Bergeron method

The Bergeron method is a graphical method of line theory to qualitatively determine the behavior of a single switching process on lossless lines in the event of reflections at non-linear, memory-free (resistive) generator and load two-poles and, as far as the accuracy allows, quantitatively. The result of the graphic process is what is known as the Bergeron diagram .

In pulse technology and digital electronics , short pulses with steep signal edges must be transmitted via connecting lines without distorting the shape if possible . In such cases, the Bergeron method can be used to analyze the reflections at non-linear circuit outputs and inputs. A typical application is the investigation of negative overshoots in TTL circuits .

The method was first published in the 1930s by Louis Bergeron (1876-1948) for the analysis of the propagation of surface waves on channels.

Basic principle

For rectangular signals with "sufficiently" steep edges and "sufficiently" long and flat roofs so that the transient process decays within a phase, the analysis of lossless lines with memory-free terminations can be traced back to a simple switching process . This can be traced back to the analysis of non-linear DC networks for both linear and non-linear degrees . The step-by-step graphical solution is possible by finding the working points for the individual time segments, each with a line transit time τ as the point of intersection between two characteristic curves .

The overlay theorem , which is also not required for the graphical solution, does not apply to non-linear degrees . Therefore, terms from linear line theory, such as the reflection factor , have no meaning for non-linear closures.

Line theory starting point

Management structure adopted for the Bergeron process

The line theory provides the definitions for the (real) wave resistance

${\displaystyle Z={\sqrt {\frac {L'}{C'}}}}$

and the (constant) phase velocity

${\displaystyle v_{0}={\frac {1}{\sqrt {L'\cdot C'}}}}$

the general solution of the line equations of the lossless line ready:

${\displaystyle u(x,t)=u_{h}\left(t-{\frac {x}{v_{0}}}\right)+u_{r}\left(t+{\frac {x}{v_{0}}}\right)}$

${\displaystyle i(x,t)={\frac {1}{Z}}\cdot \left(u_{h}\left(t-{\frac {x}{v_{0}}}\right)-u_{r}\left(t+{\frac {x}{v_{0}}}\right)\right)}$

Here and represent the forward or backward wave. In our case, both consist of ideal switching edges that move the line of length L without attenuation and distortion in time ${\displaystyle u_{h}}$ ${\displaystyle u_{r}}$

${\displaystyle \tau ={\frac {v_{0}}{L}}}$

run through.

The dynamic characteristics of the line

Total voltage and total current clearly determine both the forward and the returning (voltage) wave, because the general solution results

${\displaystyle u_{h}\left(t-{\frac {x}{v_{0}}}\right)={\frac {u(x,t)+Z\cdot i(x,t)}{2}}}$

${\displaystyle u_{r}\left(t+{\frac {x}{v_{0}}}\right)={\frac {u(x,t)-Z\cdot i(x,t)}{2}}}$

Immediately after a switching process at the input, the returning wave is stable for at least 2 · τ, because the switching edge can only act again at the input after the line has run twice as long. For this results from the above equation

${\displaystyle u_{r}\left(t\right)={\frac {u(0,t)-Z\cdot i(0,t)}{2}}}$

or reshaped

${\displaystyle u(0,t)-Z\cdot i(0,t)=2\cdot u_{r}\left(t\right)}$

This is the equation of an active two-pole , which describes the "line seen from the beginning". It is essential that the internal resistance of the two-terminal network is identical to the wave resistance of the line and is independent of its termination.

The same applies to a switching process at the end:

${\displaystyle u_{h}\left(t-{\frac {L}{v_{0}}}\right)={\frac {u(L,t)+Z\cdot i(L,t)}{2}}}$

from which the "dynamic equation of the line from the end" results:

${\displaystyle u(L,t)+Z\cdot i(L,t)=2\cdot u_{h}\left(t-{\frac {L}{v_{0}}}\right)}$

This is also the equation of an active two-pole, which describes the "line seen from the end". The internal resistance of this two-terminal pole is identical to the negative characteristic impedance of the line (due to the reverse current direction).

These "linear dynamic characteristics of the line" can be used not only in the procedure described below, but also for calculations of the switching behavior of lossless lines in the case of terminations with storage behavior.

