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State variable changes to avoid non computational issues Abstract
fer function, like the grid of a triode. But the ideas we aregoing to expose will remain the same.
This paper is about the numerical simulation of nonlinearanalog circuits with ”switch” components, such as diodes.
A ”switch” component is an electrical device that may ormay not conduct, depending on the state of the circuit. Theproblem with ”switch” components is that the topology ofthe circuit is variable and so, apparently, it is not possibleto describe the system with a single differential equationand solve it using standard numerical methods. This papershows how to choose an appropriate state variable and over-come the above difficulties.
A test example
Let’s consider the following circuit.
Figure 2: Nonlinear transfer function Let’s call id the current flowing into the diodes and vd the voltage on the diode. A standard way to proceed intothe analysis of the circuit, is to take the current i will be as the state variable of the system. The equations of the systemwill become: If the voltage on the diodes is below their threshold, no current is flowing through the diodes and so no current is flowing into the resistor and the capacitor; the circuit is open. When the voltage gets higher than the diode thresh- Where vc is the voltage on the capacitor and vd is the voltage old, the circuit becomes a standard RC filter. We have just on the diodes. Remembering that i is the state variable, in seen two different topologies that the same circuit can have, order to be integrated, the above system should be written The couple of diodes has a memory-less nonlinear trans- fer function, like the one in figure 1. Note that the cur- i(s)ds − f −1(i) = 0 rent circuit is only an example and so, instead of diodes, wecould have other switch components with a different trans- But this is not possible because id = f (vd) is not invertible.
But actually, this is not a real issue. All we have to do is the bilinear transform method. One of its main advantages choosing another state variable. Taking vd, the system can is that it is a one step method but it has quadratic order of k−ek−1−yk+yk−1 = ( Now the equation has a valid analytic form.
And, after rearranging the terms, the equation becomes: −Af (yk) − K = yk Analysis of the solution
The aim of this section is to give an example of how we can study if the solution of (1) exist and is unique. First of all (1) should be rewritten in differential form: K = ek−1 − ek + ( We can easily see that there exist one and only one so- d = 1 + f (vd)R Then we have to require that the second term of this equa- f (yk) = −A−1(yk + K) tion is continuous and satisfies the Lipschitz condition. Tohave continuity, the denominator should satisfy we note that the solution is the intersection of the graph ofy = f (x) and the line y = −A−1(x + K), as shown in the |1 + f (vd)R| > For ”clipping” transfer functions like the one of diodes, it isf (vd) 0 and so 1 + f (vd)R ≥ 1.
Instead of Lipschitz condition, we could ask that This is a stronger condition; if it is satisfied, then also Lips-chitz condition is satisfied. It is easy to see that in clippingdevices, with f (vd) 0, the above condition is true if andonly if f ∈ C1.
Considering that in real applications f is not defined an- alytically, but is often a regular function interpolating somemeasurement point, asking f ∈ C1 does not limit the valid-ity of this technique.
The numerical discretization
Because the lines have a negative slew and the nonlinear transfer function is monotone with a positive slew, there is Now we are ready to study a numerical technique to solve only one point of intersection, and so the solution is unique.
(1). Let’s consider a discretization step of h, such that A solution of (2) can be easily found with standard nu- merical methods, like Newton-Raphson or bisection algo- rithm, which globally converge thanks to the regularity and We want to find a sequence {yk} that approximates the real monotonicity of the function. A little trick to improve the speed of convergence is to use a ”hot start” at each itera-tion; this consists in initializing the iterative methods using the solution at the previous discretization point, which is afirst order approximation of the new solution.
Starting from equation (1), we have to give a numerical approximation of the integral. A common choice in the au-dio applications is the trapezoidal rule, that corresponds to Comparison with a simplified method age of 20V peak to peak. The solid line is the exact simula-
tion. The dotted line is the simulation with the approximated In the previous sections we studied a very simple example like a model; but in spite of its simplicity, the analysis andthe computational models are not elementary. Generally, inpractical applications like the signal processing in the mu-sical field, simplified models are used instead of the one presented here. Giving up the possibility to obtain an exact simulation of the circuit, they are computationally cheaper and easier to implement. For example, the circuit consid- ered in the previous sections can be approximated by a one pole high pass filter (with cutoff frequency at lowed by a nonlinear waveshaper with the transfer function of the diodes. This solution is commonly used into the DSP models in tube preamplifier simulators. So we may ask if such an approximation can give sufficiently good accuracy.
We are going to present the results of a numerical ex- periment that can be quite illuminating. The experimentis based on the same circuit of the previous sections, tak- ing R = 10KΩ, C = 22nF and the generator e as a sinesource with a frequency of 60Hz. For this circuit both the Figure 5: Simulation of the circuit with an input volt- exact and the simplified models have been built. In the first age of 0.5V peak to peak. The solid line is the exact simula- simulation, the amplitude of the source signal was 20V peak tion. The dotted line is the simulation with the approximated to peak, so the signal was heavily clipped by the diodes. In the second run, the same circuit has been simulated using aninput voltage of 0.5V peak to peak. Here are the graphicswith the results: Conclusions
In the first case (figure 4) the source signal is heavily clipped; this means that the diodes are conducing in about This paper would like to introduce, from an operative point all the period of the signal, current is flowing through the of view, some aspects of the simulation of circuits with capacitor and so the RC network is behaving like the HPF ”switching” components. It would not to be a complete the- of the approximated model. But in the second case (figure oretic treatment, but it would like to give a complete exam- 5), diodes are not conducing, so in the real case no current ple of how to build models when switching components are is flowing through the RC network and no filtering effect present, how to analyze them and give a numerical method is performed. This is a big difference between the approxi- for the simulation. This treatment can be applied without relevant changes to all the saturating components, like tran-sistor, tubes and saturating magnetic cores, which are al-most all the nonlinearities found in audio applications.
Acknowledgment
we would like to thank Gianpaolo Borin who inspired this work and Pierre Richemond for his revision.
References
John, H. Mathews, ”Numerical Methods for Mathe-matics, Sience, and Engeneering”, Prentice Hall, 1992 Morgan Jones, ”Valve Amplifiers”, Newnes, Reed Ed- ucational and Professional Publishing, 1995 Figure 4: Simulation of the circuit with an input volt- Charles Rydel, ”Simulation of Electron Tubes withSpice” in Preprint no.3965 of AES 98th Convention1995 G. Borin, G. De Poli, D. Rocchesso, ”Elimination offree delay loops in discrete-time models of nonlinearacoustic systems”, IEEE Trans. on Speech and AudioProcessing, vol 8, issue 5, September 2000, pp.597-605 T. Serafini, ”Metodi per la simulazione numericadi sistemi dinamici non lineari, con applicazioni alcampo degli strumenti musicali elettronici”, Univer-sita’ di Modena e Reggio Emilia, Italy, 2001 (MS The-sis) Foti Frank, ”Aliasing Distortion in Digital DynamicsProcessing, the Cause, Effect, and Method for Mea-suring It: The Story of ’Digital Grunge!’ ”, in Preprintno.4971 of AES 106th Convention 1999

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