© H. Heck 2008Section 3.21 Module 3:Analysis Techniques Topic 2: Bergeron Diagrams OGI EE564 Howard Heck

Bergeron Diagrams EE 564 © H. Heck 2008 Section 3.22 Where Are We? 1.Introduction 2.Transmission Line Basics 3.Analysis Tools 1.Lattice Diagrams 2.Bergeron Diagrams 4.Metrics & Methodology 5.Advanced Transmission Lines 6.Multi-Gb/s Signaling 7.Special Topics

Bergeron Diagrams EE 564 © H. Heck 2008 Section 3.23 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

Bergeron Diagrams EE 564 © H. Heck 2008 Section 3.24 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. 5V5V I V 55  Pull-up 0V0V I V 40  Pull-down 0V0V I=0 V R TT =  Receiver 5V 55  40  50  Example Circuit I Analyze the low-high transition.

Bergeron Diagrams EE 564 © H. Heck 2008 Section 3.25 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. V out [V] I out [mA] Pull-up Pull-down Rcvr

Bergeron Diagrams EE 564 © H. Heck 2008 Section 3.26 Analysis: Initial Wave 2.Start at the steady state, which is defined by the intersection of the pull-down and receiver load lines. V out [V] I out [mA] Pull-down Rcvr Pull-up V = 2.381V I = 47.6 mA V = 0.000V I = 0 mA 3.Construct the load line for the interconnect. The slope is 1/ Z 0.  The intersection of load lines for transmission line & pull-up give us the initial voltage and current on the network.

Bergeron Diagrams EE 564 © H. Heck 2008 Section 3.27 V out [V] I out [mA] Pull-down V = 0.000V I = 0 mA V = V I = 47.6 mA Rcvr Pull-up Analysis: 1 st Reflection 4.The next load line segment will have a slope of -1/ Z 0. 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 = V I = 0 mA

Bergeron Diagrams EE 564 © H. Heck 2008 Section 3.28 Analysis: Successive Reflections 5.Continue the process, alternating the load line slopes between 1/ Z 0 and -1/ Z 0, until reaching the point at which the load and source lines intersect (steady state high) V out [V] I out [mA] Pull-up Rcvr V = 4.762V I = 0 mA V = V I = 2.3 mA V = 4.989V I = 0.0 mA V = 4.994V I = 0.1 mA V = 4.999V I = 0.0 mA

Bergeron Diagrams EE 564 © H. Heck 2008 Section 3.29 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.

Bergeron Diagrams EE 564 © H. Heck 2008 Section V out [V] I out [mA] Load Pull-down High-Low Transition Example The technique works for high-low transitions. Pull-up 2.222V mA V 0.0 mA V 6.2 mA V 0.0 mA

Bergeron Diagrams EE 564 © H. Heck 2008 Section 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

Bergeron Diagrams EE 564 © H. Heck 2008 Section Non-Linear Device Analysis Let’s look at the falling edge transition: Pull-up Rcvr Pull-down

Bergeron Diagrams EE 564 © H. Heck 2008 Section Non-Linear Device Analysis #2 A closer look at the falling edge transition: Pull-up Rcvr Pull-down

Bergeron Diagrams EE 564 © H. Heck 2008 Section An Even Closer Look Rcvr Pull-down

Bergeron Diagrams EE 564 © H. Heck 2008 Section 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 time [ t d ] voltage [V] Rcvr Driver Example: PowerPC 604E

Bergeron Diagrams EE 564 © H. Heck 2008 Section 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…

Bergeron Diagrams EE 564 © H. Heck 2008 Section References S. Hall, G. Hall, and J. McCall, High Speed Digital System Design, John Wiley & Sons, Inc. (Wiley Interscience), 2000, 1 st edition. R. Poon, Computer Circuits Electrical Design, Prentice Hall, 1 st edition, “Transmission Line Effects in PCB Applications,” Motorola Application Note AN1051, W.R. Blood, MECL System Design Handbook, Motorola, Inc., 4 th edition, “The Bergeron Method: A Graphic Method for Determining Line Reflections in Transient Phenomena,” Texas Instruments, October L. Bergeron, Du Coup de Belier en Hydraulique au Coup de Foudre en Electricite, Dunod, Paris, 1949.