The Bergeron method

The initial state

“Empty” Bergeron diagram at time t <0

In the Bergeron diagram to be constructed with Cartesian coordinate axes for voltage and current, the two characteristic curves of the generator two-pole before the switching process (shown in blue in the diagram) and after the switching process (shown in brown in the diagram) as well as the characteristic curve of the load two- pole (shown in green in the diagram) ) entered. If necessary, two auxiliary lines can be drawn through the 0 point, the increases of which are determined by the positive or negative wave resistance (which was not done in the example given). They could serve as an aid to parallel translation in the following steps . ${\displaystyle i=f_{G}(u,-0)}$ ${\displaystyle i=f_{G}(u,+0)}$ ${\displaystyle i=f_{L}(u)}$

Before the switching process taking place at the time , the line and its terminations are in the steady state in terms of direct current . The lossless line has no effect and the working point - always referred to below as A (time, location) - at the beginning of the line A (−0.0) is the same as the working point at the end of the line A (0, L). It can be easily determined as the intersection of the characteristics of the two-terminal generator (before the switching process) and the two-terminal load. ${\displaystyle t=0}$

The switching process

Bergeron diagram directly after the switching process at time t = + 0

At the point in time , the characteristic of the generator dipole changes abruptly. Since the returning wave must remain constant, the new working point A (+0.0) at the beginning of the line results as the intersection of the new generator characteristic and the "dynamic characteristic of the line seen from the beginning". Since this must also go through the old working point and its increase is determined by the wave resistance (10 kΩ in the example given), it can be clearly constructed (shown in red in the diagram) and thus the intersection point A (+0.0) can be found. Because the returning wave is constant for at least two line transit times, only the value of the incoming wave changes abruptly and the resulting edge runs at the same speed as the line end. ${\displaystyle t=0}$ ${\displaystyle i=f_{G}(u,+0)}$ ${\displaystyle i=f_{G}(u,-0)}$ ${\displaystyle Z_{0}}$ ${\displaystyle 2\tau }$ ${\displaystyle v_{0}}$

The reflection at the end

Bergeron diagram at time t = τ

When this (or a later emerging) edge arrives at the end, a new operating point A (τ, L), generally A ((2i + 1) · τ, L), must also be set there. For this purpose, the "dynamic characteristic of the line seen from the end" is drawn through the last operating point at the beginning and with the increase determined by the "negative wave resistance". Their intersection with the characteristic curve of the load dipole results in the new working point at the end of the line. Because the incoming wave is constant there for two line transit times, only the value of the returning wave can change abruptly and the resulting edge travels back to the beginning of the line at the same speed . ${\displaystyle i=f_{L}(u)}$ ${\displaystyle v_{0}}$

The reflection at the beginning

Bergeron diagram at time t = 2τ

When the edge that occurs at the end of the line arrives at the beginning of the line, a new operating point A (2τ, 0), generally A (2i · τ, 0), must be set there. For this purpose, the "dynamic characteristic of the line seen from the beginning" is drawn through the last operating point at the end and with the increase determined by the (now "positive") wave resistance. Their intersection with the characteristic of the generator dipole results in the new working point at the beginning of the line. Again, the value of the returning wave changes abruptly and the resulting edge runs again at the speed to the end of the line. ${\displaystyle v_{0}}$

Bergeron diagram at time t = 7τ

Both of the above procedures are carried out multiple times. Except in special cases, the reflections are repeated infinitely often and converge at the intersection of the characteristics of the generator and load two-pole A (∞, 0) = A (∞, L). This represents the stationary operating point after the transient process. In practice, the construction is ended when the desired or the reading accuracy is achieved.

Timelines

Time curve of voltage and current "read off" from the Bergeron diagram at the beginning and end of a lossless line

If necessary, the values ​​of voltage and current at the beginning and at the end of the line can be taken from the finished Bergeron diagram as even or odd multiples of the transit time τ and represented as a (standardized) time sequence.

Example of a switching process between TTL circuits

Bergeron diagram for a line between TTL gates with a high-low edge

A typical application of the Bergeron method is the investigation of the transmission of impulses on "not properly adapted" lines between digital circuits . In this example, the high / low edge at the output of a standard TTL gate , which drives the input of a standard gate (without a "clamping diode") via a line with Z = 120 Ω, is considered. The characteristic curves correspond approximately to the actual ones, the construction was carried out and designated according to the steps outlined above.

Course of the voltage on a line between TTL gates with a high-low edge

The curve of the voltage at the input and output of the line (ie at the output or input of the TTL gate) can be "read" from the resulting Bergeron diagram. You can see the strong negative overshoot of the voltage at the output of the line, which can possibly lead to the destruction of the following circuit and was avoided in the further development of the circuit by installing a clamping diode .

See also

• Wave timetable

literature

• Wolfgang Hilberg : Impulses on lines . R. Oldenbourg Verlag GmbH, Munich 1981, ISBN 3-486-25491-X .
• Hans-Georg Unger : Electromagnetic waves on lines . Dr. Alfred Hüthig Verlag, Heidelberg 1980, ISBN 3-7785-0601-3 .
• Eberhard Kühn, Horst Schmied: Integrated circuits . Verlag Technik, Berlin 1974.

Web links

• Texas Instruments : The Bergeron Method - A Graphic Method for Determining Line Reflections in Transient Phenomena (PDF October 1996). Retrieved January 18, 2014.

Individual evidence

1. ↑ M. Louis Bergeron: Method Graphique Generale De Calcul Des Propagations D'ondes Planes . In: Bulletin de juillet-août . Extrait des Mémoires de la Société des Ingénieurs Civils de France, 1937.