
32:15
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 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 of the circuit
32:40
It sees the impedance at the input of the stub, doesn't see the far end directly. 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 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.
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It sets the timing of the wave that reflects back. We want it to equal 1.8 positive. So we start at where the open circuit is and walk all the way around our chart. (light music) And we stop at 1.8.
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When we measure the angle, we see that that's about 302 degrees. Okay, now 302 divided by 360, 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 millimeters. That should be the length of our stub. Now, that's the shortest length that works, but since the chart repeats every half wavelength, we could add another full lap, 92 millimeters, and reach exactly the same reactance.
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We're gonna start with the longer stub and then trim it down to try and tune our match. We haven't matched it yet, so we're still, we still have some of this reactance that we have to cancel out. - Exactly, yeah. And hopefully as we cut this transmission line shorter and shorter, we'll move through that point. - All right, so let's do it. - Let's do it. Let's do it. You tell, tell me what's happening. - [Professor Stepan] Okay. It's just like a little bit basically.
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- [Henry] Like that might be...
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- [Professor Stepan] Come on, gimme that. - Ooh, that's not too bad. - Yeah. - All right, here's the last little bit. - Ooh, that's not bad. - 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 stub, we've matched the impedance. Insane. Look at it. We stopped every reflection on this line with just a break in the circuit, a dangling piece of cable connected to nothing trimmed to the right length.
34:39
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 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 admittance.
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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 do every day. They play a game trying to zigzag their way to the middle of the Smith Chart.
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But all these charts have one big drawback. They only work for a single frequency, but real world signals aren't this perfect. The moment you turn them on or change them in any way, they spread across a range of frequencies. There are even radio techniques that encode information 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,
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but rather it traces out a curve across its frequency range. So instead of matching one point on a Smith Chart, 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 Philip Smith designed a chart that could revolutionize how radio engineers worked, but at first, no one really cared. It took about two years of slow responses and rejections from technical magazines
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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. (upbeat music) 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,
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scientists were building new microwave radar systems 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 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
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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 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.
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That's because while a computer 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, 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
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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 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 wanna be. - Sure. So it gives you an intuition of what direction to go. - That's right. - Okay. Before the Smith Chart, engineers were stuck using cumbersome formulas and guessing.
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It took a concerted effort by Philip Smith, Tōsaku Mizuhashi, and Amiel Volpert to make this complex world of reflections and standing waves easier to understand. For me, it brings to mind Mendeleev and the Periodice 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 innovation and discovery possible.
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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 in no small part built from this chart. So yes, it's scary, but it's also damn useful. (light music) 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,
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you also have to put down a number from one to 10 for 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 every time we play, things do get pretty heated. The game comes out later this year, so if you wanna pre-order, there's a link in the description that'll take you to the Elements of Truth website. Thank you for supporting this project,
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and as always, thanks for watching.
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Before we go, I wanted to thank Imperial College of London for having us, and especially Dr. Stepan Lucyszyn and Ian Russock for setting up the demo. And as always, I wanted to thank you for watching. (people indistinct chattering)
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