Delay - "Interconnect Delay Models" : Static Timing Analysis (STA) basic (Part 4b)

STA & SI:: Chapter 2: Static Timing Analysis
2.1	2.2	2.3a	2.3b	2.3c	2.4a
Timing Paths	Time Borrowing	Basic Concept Of Setup-Hold	Basic Concept of Setup-Hold Violation	Examples:S-H Time/Violation	Timing Path Delay
2.4b	2.4c	2.5a	2.5b	2.6a	2.6b
Interconnect Delay Models	Delay - Wire Load Model	Maximum Clock Frequency	Calculate “Max Clock Freq”-Examples	Fix Setup-Hold Violation-1	Fix Setup-Hold Violation-2
2.6c	2.7a	2.7b	2.7c	2.8
Fix Setup-Hold Violation-3	Incr/Decr Delay Method-1	Incr/Decr Delay Method-2	Incr/Decr Delay Method-3	10 ways to fix Setup-Hold Violation.

Static Timing analysis is divided into several parts:

• Part1 -> Timing Paths
• Part2 -> Time Borrowing
• Part3a -> Basic Concept Of Setup and Hold
• Part3b -> Basic Concept of Setup and Hold Violation
• Part3c -> Practical Examples for Setup and Hold Time / Violation
• Part4a -> Delay - Timing Path Delay
• Part4b -> Delay - Interconnect Delay Models
• Part4c -> Delay - Wire Load Model
• Part5a -> Maximum Clock Frequency
• Part5b -> Examples to calculate the “Maximum Clock Frequency” for different circuits.
• Part 6a -> How to solve Setup and Hold Violation (basic example)
• Part 6b -> Continue of  How to solve Setup and Hold Violation (Advance examples)
• Part 6c ->  Continue of  How to solve Setup and Hold Violation (more advance examples)
• Part 7a -> Methods for Increase/Decrease the Delay of Circuit (Effect of Wire Length On the Slew)
• Part 7b -> Methods for Increase/Decrease the Delay of Circuit (Effect of Size of the Transistor On the Slew)
• Part 7c -> Methods for Increase/Decrease the Delay of Circuit (Effect of Threshold voltage On the Slew)
• Part 8 -> 10 ways to fix Setup and Hold Violation.

In the previous post we have discussed about the way tool calculate the max and min delay in a circuit. Now we will discuss other basics of the Delay and delay calculation. During your day to day work (in Semiconductor Field) or say in different Books, you come across different terminology related to the delays. There is a long list of that.

• Input Delay
• Output Delay
• Cell Delay
• Net Delay
• Wire Delay
• Slope Delay
• Intrinsic Delay
• Transition Delay
• Connect Delay
• Interconnect Delay
• Propagation Delay
• Min/Max Delay
• Rising/Falling Delay
• Gate Delay
• Stage delay

Fortunately or say luckily out of the above mention long list few are just synonym of other and few are interrelated to each other . Like Net delay also know as Wire Delay , Interconnect delay. Broadly we can divide this Long List into 2 type of delay.  Net Delay (Wire delay) and Cell Delay.  ( Note : Stage Delay = Net delay + Cell Delay. )

So let’s discuss these one by one. In digital design, a wire connecting pins of standard cells and blocks is referred to as a NET. A net

• Has only one driver
• Has a number of fanout cells or blocks.
• Can travel on multiple metal layers of the chip.

“Net Delay” refers to the total time needed to charge or discharge all of the parasitic (Capacitance / Resistance / Inductance) of a given Net. So we can say that Net delay is a function of

• Net Resistance
• Net Capacitance
• Net Topology

Now to calculate the Net delay, the wires are modeled in different ways and there are different way to do the calculation. Practically, when you are applying a particular delay model in a design , then you have to apply that to all cells in a particular library. You cannot mix delay models within a single library. There are few recommendations provided by experts or say experienced designer regarding the application of a particular Delay model in a design and that depends on

• Technology of design.
• At what stage you are ? Or say at what stage you want to apply a delay model. 
• How accurately you want to calculate the delay.

Note : Ideally Till the physical wire is not present in you design, you cannot calculate the Net delay. Reason is ... If wire is not present , you have no idea about the Length/Width of the wires. SO YOU CANN'T CALCULATE THE ACCURATE VALUES OF PARASITIC OR SAY DELAY VALUE OF THE WIRE. But here main point is  accurate value, means there is possibility of inaccurate or say approximate value of delay value before physical laying of wire in a design.

