APPROXIMATING NON-LINEAR INDUCTORS USING TIME-VARIANT LINEAR FILTERS

Transcription

1 Proc. of the 18 th Int. Conference on Digita Audio Effects (DAFx-15), Trondheim, Norway, Nov 3 - Dec 3, 215 APPOXIMATING NON-LINEA INDUCTOS USING TIME-VAIANT LINEA FILTES Giuio Moro Centre for Digita Music Queen Mary University of London London, United Kingdom g.moro@qmu.ac.uk Andrew P. McPherson Centre for Digita Music Queen Mary University of London London, United Kingdom a.mcpherson@qmu.ac.uk ABSTACT In this paper we present an approach to modeing the non-inearities of anaog eectronic components using time-variant digita inear fiters. The fiter coefficients are computed at every sampe depending on the current state of the system. With this technique we are abe to accuratey mode an anaog fiter incuding a noninear inductor with a saturating core. The vaue of the magnetic permeabiity of a magnetic core changes according to its magnetic fux and this, in turn, affects the inductance vaue. The cutoff frequency of the fiter can thus be seen as if it is being moduated by the magnetic fux of the core. In comparison to a noninear mode, the proposed approach has a ower computationa cost whie providing a reasonaby sma error. 1. INTODUCTION This work investigates how it is possibe to use inear, time variant fiters in order to introduce non-inearities in a digita signa processing system. The idea is to use time-varying infinite impuse response fiters whose coefficients are updated at every time sampe according to the state of the system at the previous time sampe. This approach is appied here to sove eectronic circuits with non-inear components in the digita domain and can be used as a buiding bock for Virtua Anaog appications. Non-inear DSP systems are governed by non-inear equations that have to be soved iterativey at a non-negigibe computationa cost [1, 2]. Adding a new non-inear equation to an existing system increases the computationa cost even further and requires sometimes to partiay re-design the existing system [3]. Noniterative ways to sove such systems have been proposed which rey on pre-computed tabes [4]. The ookup tabes method does not scae up easiy to systems with mutipe non-inearities as this increases the dimensionaity of the tabe, increasing memory usage and computationa cost. The method presented in this paper repaces the non-inear equations with a time-varying inear fiter, which is by itsef ess expensive in terms of computations and can, potentiay, be expanded to repace systems with mutipe non-inearities with higherorder fiters. We do not expect the output of our mode to be a sampe-by-sampe repica of the resuts obtainabe with more accurate simuations, but we expect it to be cose enough that the oss in accuracy is justified by improvements in execution speed and scaabiity. Whie genera criteria that determine the stabiity of stationary recursive fiters are we defined in the iterature [5], criteria to assess the stabiity of time-variant inear fiters can be studied and defined ony under some specific conditions. Existing work mainy focuses on bounded input-bounded output stabiity [6, 7] and transient suppression [8]. However, these methods are not readiy appicabe when fiter coefficients are changing at every sampe. ecent work proves that stabe time-varying behavior can be obtained using state variabe fiters [9]. However, as in this paper we dea with passive first-order fiters, we chose to use a different fiter topoogy. In [1], time-varying coefficients are used to introduce a cipping function in the feedback oop of an II fiter, in order to reproduce the behaviour of anaog votage controed fiters. In [11] it is shown that this method affects the frequency response of a resonant fiter by increasing its bandwidth and moving its centre frequency. The use of II coefficients varying on a sampe-bysampe basis has been expoited previousy in [12] where feedback ampitude moduation is used for sound synthesis