© H. Heck 2008 Section 3.2 1

Module 3: Analysis TechniquesTopic 2: Bergeron Diagrams

OGI EE564

Howard Heck

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Where Are We?

1. Introduction

2. Transmission Line Basics

3. Analysis Tools1. Lattice Diagrams

2. Bergeron Diagrams

4. Metrics & Methodology

5. Advanced Transmission Lines

6. Multi-Gb/s Signaling

7. Special Topics

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Contents

Analysis Process Approach Load Line Setup Initial Wave First Reflection Successive Reflections

High-Low Transition Example Application to Non-Linear Devices Conversion to Waveforms Summary References

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Analysis Process: Approach

1. Plot the I-V curves for the transmitter (source) and the receiver (load).a) Write the Ohm’s law equations for the equivalent circuits.

5V

I

V

55

IV 555

Pull-up

IV 40

0V

I

V

40

Pull-down

0I

0V

I=0

V

RTT=

Receiver

5V

55

40

50

Example Example CircuitCircuit

I

Analyze the Analyze the low-high low-high transition.transition.

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Analysis Process: Load Line Setup

b) Plot V as a function of I for each equivalent circuit. This gives us the load lines for the transmitter and receiver.

Vout [V]

-100

-50

0

50

100

150

200

-3.0

-2.0

-1.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

I out [

mA

]Pull-up

IV 555

Pull-down

IV 40Rcvr 0I

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Analysis: Initial Wave2. Start at the steady state, which is defined by the intersection of

the pull-down and receiver load lines.

Vout [V]

-100

-50

0

50

100

150

200-3

.0

-2.0

-1.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

I out [

mA

]

Pull-down

Rcvr

Pull-up

V = 2.381VI = 47.6 mA

V = 0.000VI = 0 mA

3. Construct the load line for the interconnect. The slope is 1/Z0. The intersection of load lines for transmission line & pull-up give

us the initial voltage and current on the network.

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Vout [V]

-100

-50

0

50

100

150

200

-3.0

-2.0

-1.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

I out [

mA

]

Pull-down

V = 0.000VI = 0 mA

V = 2.381 VI = 47.6 mA

Rcvr

Pull-up

Analysis: 1st Reflection4. The next load line segment will have a slope of -1/Z0. The

intersection of the load line for the transmission line with the load line for the load resistor defines the voltage and current at the receiver after the first time-of-flight delay:

V = 4.762 VI = 0 mA

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Analysis: Successive Reflections5. Continue the process, alternating the load line

slopes between 1/Z0 and -1/Z0, until reaching the point at which the load and source lines intersect (steady state high).

0.0

0.5

1.0

1.5

2.0

2.5

4.75

4.80

4.85

4.90

4.95

5.00

V out [V]

I ou

t [m

A]

Pull-up

RcvrV = 4.762VI = 0 mA

V = 4.975 VI = 2.3 mA

V = 4.989VI = 0.0 mA

V = 4.994VI = 0.1 mA

V = 4.999VI = 0.0 mA

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Why Does This Work?

A linear driver follows Ohm’s law. So does a transmission line. They are connected. At the point of connection, we

have to satisfy Kirchoff’s current and voltage laws. This effectively leaves us with two equations in two

unknowns. The Bergeron diagram is simply a graphical way of

solving the simultaneous equations.

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-1.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

Vout [V]

-100

-80

-60

-40

-20

0

20

40

60

80

100I ou

t [m

A]

Load

Pull-down

High-Low Transition Example

The technique works for high-low transitions.

Pull-up

2.222V-55.6 mA

-0.556V0.0 mA

-0.247V6.2 mA

-0.062V0.0 mA

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Application to Non-Linear Devices

We can also use the Bergeron diagram with non-linear sources and loads.

For example, consider the following network:

70 PowerPC

604E

I

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-1.5

-1.0

-0.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

V [V]

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

I [

A]

Non-Linear Device Analysis

Let’s look at the falling edge transition:

Pull-up

Rcvr

Pull-down

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Non-Linear Device Analysis #2

A closer look at the falling edge transition:

-0.015

-0.005

0.005

0.015

0.025

0.035

0.045-1

.5

-1.0

-0.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

V [V]

I [

A]

Pull-up

Rcvr Pull-down

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An Even Closer Look

-0.2

5

-0.2

0

-0.1

5

-0.1

0

-0.0

5

0.00

0.05

0.10

0.15

0.20

0.25

V [V]

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

I [

A] Rcvr

Pull-down

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Conversion to Waveforms

The Bergeron diagram can be used to construct the voltage and current waveforms. Just read off the voltage (or current) at the intersections. First intersection gives the value at the driver, second at the

receiver, etc.

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14

time [t d]

volt

age

[V]

Rcvr

Driver

Example: Example: PowerPC 604EPowerPC 604E

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Summary

Bergeron diagrams are another useful interconnect analysis tool.They provide a graphical solution to the circuit

equations.Use them to analyze voltage and current.

Bergeron diagrams can be used with non-linear elements.

Bergeron diagrams also comprehend the effects of static current on the behavior of our circuits.We’ll explore this in the homework…

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References S. Hall, G. Hall, and J. McCall, High Speed Digital

System Design, John Wiley & Sons, Inc. (Wiley Interscience), 2000, 1st edition.

R. Poon, Computer Circuits Electrical Design, Prentice Hall, 1st edition, 1995.

“Transmission Line Effects in PCB Applications,” Motorola Application Note AN1051, 1990.

W.R. Blood, MECL System Design Handbook, Motorola, Inc., 4th edition, 1988.

“The Bergeron Method: A Graphic Method for Determining Line Reflections in Transient Phenomena,” Texas Instruments, October 1996.

L. Bergeron, Du Coup de Belier en Hydraulique au Coup de Foudre en Electricite, Dunod, Paris, 1949.