There are several delay models. Those which can provide more accurate result, takes more runtime to do the calculation and those which are fast provides less accurate value of delay. Lets discuss few of them. Most popular delay models are - 

• Lumped Capacitor Model
• Lumped RC model
• Distributed RC model
• Pi RC network
• T RC network
• RLC model
• Wire Load model
• Elmore Delay model
• Transmission Line Model

Lumped Capacitor Model.

• Model assume that wire resistance is negligible.
• Source driver sees a single loading capacitance which is the sum of total capacitance of the interconnect and the total loading capacitance at the sink.
• In past (higher technology-350nm and so), capacitor was dominating and that’s the reason in the model we have only capacitance.
• Older technology had wide wires, 
•  More cross section area implies less resistance and more capacitance.
• So Wire model only with capacitance.
•  In the Fig R=0

Lumped Capacitor Model

Lumped RC (Resistance Capacitance) model:

• As the feature size decreases to the submicron dimensions, width of the wire reduced.
• Resistance of wire is no longer negligible.
• Have to incorporate the resistance in our model. And that’s the reason Lumped RC model (or say RC tree) comes into picture.

In lumped RC model the total resistance of each wire segment is lumped into one single R, combines the global capacitive into single capacitor C.

Lumped RC Model

Distributed RC model:

Distributed means RC is distributed along the length of the wire. The total resistance (Rt) and capacitance (Ct) of a wire can be expressed as

Rt = Rp * L

Ct = Cp * L

Where

Cp and Rp are Capacitance and Resistance per unit length.

L is the length of the wire.

Ideally, distributing the resistance and capacitance of a wire in very small portion of the wire (say delta) give you the better performance. Now to find out the total capacitance and resistance we use the differential equation. Distributed RC model provides better accuracy over lumped RC model. But this type of model is not practically possible.

Distributed RC Model

  

The distributed RC model can be configured by 2 ways based on the structure or say shape (pi and T). Following is the pictorial view.

T model:

• Ct is modeled as a half way of the resistive tree.
• Rt is broken into 2 sections (each being Rt/2 )

Pi Model:

• Ct is broken into 2 sections (each being Ct/2) are connected on either side of the resistance.
• Rt is in between the capacitances.

For practical purpose, wire-models with 5-10 elements/nodes are used to model the wire.  It will provide the more accurate result. For N element section  

For T network:

• Intermediate section of resistance are equal to Rt/N.
• Intermediate section of Capacitance are modeled by Ct/N
• End section of Resistance are equal to Rt/(2N).
• This T Network is represented as TN model.

For Pi network:

• Intermediate section of resistance are equal to Rt/N.
• Intermediate section of Capacitance are modeled by Ct/N
• End section of Capacitance are equal to Ct/(2N).
• This Pi Network is represented as PiN model.

2 Types of Distributed RC Model (Pi- Model and T- Model)

Note: Lumped Vs Distributed RC wire:

Following is the comparison between the Lumped and distributed RC network. It will help you to understand in terms of uses of the both type of network in terms of accuracy and runtime.

Following is the Step Response of Lumped Vs Distributed RC line.

Step Response Of Lumped and Distributed RC Network

Below comparison Table will give you more accurate picture.

Output Potential range	Time Elapsed
	Distributed RC Network	Lumped RC network
0 to 90%	1.0RC	2.3RC
10% to 90% (rise time)	0.9RC	2.2RC
0 to 63%	0.5RC	1.0RC
0 to 50%	0.4RC	0.7RC
0 to 10%	0.1RC	0.1RC

RLC model

In the past since the design frequency was low so the impedance (wL) was dominated by Resistance (wL << R). So we are not caring “L”. However if you are operating at higher frequency and use the wider wire that reduce the resistivity then we have to take account the inductance into our modeling.

Distributed RLC Model



In next part we will discuss Wire Load Delay Model...