and in [13] where time-varying fractiona deays are used to mode non-inear vibrating strings. In this paper we wi present a physicay-informed mode for an inductor, with its characteristic non-inearity caused by the saturation of the magnetic core [14]. Non-inear differentia equations and a state-space mode to sove the non-inear transformer are presented in [15], whereas a Wave Digita Fiter approach can be found in [16]. From a physica standpoint, the saturation of the core in an inductor affects the present vaue of its inductance. From this consideration we wi buid an infinite impuse response (II) inear fiter whose coefficients are updated at every time step using the actua vaue of the inductance given by the current saturation state of its core. This wi produce a deay-free oop which we wi address using a variation on the cassic 1-sampe deay approach, widey used in the iterature ([17, 18]), and inearizing the system around the operating point. D angeo recenty discussed the inearization of a non-inear system around an operating point to sove a transistor adder fiter [19], generaizing the deay-free oops resoution method in [2]. The physics of the non-inear inductor is reviewed in Section 2. Section 3 wi present a non-inear discrete-time mode for the inductor which wi be used to evauate the mode presented in Section 4. esuts and discussion foow in Section 5 and Section 6 respectivey. 2. PHYSICS OF INDUCTOS An inductor is a passive component with inductive behavior, usuay buit using a coi of wire winded on a core made of ferromagnetic materia, such as ferrite. Whie inductors are often modeed as inear components, most rea inductors exhibit non-inear DAFX-1

2 B[T] Proc. of the 18 th Int. Conference on Digita Audio Effects (DAFx-15), Trondheim, Norway, Nov 3 - Dec 3, 215 behavior caused primariy by the progressive saturation of their ferromagnetic core. The distortion caused by the core occurs primariy for signas with arge currents and ow frequencies. We present here a simpified mode for the inductor which modes the saturation of the core but does not take into account osses, hysteresis and parasitic parameters. From Faraday s aws we have, for a soenoid: db dt = VL NS where B is the magnetic fux density in the inductor core, V L is the votage across the inductor, S is the area of the section of the core and N is the number of turns in the inductor. Ampere s aw gives the magnetizing force H for a soenoid traversed by a current I L as: [21] H = NIL and is the ength of the induction path. In an idea inductor there is a inear reation between the fux density and the magnetizing force: (1) (2) B = µh (3) where µ is the absoute magnetic permeabiity of the core, defined as: µ = µ µ i (4) where µ is the vacuum permeabiity and µ i 1 is the reative permeabiity of the magnetic core. In the case of a ferromagnetic core, however, the magnetic fux density cannot be increased above a certain vaue. This vaue is caed magnetic fux density saturation and depends on the materia and geometry of the core. For ow fux density eves, Eq. (3) is vaid and the inductor can be considered as a inear component. As the core approaches the saturation eve B sat, the reation between H and B becomes non inear, the magnetic characteristics of the core change from those of a ferromagnetic materia to those of a paramagnetic materia and the vaue of µ progressivey changes from being µ = µ i µ when B = to being approximatey µ = µ when B = B sat. The Fröhich-Kenney reation gives the foowing reation between B and H for a ferromagnetic core: [22] B = H c + b H where b and c are defined as 1 b = B sat c = 1 µ µ i The B-H reation described by these formuas is shown in Fig. 1. For sma vaues of H and/or arge vaues of B sat, Eq. (5) is equivaent to Eq. (3), thus expaining the inear behaviour at ow currents. The inductance L of a soenoid is derived by Ampere s aw as: [21] 1 µ i L = µn 2 S (5) (6) H[Am -1 ] #1 5 Figure 1: Anhysteretic B-H curve according to the Fröhich mode for an inductor with a ferrite core (u i = 4, B sat = 1.3) where N, S,, µ are the physica parameters of the inductor described above. As denoted by Eqs. (3) and (5), µ is not constant and its vaue can drop by severa orders of magnitude as the core progressivey saturates and this is refected directy on the vaue of the inductance through Eq. (6). 