So, just to sort of, you know, summarize in a sort of a cartoon.

I've got a wire, i've got a, a gate driving a wire, driving another couple of

gates. I turn that into a tree, it's a tree of

resistors. In this case, a, b, c, d, e, f is the

nodes. And at every intermediate node, a

capacitor hanging off of it. I add to that another resistor and a

voltage source. That models the gate that's driving

things. I add to that also, another capacitor for

every gate that is, being driven by this wire.

And what I'm actually going to be bale to do is a very simple computation on the

tree to calculate a delay number for each output of the tree.

And one of the things that's really great is I'm going to get a unique number for

every output. So I'm going to really physically model

the shape of the wire. And so I'm going to get maybe a long

delay to some outputs, and a short delay to other outputs, depending on what is

correct and physical for the shape of the wire.

So this is this sort of very famous result, we're heading toward and this has

a name. It's called the Elmore delay.

So it's named after a physicist, famous Physicist Elmore.

this was derived, amazingly, way back in the 1940's for certain kinds of circuit

applications. And then it was sort of resurrected, like

40 years later, by Penfield, Rubinstein, and Horowitz for these RC trees because

it turns out to be useful for, for you know designing modern integrated

circuits. It's famous enough that if you type

Elmore delay into Wikipedia, you will get an article on it and you can see all the

original references and things like that. For our purposes, what's wonderful is

that it's very simple and very useful and has a very easy computational recipe.

So, here's the tree again, and I've got the input driven gate modeled by the

resistor with a 5 on it. And I've also got the 2 capacitors at

nodes e and f at the output added. So I've got the driving gate and the

driven gates and it's, again, a little tree of resistors with nodes a, b, c, d

and e. There is a resistor of size 2 which we

know is a and b resistor 4 between b and c resistor 1 between c and e resistor 1

between b and d resistor 3 between d and f.

And there's a capacitor of size 1 at node a, 2 and node b, 1 at node c, 1 at node

d, 3 and node d, and 1 at node f. So, this is the example, we're going to

see this several times and there's a very simple computations, so the Elmore delay

is tau to time. We said tau is to be 0 and you walk down

the path of resistors from the root to the leaf where you want to calculate the

delay. And at every resistor you do the

following, you take the resistor value, you are on and you[COUGH] multiply it by

a number. And that number is the sum of all the

capacitors you can see downstream. And we are going to talk about what this

means. what's a downstream capacitor?

it's any capacitor reachable in the tree below this resistor.

So if current is going through this resistor and heading to the bottom of the

tree it's any capacitor that can take that current.

Alright, so let's be concrete on this example.

I've got resistor Ri equals 2 between nodes a and b.

Alright, what are the downstream resistors for node Ri?

And the answer is, the capacitors at node b, c, d, e, and f.

So, the resistors at node is between nodes a and b, what is the downstream

capacitance? Alright, any current going through node

through resistor Ri equals 2 could end up on capacitors at nodes b, c, d, e or f.

So you're going to take Ri is 2, and you're going to multiply it by 2 plus 1

plus 1 plus 1 plus 3. You're going to add that term and you're

going to keep doing this computation. So, simple as that.

So this is just a version in words. You know, Ri is equal to Ri is to the

resistor in the tree. The term we would add to the Elmore delay

is 2 times 2, plus 1, plus 1, plus 1, plus 3, okay?

So let's just take a look at it. So here's the tree again, a, b, c, d, e,

f, is the nodes. we want to calculate the delay to a

particular point in this tree. So, where do we, where are we looking?

we want to go down the resistor, the 5, the resistor that goes into node a, a to

b, b to c, c to e. We want to calculate the delay for the

capacitor that's at node e, alright? and so we just use the formula.

And the formula says you set tau to be equal to zero and then, you start walking

down the resistors. So you walk down the resistor that's a

value five. And you add together all the capacitors

down stream, which in this case is all the capacitors, 1 plus 2, plus 1, plus 1,

plus 3, plus 1, okay? And then you go further, and you go to

the resistor with a value of 2 between a and b, and you add all those resistors

down stream, 2 plus 1, plus 1, plus 3, plus 1.

and then you keep going, so we're going to go down the resistor of value 4,

between b and c. While the resistor, the capacitors you

see now in front of you are 1 and 3, so 4 times 1 plus 3.

And then you get the resistor of value 1 between c and d, multiply that by the

capacitor 3. 45 plus 16, plus 16, plus 3, if I did it

right, it's 80, 80 is the delay to that note, it's as simple as that.

if you're not, very knowledgeable about the circuits let me offer you an

alternative analogy. So this is like a, like a branching

stream. So I've got a little picture of a stream

here, sort of dividing, and dividing, and dividing.

An the goal is that you are downstream, and you are trying to fill your bucket,

okay? And what you care about, is how fast you

can fill your bucket. Now unfortunately, at every branch point,

somebody else has a bucket, and so the farther you are downstream, the less

water you can get. Alright, so just to sort of draw this,

there's water going in, it's a fixed supply up at the you know mouth of the

river, the source of the river. And there you are with your bucket way,

way down at the, one of the ends of the branch points of the river.

And unfortunately, everybody else at each one of those branch points has a bucket

and so the water that's going in that's a fixed limited supply is filling up

everybody's bucket. What you want to know is how fast you can

fill your bucket. All right, how fast can you do that?

So what matters? What matters is the width of the upstream

branches, because that's going to tell sort of like how much water can go down

those branches. And the size of everybody else's bucket.