I/O CIRCUITS AND MODELS - Advanced Signal Integrity for High-Speed Digital Design - page 461

11.6 BERGERON DIAGRAMS

At this point we take a slight detour from our survey of transceivers to introduce
the Bergeron diagram, a technique that will allow us to analyze the behavior of
interconnects with nonlinear transmitter and receiver characteristics by graphi-
cally solving the simultaneous current-voltage relationships of the components
of a signaling system. We introduce this technique as it is useful for furthering
our understanding of transmission-line basics. We demonstrate and develop the
technique through a simple example.

Example 11-3 Bergeron Diagram for a Linear Interconnect Circuit In this
example we analyze the rising behavior for the circuit shown in Figure 11-18.
The transmitter is a 2.5-V push-pull driver with symmetrical pull-up and
pull-down impedances. The receiver has a resistive termination to a 2.5-V
termination supply.
We start the analysis by plotting the current versus voltage relationships for
the transmitter and receiver. The easiest way to do so is to draw the equivalent
circuits and write the Ohm's law expressions. This is shown in Figure 11-19
along with the load-line plot. Note that we were careful to account correctly for
the direction of the current flow in all of the equivalent circuits and that we
chose to use milliamperes as the units for the y -axis. Note from the plot that
the intersection of the transmitter pull-down and the receiver load lines gives the
voltage and current (0.357 V and 28.6mA) for the circuit at the steady state when
driven low. Since the circuit is at steady state, this is the potential and current
flow at all points on the transmission line. This gives us the starting point for
our analysis, since we are studying a rising-edge transition.
Our next step is to find the initial voltage and current for the rising edge
at the transmitter. We do this by drawing a load line representing the 50-
transmission line, starting at the intersection of the transmitter pull-down and
receiver (steady-state low) and extending until it intersects with the load line
for the transmitter pull-up. This load line has a slope equal to 1 /Z 0 , since the

transmission line also obeys Ohm's law and the load line is a plot of current
versus voltage. In effect, the load line for the transmission line is graphically
depicting the Ohms' law equation:

−

v − v 0 = Z 0 (i − i 0 ) (11-5)

2.5 V
75

Ω

i

12.5 Ω

50 Ω , 2 ns

0 V 2.5 V

t r = 100 ps

Figure 11-18 Interconnect circuit for Example 11-3.