3. DISCETIZATION OF THE NON-LINEA INDUCTO In order to sove a circuit incuding a non-inear inductor in the discrete-time domain, Eqs. (1), (2) and (5) have to be discretized. This is straightforward for Eqs. (2) and (5): H[n] = N IL[n] (7) B[n] = H[n] c + b H[n] whie Eq. (1) requires an integration formua. The soution to an equation of the form: dx dt (8) = f(x, t) (9) is given in the discrete-time domain by the backward Euer formua as: [23] x[n] = x[n 1] + T f(x[n]) (1) where T is the samping period of the discrete-time system. This formua, when appied to Eq. (1), yieds: B[n] = B[n 1] + T VL[n] (11) NS Combining Eqs. (7), (8) and (11), we obtain: ( V L[n] = NS N IL[n] ) B[n 1] T c + bn I L[n] (12) Where V L[n] is the votage across an inductor at a time instant n given the current through it I L[n] and the magnetic fied at the previous time instant B[n 1]. DAFX-2

3 Proc. of the 18 th Int. Conference on Digita Audio Effects (DAFx-15), Trondheim, Norway, Nov 3 - Dec 3, High pass fiter We now consider the circuit in Fig. 2. In this circuit ow frequencies from the input V i wi find an easier path to ground through the inductor than high frequencies, therefore, considering node V o as the output, the circuit wi act as a high pass fiter. For the circuit in Fig. 2, the current through the inductor is: I L[n] = Vi[n] Vo[n] whie the votage across the inductor is: (13) V L[n] = V o[n] (14) Substituting these vaues in Eqs. (7), (11) and (12) gives: V o[n] = NS T B[n] = B[n 1] + ( N c + bn V i [n] V o[n] V i[n] V o[n] T Vo[n] (15) NS B[n 1] ) (16) Combining Eqs. (15) and (16) we obtain the foowing system equation for the discretized version of the circut in Fig. 2 with a non-inear inductor. with β[n] = bnt k[n] V 2 o [n]β[n] V o[n]γ[n] δ[n] = (17) γ[n] = ct + bnt k[n]v i[n] + N 2 S bn 2 Sk[n]B[n 1] δ[n] = cnsb[n 1] bn 2 Sk[n]B[n 1]V i[n] + N 2 SV i[n] in which, k is the sign of the current through the inductor at time instant n, V i[n] is the votage input to the system and B[n 1] is computed at each time step using Eq. (15). Eq. (17) is stricty speaking a non-poynomia equation as it contains k = sign(v o V i) and shoud then be soved using iterative numerica approaches (e.g. Newton Method). On the other hand, it can be considered as two distinct second-order poynomias one with k = 1 and one with k = 1. Soving these two poynomias wi produce four soutions for V o, of which one and ony one wi be rea and therefore acceptabe. The foowing schedue can thus be used to find the output V o[n] of the system for every n: 1. Sove Eq. (17) as expained above to obtain V o[n]. For n = assume B[n 1] = 2. Compute B[n] using Eq. (15) 3.2. Low pass fiter We now consider the circuit in Fig. 3. In this circuit high frequencies from the input V i wi be attenuated whie passing through the inductor more than ow frequencies, therefore, considering node V o as the output, the circuit wi act as a ow pass fiter. In this case the current through and the votage across the inductor are, respectivey: I L[n] = Vo[n] (18) V L[n] = V i[n] V o[n] (19) Substituting these vaues in Eqs. (7) and (11) and going through passages simiar to those described in Section 3.1 we obtain: with ɛ[n] = bnt k[n] V 2 o [n]ɛ[n] + V o[n]ζ[n] + η[n] = (2) ζ[n] = bn 2 Sk[n]B[n 1] ct + bnt k[n]v i[n] N 2 S η[n] = cnsb[n 1] + ct V i[n] where k[n] = sign(v o[n]) and for which the same considerations made above for the resoution of Eq. (17) are vaid. 4. APPOXIMATION OF THE NON-LINEA INDUCTO WITH A VAIABLE INDUCTANCE The mode presented in Section 3 soves the non-inear eectronic