- Hey everybody. Thanks for tuning in. I'm Zach Peterson. I'm a Technical Consultant with Altium and today we're gonna continue talking about transmission lines and specifically, we're gonna talk about input impedance. So if you remember the previous two videos that we've been doing on these topics, first, we looked at the
difference between characteristic and equivalent impedance. They're not the same thing. Then we looked a little deeper at just the characteristic
impedance of a transmission line. So characteristic impedance
doesn't tell the whole story. What you really need to look
at is the input impedance in a transmission line. And then specifically what happens at the end of a transmission
line where it reaches its load. So there's a lot to get through, let's go ahead and get started. (bright upbeat music) To get started with
looking at input impedance, I'm just gonna redraw
the typical circuit model that's used to represent
a transmission line. So here just for the moment
let's consider an AC source and then I've got my R and I've got my L and then I've got my G, which represents by dielectric losses. And then I've also got
my C for my capacitance in my transmission line. And then down here at the other end we have a load impedance. I've drawn it as a
resistor but in general, it's an impedance. We'll just call it Z sub L. So this circuit model is
our transmission line. In a transmission line,
the line is long enough. This is what makes transmission
line of transmission line. The line is long enough that
we need to worry about the propagation of a signal, and I'm drawing a digital signal here, but could be an analog signal. It could just be an AC
signal being sourced here from this source and how
it travels down this line and interacts with all
the different portions of the line along the way to eventually reach the load. What is the impedance
that affects this signal once it's injected into the
line right here from the source? Is it the characteristic
impedance Z sub zero equals R plus I omega L over G plus I omega C? Is it this this value? Is it really just the load impedance? So Z sub L? Or is it something else? Well, the answer is this
new concept that we're going to introduce called the input impedance. So the input impedance is
just the impedance seen here at this interface between
wherever my source comes into the line and then the line itself. And so the input impedance
depends on the load impedance Z sub L, but it also depends on the
characteristic impedance Z, or more generally written Z sub zero. This is important because it brings up another important concept
called termination. And so as we'll see in just
a moment when the load input impedance matches the
characteristic impedance, then we don't really have to worry about the input impedance. The input impedance is
going to be the same across these two values. So it's going to be equal to Z sub zero, which also just happens
to be equal to Z sub L. So let's look at input impedance and look at the equation specifically
for input impedance, and then hopefully we'll be able to see why termination is important. When I have my source here
and I have some impedance and maybe it's connected
to another impedance here, we'll just call this our
load, coming back around. I need to define my input
impedance seen right here at this interface. So I have my source and I want
to know what the impedance is seeing by a signal entering
right here at the source and traveling into my line, and then eventually over to my load. So the input impedance
is defined as Z sub zero equals the load impedance plus
the characteristic impedance. And it's multiplied by this
hyperbolic tangent term, and this is where we now
need to bring back in the propagation constant
and the length of the line. So here we have our length
of our line L and then here we have some terms in
the denominator, Z sub L. And then we have another
hyperbolic tangent term. If you remember from the previous clip, you'll know that I said that
when Z sub zero equals Z sub L, what do we have here
for the input impedance? Z sub in is going to be equal
to, we have a Z sub zero, and then we have another Z sub zero. So these terms, and these
terms in the denominator will all cancel out. So we've got a Z sub zero
here, a Z sub zero here. This entire fraction is equal to one. And so what will happen
here is you'll have Z sub in equal to Z sub zero equal to Z sub L. So this condition is sometimes
called impedance matching. It's also just sometimes
called termination. So what this tells you is that as long as my source impedance
Z sub S is also equal to Z sub zero and Z sub zero is also equal to Z sub L. There won't be any
reflection here in the signal that I try and inject into
this portion of the system. Now, similarly, once the
signal gets over here to the interface between
the line and my load, there also won't be any
reflection here at this point. So remember when you source
a signal into a circuit, and when you source it
into a transmission line, you want the signal, meaning all of that power
contained in that signal to get over here, to the load. You want all of that
power to enter the load. That means you don't
want any reflection here, and you don't want any
reflection here, ideally. So this is why this condition is important because it eliminates these
reflections here and then here. In a real system involving
digital components, what you generally have is this
value is actually very high. So it's something on
the order of mega ohms, so millions of ohms. But Z sub zero is normally not mega ohms, it's normally 50 ohms. How do we make it so that
this interface right here, where we have this
interface between the end of the transmission line and the load, how do we make it so that this
impedance looks like 50 ohms instead of mega ohms? Well, what we do is we apply termination. So termination just simply
means either I add in some resistance in series or in parallel, so that the equivalent
impedance here seen at the input of the load is now equal to
my transmission line impedance of 50 ohms. If you go back to your
basic circuit theory where you're doing equivalent
circuits in parallel, you should know that if
I have a impedance here and I have another impedance in parallel, and this impedance is let's
say on the order of mega ohms, what does this impedance
need to be in order for the equivalent arrangement
here to be equal to 50 ohms? Well, this resistance
just basically needs to be about 50 ohms. And you might say, well,
hey, Zack, I mean, you know, I'm putting 50 ohms in
parallel with mega ohms, why does that get me back to 50 ohms? And that's because you have
these two loads in parallel, and you should know that
the equivalent impedance at the input of the load Z
eq is equal to one over 50 plus one over, say, it's a mega ohm. So this is .0000001. This is an extremely small number, compared to this, which is a 0.02. If you just crank this, crank out this math on your calculator, and then invert this to get Z eq, this comes out to be like 50.000. It's just small fraction at the end of it. Who cares? It's not even important. It's so small it doesn't even matter. So this gives basically
gets you back to 50 ohms by having this parallel termination. So this is one of the reasons
why with differential pairs and then with single ended signals, what you'll see at the
load end of the line with a termination resistor is applied is that it is applied in parallel. And that's the reason for doing this. That's what then causes your
