Series and Parallel Resistance
C H Gough
Introduction
This Note is one of a series, documenting the development of the five Storage Ring Kicker systems for the SLS. Early experience with the prototype pulser showed that it was relatively easy to obtain a half-sine pulse of 4kA and 5s, using semiconductor switches, to give +/-1% matching. For better performance, far more care was needed to consider component tolerances, in particular during the turn-off transient in the pulser diodes.
First measurements showed that there were large differences in the resistive component due to the metallised ceramic chambers.
A resistance across the magnet terminals seemed to give exact trimming for the magnetic field differences due to the metallisation.
Main Pulse Circuit
The basic circuit is shown in Fig.1. The half-sine resonance has a frequency
,
The amplitude A is found from comparing peak energy in the capacitor and the inductor:
giving
For the present application, the absolute difference is the interesting parameter, since this gives an absolute deflection to the stored electron beam. However, experience shows that the tracking is mostly proportional to the magnitude, and proportional measurements are slightly easier to make.
The proportional difference between two sinewaves is:
The worst case is when the reference waveform, Iref, is either I1 or I2. The best case is when the reference waveform is the mean of I1 and I2.
To simplify the treatment, the amplitude and frequency differences are dealt with separately. Firstly, the frequency difference
In the best case that Cref=Cmean, (A+B)/2=R, and the equation reduces to
Again taking the best case that ref=(a+b)/2
To express this in words, the proportional difference increases linearly with time and frequency, and is approximately linear to the proportional frequency difference between resonators.
Next the amplitude error
In words, the proportional error due to amplitude is the difference of the proportional amplitude error.
Summing the errors independently,
Complete nonsense - fix later
Circuit Behaviour
The circuit in Fig.1a-c shows two alternatives of the same basic circuit, without parasitics. An extra shunt or series component must be used to give sense to the analysis, otherwise infinite dV/dt can occur with finite value of TT; the resistor R3 is used for this purpose. In Fig.1, the difference between the two circuits is in the "low frequency" response; with the series circuit (resistor
shunting the diode), the decaying tail is driven by the negative voltage stored in the capacitor; with the parallel resistance, the inductor has decaying current tail with zero driving voltage.
The diode parameters can be adjusted in the PSpice library. If the diode transit time parameter TT is reduced to zero, then a perfect half-sine pulse is produced. With finite TT, some reverse current flows through the diode [Ref1-3]. The default parameters used were:
Zero-bias capacitance Cjo=5nF Transit time TT=100ns
p-n grading exponent m=0.5 recombination current ISR=0
Forward Bias depletion capacitance FC=0.5 Emission coefficient N=1.0
Fig.1a The two alternatives of the basic circuit.
Fig.1b Response of the shunt circuit. The inductor current peak is reduced, and at turn on the switch must provide current instantaneously to Rp.
Fig.1c Response of the series circuit. The inductor current peak is higher, but there is a long current tail which lasts until C2 is fully discharged.
There are two different modes of turn-off, according to the value of resistor R3. If the diode current exceeds the resistor current at the time TT after zero-crossing, there is a fast positive voltage swing, with dV/dt limited only by the stray capacitance of the diodes. If the diode current is smaller than the resistor current, the voltage transition is slower and has an exponential tail.
For a low-loss circuit, the resistor current at turn-on is the same as for turn-off
Equating these two gives
For the values in the circuit of Fig.1, and TT=100ns, the critical resistance value is 20. The different responses are seen in detail in Fig.2, for resistances of 10, 20 and 30.
Fig.2a Response for resistances above and below the critical value in the parallel circuit. The main current is dependant of Rp.
Fig.2b Response for resistances above and below the critical value in the series circuit. The main current is independent of Rs.
Oscillation Damping
In Fig.2, the high resistance did not give and oscillatory response, but this was by chance. For the present application, the magnet current can be many shapes, including with a large oscillatory tail; the requirement is that the magnet currents track each other in time. However, in general, an oscillation requires two reactances, and a resistance. The tolerances for these components is increasingly difficult as the oscillation frequency increases, or the damping decreases. A heavily damped characteristic with one reactive element and one resistance reduces by one the number of precision components.
A single power diode may have 5nF capacitance at zero bias, decreasing to 100pF at high reverse bias. The magnet terminations may also have 100pF of stray capacitance. It seems better to swamp the non-linear resistance and capacitance of the diode with a precise external component.
The basic circuit can be modelled as a simple RLC parallel or series circuit, as in Fig.4a-c. The capacitance comes from the reverse-biased diode.
Fig.4a Series and parallel resonators, with capacitance values for critical damping.
Fig.4b Series circuit with Cs=10, 20 and 30nF
Fig.4c Parallel circuit with Cp=0.25, 1.25 and 2.25nF.
With the series circuit then, a large capacitance is needed to avoid oscillation, and this would seem the better choice for swamping the non-linear capacitance characteristic of the diode.
Conclusion
This Note has given a systematic method for designing a pulser for best tracking performance. A parallel resistance has the following characteristics:
the tail current reduces quickly to zero without additional capacitors
additional parallel trimming resistances in the range 10-50 may be added to trim the differences between magnets low values of stray parallel capacitance tend to give oscillations
A series resistance has the following characteristics:
the main current is independent of the resistance value
a pure series circuit has no additional circuit loops to give multiple resonances large values of capacitor can be used across the diodes without oscillation.
References
1. "Fast, Faster, Fastest -Optimized Diodes for Switching Applications", U Steinebrunner, IXYS Semiconductor GmbH.
2. "Fast Recovery Expitaxial Diodes (FRED), Characteristics, Applications and Examples", R Bürkel, T Schneider, IXYS Semiconductor GmbH.
3. "Ultrafast Recovery Diodes Meet Today's Requirements for High Frequency Operation and Power Ratings in SMPS Applications", A Guerra, K Andoh and S Fimiani, International Rectifier.