circuits proposed using non-inear equations. A different approach is presented in this section which soves the same circuits using time-varying inear fiters informed by the physica behaviour of the non-inearity under exam. In Section 2 we showed that the change in the permeabiity of the core of an inductor as it approaches saturation affects the effective inductance of the core. The non-inear behaviour is modeed here using a time-varying vaue for the inductance which is, for each time instant, determined by the current vaue of the core permeabiity. The incrementa magnetic permeabiity of a ferromagnetic materia is the rate of change of magnetic fux density with respect to the magnetizing force and is given, in its differentia definition, by: [24] µ inc = db dh (21) Figure 2: Passive high pass fiter Figure 3: Passive ow pass fiter DAFX-3

4 Proc. of the 18 th Int. Conference on Digita Audio Effects (DAFx-15), Trondheim, Norway, Nov 3 - Dec 3, 215 For a rea inductor, according to the Fröhich mode, µ inc is given combining this with Eq. (5): µ inc = db dh = c (22) (c + b H ) 2 emembering Eqs. (7) and (22), the time-discrete formuation for µ inc is: µ inc[n] = c (c + bn I L[n] ) 2 (23) The equation of the inductance of a soenoid as given by Eq. (6) is discretized as: L[n] = µinc[n]n 2 S Which, combined with Eq. (23), gives: 4.1. High pass fiter L[n] = (24) cn 2 S (c + bn I L[n]) 2 (25) If we consider the inductor in Fig. 2 to be idea, the cutoff frequency F c of the fiter can be computed from the vaues of its eectronic components as: F c = 2πL (26) We can discretize the circuit under exam using the we-known biinear transform. If the component vaues were time-invariant, the z-transform of the first-order highpass fiter woud be: H(z) = 2L 2Lz 1 T + 2L + z 1 (T 2L) (27) If we now consider the inductor to have a saturating core, the vaue of L actuay changes at every sampe, according to Eq. (25). Comparing Eqs. (25) and (26) it emerges that as the current through the inductor increases, the actua inductance vaue is decreased and consequenty the cutoff frequency increases. As a consequence, the fiter coefficients in Eq. (27) wi aso change over time. Considering the time-varying eements, the finite difference equation for this system is, therefore: V o[n] = b [n]v i[n] + b 1[n]V i[n 1] a 1[n]V o[n 1] (28) where: b [n] = 2L[n]/(T + 2L[n]), b 1[n] = 2L[n]/(T + 2L[n]), a 1[n] = (T 2L[n])/(T + 2L[n]) (29) For the circuit in Fig. 2, L[n] depends on the instantaneous current through the inductor, as given by Eq. (13), which, in turn, depends on V o[n]. Therefore L[n] cannot be computed before V o[n] and it cannot appear in the right hand side of Eq. (28). This constraint eads to an uncomputabe oop, aso known as deay-free oop [25]. To eiminate the deay free oop we must use an approximate vaue for I L[n] which does not depend on V o[n]. We can estimate a vaue V o[n] V o[n] by inearizing the output signa around time instant n 1 and estimating the vaue of V o[n] using inear extrapoation. Given the discrete-time differentiation of V o[n] we can write V o[n] = V o[n] V o[n 1] (3) V o[n] = V o[n 1] + V o[n 1] (31) Given a parameter α [, 1], we can define an estimated vaue V o(n, α) for the output votage at every time instant between n 1 and n by ineary interpoating between V o[n 1] and V o[n] as: V o(n, α) = α V o[n 1] + (1 α) V o[n] (32) which, for α = 1 equas V o[n 1] and for α = equas V o[n]. In order to appropriatey compute the current through the inductor, the input and output votage must be considered at the same instant in time, therefore we aso define V i (n, α) as the inear interpoation between V i[n 1] and V i[n], parametrized by α: V i (n, α) = α V i[n 1] + (1 α) V i[n] (33) Now we can compute approximate vaue for I L[n] parametrized by α by repacing V o with V o(n, α) and V i with V i (n, α) in Eq. (13): 4.2. Low pass fiter I Lα [n] = V i (n, α) V o(n, α) The z-transform of the ow pass fiter in Fig. 