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Approximating Non-Linear Inductors Using Time-Variant Linear Filters

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Approximating Non-Linear Inductors Using Time-Variant Linear Filters Proc. of the 18th Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, Nov 30 - Dec 3, 2015 APPROXIMATING NON-LINEAR INDUCTORS USING TIME-VARIANT LINEAR FILTERS Giulio Moro Andrew P. McPherson Centre for Digital Music Centre for Digital Music Queen Mary University of London Queen Mary University of London London, United Kingdom London, United Kingdom [email protected] [email protected] ABSTRACT mainly focuses on bounded input-bounded output stability [6, 7] and transient suppression [8]. However, these methods are not In this paper we present an approach to modeling the non-linearities readily applicable when filter coefficients are changing at every of analog electronic components using time-variant digital linear sample. Recent work proves that stable time-varying behavior can filters. The filter coefficients are computed at every sample de- be obtained using state variable filters [9]. However, as in this pending on the current state of the system. With this technique paper we deal with passive first-order filters, we chose to use a we are able to accurately model an analog filter including a non- different filter topology. linear inductor with a saturating core. The value of the magnetic In [10], time-varying coefficients are used to introduce a clip- permeability of a magnetic core changes according to its magnetic ping function in the feedback loop of an IIR filter, in order to re- flux and this, in turn, affects the inductance value. The cutoff fre- produce the behaviour of analog voltage controlled filters. In [11] quency of the filter can thus be seen as if it is being modulated by it is shown that this method affects the frequency response of a the magnetic flux of the core. In comparison to a reference non- resonant filter by increasing its bandwidth and moving its centre linear model, the proposed approach has a lower computational frequency. The use of IIR coefficients varying on a sample-by- cost while providing a reasonably small error. sample basis has been exploited previously in [12] where feed- back amplitude modulation is used for sound synthesis and in [13] 1. INTRODUCTION where time-varying fractional delays are used to model non-linear vibrating strings. This work investigates how it is possible to use linear, time vari- In this paper we will present a physically-informed model for ant filters in order to introduce non-linearities in a digital signal an inductor, with its characteristic non-linearity caused by the satu- processing system. The idea is to use time-varying infinite im- ration of the magnetic core [14]. Non-linear differential equations pulse response filters whose coefficients are updated at every time and a state-space model to solve the non-linear transformer are sample according to the state of the system at the previous time presented in [15], whereas a Wave Digital Filter approach can be sample. This approach is applied here to solve electronic circuits found in [16]. with non-linear components in the digital domain and can be used From a physical standpoint, the saturation of the core in an in- as a building block for Virtual Analog applications. ductor affects the present value of its inductance. From this consid- Non-linear DSP systems are governed by non-linear equations eration we will build an infinite impulse response (IIR) linear filter that have to be solved iteratively at a non-negligible computational whose coefficients are updated at every time step using the actual cost [1, 2]. Adding a new non-linear equation to an existing sys- value of the inductance given by the current saturation state of its tem increases the computational cost even further and requires core. This will produce a delay-free loop which we will address sometimes to partially re-design the existing system [3]. Non- using a variation on the classic 1-sample delay approach, widely iterative ways to solve such systems have been proposed which used in the literature ([17, 18]), and linearizing the system around rely on pre-computed tables [4]. The lookup tables method does the operating point. D’angelo recently discussed the linearization not scale up easily to systems with multiple non-linearities as this of a non-linear system around an operating point to solve a transis- increases the dimensionality of the table, increasing memory usage tor ladder filter [19], generalizing the delay-free loops resolution and computational cost. method in [20]. The method presented in this paper replaces the non-linear The physics of the non-linear inductor is reviewed in Sec- equations with a time-varying linear filter, which is by itself less tion 2. Section 3 will present a reference non-linear discrete-time expensive in terms of computations and can, potentially, be ex- model for the inductor which will be used to evaluate the approxi- panded to replace systems with multiple non-linearities with higher- mated model presented in Section 4. Results and discussion follow order filters. We do not expect the output of our approximated in Section 5 and Section 6 respectively. model to be a sample-by-sample replica of the results obtainable with more accurate simulations, but we expect it to be close enough that the loss in accuracy is justified by improvements in execution 2. PHYSICS OF INDUCTORS speed and scalability. While general criteria that determine the stability of station- An inductor is a passive component with inductive behavior, usu- ary recursive filters are well defined in the literature [5], criteria ally built using a coil of wire winded on a core made of ferro- to assess the stability of time-variant linear filters can be studied magnetic material, such as ferrite. While inductors are often mod- and defined only under some specific conditions. Existing work eled as linear components, most real inductors exhibit non-linear DAFX-1 Proc. of the 18th Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, Nov 30 - Dec 3, 2015 behavior caused primarily by the progressive saturation of their 1.5 ferromagnetic core. The distortion caused by the core occurs pri- marily for signals with large currents and low frequencies. 1 We present here a simplified model for the inductor which models the saturation of the core but does not take into account 0.5 losses, hysteresis and parasitic parameters. From Faraday’s laws we have, for a solenoid: 0 dB V B[T] = L (1) dt NS -0.5 where B is the magnetic flux density in the inductor core, VL is the voltage across the inductor, S is the area of the section of the core -1 and N is the number of turns in the inductor. Ampere’s law gives the magnetizing force H for a solenoid traversed by a current IL -1.5 as: [21] -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 H[Am -1] #10 5 NI H = L (2) l Figure 1: Anhysteretic B-H curve according to the Fröhlich model and l is the length of the induction path. In an ideal inductor there for an inductor with a ferrite core (ui = 400, Bsat = 1:3) is a linear relation between the flux density and the magnetizing force: where N, S, l, µ are the physical parameters of the inductor de- B = µH (3) scribed above. As denoted by Eqs. (3) and (5), µ is not constant and its value can drop by several orders of magnitude as the core where µ is the absolute magnetic permeability of the core, defined progressively saturates and this is reflected directly on the value of as: the inductance through Eq. (6). µ = µ0µi (4) where µ0 is the vacuum permeability and µi ≥ 1 is the relative 3. DISCRETIZATION OF THE NON-LINEAR INDUCTOR permeability of the magnetic core. In the case of a ferromagnetic core, however, the magnetic flux density cannot be increased above In order to solve a circuit including a non-linear inductor in the a certain value. This value is called magnetic flux density satura- discrete-time domain, Eqs. (1), (2) and (5) have to be discretized. tion and depends on the material and geometry of the core. This is straightforward for Eqs. (2) and (5): For low flux density levels, Eq. (3) is valid and the inductor can N be considered as a linear component. As the core approaches the H[n] = I [n] (7) l L saturation level Bsat, the relation between H and B becomes non H[n] linear, the magnetic characteristics of the core change from those B[n] = (8) of a ferromagnetic material to those of a paramagnetic material and c + bjH[n]j the value of µ progressively changes from being µ = µ · µ when i 0 while Eq. (1) requires an integration formula. The solution to an B = 0 to being approximately µ = µ when B = B . The 0 sat equation of the form: Fröhlich-Kennelly relation gives the following relation between B and H for a ferromagnetic core: [22] dx = f(x; t) (9) H dt B = (5) c + bjHj is given in the discrete-time domain by the backward Euler for- mula as: [23] where b and c are defined as x[n] = x[n − 1] + T f(x[n]) (10) q 1 1 − µ b = i where T is the sampling period of the discrete-time system. This Bsat formula, when applied to Eq. (1), yields: 1 c = T µ0µi B[n] = B[n − 1] + V [n] (11) NS L The B-H relation described by these formulas is shown in Fig.
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