"Delay - Timing path Delay" : Static Timing Analysis (STA) basic (Part 4a)

STA & SI:: Chapter 2: Static Timing Analysis
2.1	2.2	2.3a	2.3b	2.3c	2.4a
Timing Paths	Time Borrowing	Basic Concept Of Setup-Hold	Basic Concept of Setup-Hold Violation	Examples:S-H Time/Violation	Timing Path Delay
2.4b	2.4c	2.5a	2.5b	2.6a	2.6b
Interconnect Delay Models	Delay - Wire Load Model	Maximum Clock Frequency	Calculate “Max Clock Freq”-Examples	Fix Setup-Hold Violation-1	Fix Setup-Hold Violation-2
2.6c	2.7a	2.7b	2.7c	2.8
Fix Setup-Hold Violation-3	Incr/Decr Delay Method-1	Incr/Decr Delay Method-2	Incr/Decr Delay Method-3	10 ways to fix Setup-Hold Violation.

Static Timing analysis is divided into several parts:

• Part1 -> Timing Paths
• Part2 -> Time Borrowing
• Part3a -> Basic Concept Of Setup and Hold
• Part3b -> Basic Concept of Setup and Hold Violation
• Part3c -> Practical Examples for Setup and Hold Time / Violation
• Part4a -> Delay - Timing Path Delay
• Part4b -> Delay - Interconnect Delay Models
• Part4c -> Delay - Wire Load Model
• Part5a -> Maximum Clock Frequency
• Part5b -> Examples to calculate the “Maximum Clock Frequency” for different circuits.
• Part 6a -> How to solve Setup and Hold Violation (basic example)
• Part 6b -> Continue of  How to solve Setup and Hold Violation (Advance examples)
• Part 6c ->  Continue of  How to solve Setup and Hold Violation (more advance examples)
• Part 7a -> Methods for Increase/Decrease the Delay of Circuit (Effect of Wire Length On the Slew)
• Part 7b -> Methods for Increase/Decrease the Delay of Circuit (Effect of Size of the Transistor On the Slew)
• Part 7c -> Methods for Increase/Decrease the Delay of Circuit (Effect of Threshold voltage On the Slew)
• Part 8 -> 10 ways to fix Setup and Hold Violation.

This particular post is inspired by a question asked by Lalit. And Frankly speaking I am not able to resist myself to write a blog on this. I was thinking to capture all this since long but every time because of work  I have to drop my thoughts.. But today after reading his question.. I am not able to control myself. :)

So the Question is: (original question)

I have a doubt regarding how delay is calculated along a path.i think there are two ways

1) to calculate max delay and min delay, we keep adding max delays and min delays of all cells(buffer/inverter/mux) from start point to end point respectively.

2)in other way, we calculate path delay for rising edge and falling edge separately. we apply a rise edge at start point and keep adding cell delay. cell delay depends upon input transition and output fanout. so now we have two path delay values for rise edge and falling edge. greater one is considered as Max delay and smaller one is min delay.

which one is correct ?

Short Ans is .. both are correct and you have to use both. May be you all become confuse, so let me give you few details.

As I have mention that for Setup and Hold calculation , you have to calculate the Delay of the Timing path (capture path or launch path). Now in a circuit there are 2 major type of Delay.

1. CELL DELAY
• Timing Delay between an input pin and an output pin of a cell.
• Cell delay information is contained in the library of the cell. e.g- .lef file
2. NET DELAY.
• Interconnect delay between a driver pin and a load pin.
• To calculate the NET delay generally you require 3 most important information.
• Characteristics of the Driver cell (which is driving the particular net)
• Load characteristic of the receiver cell. (which is driven by the net)
• RC (resistance capacitance) value of the net. (It depends on several factor- which we will discuss later)

Both the delay can be calculated by multiple ways. It depends at what stage you require this information with in the design. e.g During pre layout or Post layout or during Signoff timing. As per the stage you are using this, you can use different ways to calculate these Delay. Sometime you require accurate numbers and sometime approximate numbers are also sufficient.

Now lets discuss this with previous background and then we will discuss few new concepts.

Now in the above fig- If I will ask you to calculate the delay of the circuit, then the delay will be

Delay=0.5+0.04+0.62+0.21+0.83+0.15+1.01+0.12+0.57=4.05ns (if all the delay in ns)

Now lets add few more value in this. As we know that every gate and net has max and min value, so in that case we can find out the max delay and min delay. (on what basis these max delay and min delay we are calculating .. we will discuss after that)

So in the above example, first value is max value and 2nd value is min value. So

Delay(max)= 0.5+0.04+0.62+0.21+0.83+0.15+1.01+0.12+0.57=4.05ns

Delay(min)= 0.4+0.03+0.6+0.18+0.8+0.1+0.8+0.1+0.5=3.51ns

Till now every one know the concept. Now lets see what's the meaning of min and max delay.

