Bergeron diagram

The Bergeron method is a graphical method of management theory, to high and as far as to determine on lossless lines, the behavior of a single shift operation with reflections on nonlinear memory free ( resistive ) generator and load Two Poland, the accuracy allows quantitative. As a result of the printing process creates the so-called Bergeron diagram.

In the pulse technique and digital electronics, short pulses with steep signal edges via connecting lines must be transmitted if possible without shape distortion. In such cases, the Bergeron process used to analyze the reflections from the non-linear Schaltkreisaus and inputs. A typical application is the study of the negative overshoot 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 " enough " steep flanks and " sufficient " long, flat roofs, so that the transient decays within a phase, the analysis of lossless lines can be traced back in memory free accounts to a simple shift. This can be attributed both linear as well as nonlinear degrees to the analysis of non-linear direct current networks. The gradual graphical solution is τ by finding the operating points for each of the periods of each line of a run-time possible as the intersection between two curves.

For non-linear financial statements does not apply the superposition theorem, which is also not required for the graphical solution. Why have terms from the linear transmission line theory, such as the reflection factor for non-linear financial statements no meaning.

Theoretical line starting point

The line theory provides the definitions for the (real) impedance

And the (constant ) phase velocity

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

It questions and the leading or returning wave dar. Both consist in our case from ideal switching flanks which the line of length L without attenuation and distortion in the time

Through.

The dynamic characteristics of the line

Total voltage and total current determine both the upstream and the returning (voltage) wave clearly, because of the general solution is obtained

Directly after a shift at the entrance is the reflected wave for at least 2 · τ stable because the switching edge only after twice the propagation time of the line again may act at the entrance. This follows from the above equation

Or transformed

This is the equation of an active two-terminal network, which " saw line generations from the beginning " that describes. It is essential that the internal resistance of the dipole is identical and independent of their degree with the characteristic impedance of the line.

The same applies to a switching operation at the end:

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

This is also the equation of an active two-terminal network, which " saw line from the end of her" describes. The internal resistance of this dipole is the same as the negative (due to the reverse flow direction) impedance of the line.

This " linear dynamic characteristics of the line " are not only in the method described below, but also for calculations of the switching behavior of lossless lines on degrees which memory behavior usable.

The Bergeron process

The initial state

First, the two characteristics of the Generatorzweipols before the shift (in the diagram shown in blue ) and ( shown in brown in the diagram) after the shift in the to be constructed Bergeron diagram with Cartesian coordinate axes for voltage and current as well as the characteristic of the Lastzweipols ( shown in green in the diagram ) registered. If necessary, two auxiliary lines can still be drawn through the 0-point, whose increases are determined by the positive or negative wave resistance ( which was not done in the example ). You could serve in the following steps as an aid to parallel shift.

Prior to the time shift taking place are the management and their financial statements in terms of direct current steady state. The lossless line has no effect and thereby the operating point - hereinafter always referred to as A ( time, place ) - at the beginning of the line A ( -0,0 ) is equal to the operating point at end of line A ( 0, L ). It is simply the intersection of the characteristic curves of Generatorzweipol determine ( before the shift ) and Lastzweipol.

The switching process

At the time, the characteristic of the Generatorzweipols changes abruptly. Since the reflected wave must remain constant, the new operating point A ( 0.0 ) results at the lead as the intersection of the new generator characteristic and the " dynamic characteristic of the circuit from the beginning as seen ". Since this must also go through the old operating point and her rise through the impedance ( in the example 10 k ) is determined, it can be clearly construct ( shown in red in the diagram) and thus the intersection of A ( 0.0 ) find. Because the reflected wave is constant for at least two wire delays, only the value of the departing wave changes abruptly and the resulting edge runs at the speed the end of the line.

The reflection at the end

If this (or a later will ) edge arrives at the end, there must be also a new operating point A ( τ, L), in general A ( (2i 1) · τ, L) to adjust. For this purpose, the " dynamic characteristic of the circuit seen from the end " is drawn by the last working point at the beginning and the " negative impedance " by the particular rise. It intersects the characteristic curve of Lastzweipols results in the new operating point at the line end. Because that's where the forward wave for two wire delays is constant, only the value of the returning wave can change abruptly and the resulting edge runs at the speed back to the beginning of the line.

The reflection at the beginning

If the resulting end of line edge arrives at the lead again, there must be, in turn, a new operating point A ( 2τ, 0), generally set A ( 2i · τ, 0). For this purpose, the " dynamic characteristic of the circuit as seen from the beginning here," drawn by the last working point at the end and with the ( "positive" now ) by the characteristic impedance given rise. It intersects the characteristic curve of Generatorzweipols results in the new operating point at the beginning of the line. Again the value of the returning wave changes abruptly and the resulting edge running again with the speed of the line end.

Both the above procedures are executed several times. Except in special cases, the reflections repeat infinitely often and converge at the intersection of the characteristic curves of the generator and Lastzweipol A ( ∞, 0) = A ( ∞, L). This represents the steady state after the transient operating dar. Practically, the construction is completed when the desired or the accuracy of reading has been reached.

Timings

From the finished Bergeron diagram, the values ​​of voltage and current can be when needed at the beginning and end of the line refer to even or odd multiples of the duration τ and represent as a (normalized ) time sequence.

For example for a switching operation between TTL circuits

A typical application of the Bergeron process is the study of the transmission of impulses to "not clean custom " lines between digital circuits. In this example, the high / low edge at the output of a standard TTL gate is that Ω = 120 (without " Clampingdiode " ) drives, via a line to the Z input of a standard gate is considered. The curves correspond approximately to the actual construction was carried out and indicated according to the steps outlined above.

From the resulting Bergeron chart the course of the voltage at the input and output of the line can be (ie at the output or input of the TTL gate ) "read". One can see the strong negative overshoot of the voltage at the output of the line, which eventually can lead to the destruction of the following circuit and was avoided in the development of circuits by installing a Clampingdiode.