Z sub L to be now Z sub zero, this to be Z sub zero, and then this whole fraction
to then reduce back to one. And now you're back at this condition. Now, what about at the source end? If you know anything about CMOS circuits, then at the source end
you'll know that Z sub s, it's usually about zero ohms. It's actually very low impedance. So they have low impedance sources. You have high impedance input at a load, how would you resolve this? Well, the typical strategy
then to increase Z sub zero or sorry, Z sub S at
zero ohms up to 50 ohms, it would be to add in some resistance. How do you add in some resistance? You just put it in series. So here at the input, I'll just redraw this momentarily. I would want to put a
series resistor right here. So this would be 50 ohms. So now my Z sub source is zero ohms, plus 50 ohms, gets me to 50 ohms. So now I have perfect
matching here at the input, and then at the load end. In reality, with modern components, you don't always have to
apply this 50 ohm resistor at the source, and you don't always have to
apply this 50 ohm resistor or in the case of differential pairs, maybe it's a hundred ohm
resistor at the load. Some components use what's
called on die termination. So the termination resistor
is already built into the chip and the output impedance at
the source is already set to 50 ohms. So I'm a great example is like with some Texas Instruments
microcontrollers, like on the wireless output, going for an RF signal
over to like an antenna, I bring that specific component
up because I work with it all the time on the client design, the output impedance on that
RF source is already set to 50 ohms. So I don't need to add in
this resistor on the output. So this is one of those
things where you just need to check the data sheet to make
sure that you don't need to necessarily add in these
resistors on the output. Sometimes what they'll do is
they'll have a certain guide on what resistor you need to calculate. And it really just depends
on the signaling standard that you're working with. And it depends on what the
impedances are on the die on that component versus the
target that you're trying to hit for your specific
transmission line. So far, we've been talking
all about digital signals and with digital signals,
you don't generally use this equation. The reason is because gamma
here is generally only defined for a single frequency. So if you go back to the previous video on transmission line impedance
on characteristic impedance, I talked briefly about
the propagation constant. Now the propagation constant
is actually a function of frequency, just like the characteristic
impedance doesn't saturate to square root over L squared of L over C, the propagation constant
doesn't saturate to square root of LC. So it is also not constant. It is a function of frequency. So because of that, this is actually an equation
that's generally used with RF design because in RF designs, you're generally working
with just one frequency or a very narrow range of frequencies. So it's much easier to
calculate one specific value for Z sub in. And that's really important
because sometimes what you do with RF designs is you
actually have a different transmission lines
sections that are cascaded, and you need to know the
input impedance at each of those interfaces
between different lines. That's going to determine
whether or not your signal reflects off of those interfaces. Same thing with digital circuits, we care about the reflection, just like we do in RF circuits. Hopefully that clears up
some of the terminology and really illustrates
what happens when a signal actually travels onto the line. It's not just interacting with
this characteristic impedance of the transmission line. It's actually also
interacting with the load. So last thing to consider
here is what happens with the hyperbolic tangent
function and specifically what happens to the line
when it's very, very long. So let me just rewrite this
equation because I got a little messy there for a moment. So one of the things that you'll
hear brought up quite often is what happens when
the line gets very long. So what happens when my
transmission line gets to L equals infinity? Hyperbolic tangent is defined
as hyperbolic sine divided by hyperbolic cosign. What that means is that
if my line gets very, very long, this value or this function, hyperbolic tangent evaluates to one. Same thing down here. Hyperbolic tangent will evaluate to one. And so I can essentially just ignore those on very long lines. And so what happens here? Well, I have ZL plus Z zero
divided by ZL plus Z sub zero. So again, this entire fraction
just evaluates to one. Just ignore it. So in a very long transmission line, what happens is the input
impedance is really just the characteristic impedance. We don't even care what's
happening at the load right now. The input impedance is just
the characteristic impedance. And that's because the load
end of the line is so far away that when the signal enters the line, the only thing that it's interacting with is the transmission line. And so that's why only the
characteristic impedance appears in this input impedance
when the line reaches out to infinity. So this again, should
illustrate why we care so much about the propagating behavior
of an electrical signal. When there's propagation
at a finite velocity, which is exactly how
electrical signals move around the board. We only care about the
amount of the impedance that is within the spatial
extent of the signal. So if the signal only exists
at the beginning of the line, the only impedance that
matters is the impedance at the beginning of the line. And so this is why we often
talk about transmission lines in terms of their length. The length does matter, and it is going to determine
the input impedance. One thing that you can do
with that equation is you can actually plug in different
values for propagation constant and load impedances and
characteristic impedances, and you can figure out some special cases of when the input impedance
is zero, when it's infinity, when it's exactly Z sub
zero or some fraction of it. And those are all important
conditions in RF design. We have a lot of videos to
get through because you know, everyone's been leaving
so many great questions, but one of the things
I will try and get to in an upcoming video
is looking at RF design and how input impedance
plays an important role in RF design. If you want to learn more
about input impedance and specifically the input impedance of cascaded circuit networks and cascaded transmission lines, go look in the description. There's a link to a blog. That link will contain
a bit of an explanation about how input impedance
is used in cascaded networks and cascaded transmission lines. Meaning like I have different
transmission lines arranged one after the other, on my circuit board or a different circuit
networks arranged one after the other, on my circuit board. Okay, everybody, thanks for watching. If you liked this video,
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because everybody's been asking such great questions and
we'll do our best to fit them into our schedule. For now go into the
links in the description. You'll see a free trial
link for Altium designer. You can download it, you can try it out, see if it works for you. Thanks everybody. And don't forget to call your fabricator. (Techno cymbal chime)