The delay of a cell or net depends on various parameters. Few of them are listed below.

• Library setup time
• Library delay model
• External delay
• Cell load characteristic
• Cell drive characteristic
• Operating condition (PVT)
• Wire load model
• Effective Cell output load
• Input skew
• Back annotated Delay

If any of these parameter vary , the delay vary accordingly. Few of them are mutually exclusive. and In that case we have to consider the effect of only one parameter at a time. If that's the case , then for STA, we calculated the delay in both the condition and then categorize them in worst (max delay) condition or the best condition (min delay). E.g- if a cell has different delay for rise edge and fall edge. Then we are sure that in delay calculation we have to use only one value. So as per their value , we can categorize fall and rise delay of all the cell in the max and min bucket. And finally we come up with max Delay and min delay.

Information used in Cell and net delay calculation (Picture Source - Synopsys)

The way delay is calculated also depends which tool are you using for STA or delay calculation. Cadence may have different algorithm from Synopsys and same is the case of other vendor tools like mentor,magma and all. But in general the basic or say concepts always remain same.

I will explain about all these parameter in detail in next of few blogs, but right now just one example which can help you to understand the situation when you have a lot of information about the circuit and you want to calculate the delay.

In the above diagram, you have 2 paths between UFF1 and UFF3. So when ever you are doing setup and hold analysis, these path will be the part of launch path (arrival time). So lets assume you want to calculate the max and min value of delay between UFF1 and UFF2.

Information1:

	UOR4	UNAND6	UNAND0	UBUF2	UOR2
DELAY(ns)	5	6	6	2	5

Calculation:

Delay in Path1 : 5+6=11ns,           

Delay in Path2:  6+2+5+6=19ns,    

So

Max Delay = 19ns - Path2 - Longest Path - Worst Path

Min Delay = 11ns - Path1 - Smallest Path - Best Path

Information2:

	UOR4	UNAND6	UNAND0	UBUF2	UOR2
Rise Delay (ns)	5	6	4	1	1
Fall Delay (ns)	6	7	3	1	1

Calculation:

Delay in Path1 :        Rise Delay : 5+6=11ns,              Fall Delay: 6+7=13ns

Delay in Path2:         Rise Delay : 4+1+1+6=12ns,      Fall Delay: 3+1+1+7=12ns

So

Max Delay = 13ns -Path1 (Fall Delay)

Min Delay = 11ns - Path1 (Rise Delay)

Note: here there are lot of more concepts which can impact the delay calculation sequence, like unate. We are not considering all those right now. I will explain later.

Information3:

Library	Delay	UOR4	UNAND6	UNAND0	UBUF2	UOR2
Library 1	Rise Delay (ns)	5	6	4	1	1
	Fall Delay (ns)	6	7	3	1	1
Library 2	Rise Delay (ns)	5.5	6.5	4.5	1.5	1.5
	Fall Delay (ns)	5.5	6.5	2.5	0.5	0.5

Calculation:

For Library1:

Delay in Path1 :        Rise Delay : 5+6=11ns,              Fall Delay: 6+7=13ns

Delay in Path2:         Rise Delay : 4+1+1+6=12ns,      Fall Delay: 3+1+1+7=12ns

For Library2:

Delay in Path1 :        Rise Delay : 5.5+6.5=12ns,              Fall Delay: 5.5+6.5=14ns

Delay in Path2:         Rise Delay : 4.5+1.5+1.5+6.5=14ns,      Fall Delay: 2.5+0.5+0.5+6.5=10ns

So

Max Delay = 14ns- Path1(Fall Delay)/Path2(Rise Delay)

Min Delay = 10ns - Path2(Fall Delay)

As we have calculated above, STA tool also uses similar approach for finding the Max delay and Min Delay. Once Max and Min delay is calculated then during setup and hold calculation, we use corresponding value.

Once again I am mentioning that all these values are picked randomly. So it may be possible that practically the type/amount of variation in value is not possible.

In next part we will discuss these parameter in detail one by one.