3: H(z) = T + T z 1 T + 2L + z 1 (T 2L) (34) (35) We can derive the equations for the fiter coefficients simiary to what has been done in the previous paragraph. In the case of a rea inductor, the vaue of L[n] is, again, time-varying and it depends on the current through the inductor I L[n] for each instant n. As such current is not known in advance, we need to use an approximate vaue for I L[n] when computing the fiter coefficients. Anaogousy to what has been done for the high pass fiter in Section 4.1, using the same formuas for inear extrapoation as in Eq. (31) and inear interpoation as in Eq. (32) to obtain V o(n, α), we can write the estimated current through the inductor, parametrized by α, by repacing V o with V o(n, α) in Eq. (18): I Lα [n] = V o(n, α) 5. ESULTS (36) For the evauation of the saturating inductor mode we created a digita mode of the high pass circuit in Fig. 2 and soved it using both the mode invoving non-inear equations described in Section 3, used as a, and the approximate mode introduced in Section 4. esuts for the ow pass circuit in Fig. 3 are not expicity reported here for brevity, but the findings are very simiar to those outined beow for the high pass fiter. We performed our tests using the parameter vaues isted in Tabe 4 over a of the possibe combinations of the foowing parameters: DAFX-4

5 Ampitude [V] Ampitude [V] Proc. of the 18 th Int. Conference on Digita Audio Effects (DAFx-15), Trondheim, Norway, Nov 3 - Dec 3, 215 Input eve [V] Input signa 15Hz 45Hz 89Hz 179Hz 238Hz 953Hz 381Hz 762Hz 1524Hz 195Hz Noise Bass Guitar Tabe 1: High-pass fiter: frequency-domain MS error (%) for different signas and input votages, with α = 1 and samping rate=48khz Samping rate [khz] Input signa 15Hz 45Hz 89Hz 179Hz 238Hz 953Hz 381Hz 762Hz 1524Hz 195Hz Noise Bass Guitar Tabe 2: High-pass fiter: frequency-domain MS error (%) for different signas and samping rates with α = 1 and input eve=2v Input signa α 15Hz 45Hz 89Hz 179Hz 238Hz 953Hz 381Hz 762Hz 1524Hz 195Hz Noise Bass Guitar Tabe 3: High-pass fiter: frequency-domain MS error (%) for different signas and vaues of α with samping rate=48khz and input eve=2v 1. samping frequencies: 12kHz, 24kHz, 48kHz, 96kHz, 192kHz, 384kHz. 2. audio signas: (a) Sine waves at frequencies: 15Hz, 45Hz, 89Hz, 179Hz, 238Hz, 953Hz, 381Hz, 762Hz, 1524Hz, 195Hz, ength 1 seconds (b) White noise sampe, ength 1 seconds (c) Eectric bass guitar sampe, ength 6.6 seconds (d) Eectric guitar sampe, ength 3.1 seconds 3. a set of ampitudes: 1V, 1V, 5V, 1V, 2V 4. a set of vaues for the inear interpoation parameter α:.25,.5,.75, 1 We skipped tests on sine waves whose frequency was above the Nyquist frequency of the samping rate. For samping frequencies of 48kHz and above the resut signas have been bandimited to a maximum frequency of 2kHz before computing error figures. Given a samping rate, an input signa and an ampitude, the outputs of the approximate mode for each different α have been compared to the output of the mode. [Ω] µ i B sat[t ] N S[cm 2 ] [cm] Tabe 4: Physica parameters used in the simuation Time [s] Time [s] #1-3 Figure 4: Time domain waveforms for the and high pass fiter for samping frequency 48kHz, signa frequency 15Hz, input eve 2V, α = 1. Large time scae (top) and detai (bottom) Fig. 4 dispays the time domain votage signa of a 15Hz sinusoid of peak ampitude 2V processed through the high pass fiter. The time domain waveforms are very simiar. Fig. 4 (bottom) shows that ony by zooming in on the time axis we can notice the difference: the drop in votage caused by the saturation of the core is sighty deayed in the waveform. This can be easiy expained considering that in the mode the vaue for the fux density is computed based on the vaue of the current at the previous time sampe, which causes an inherent deay in the response. Fig. 5 shows that the fux density is in fact DAFX-5

6 Ampitude [db] Ampitude [db] Fux density [T] Fux density [T] Ampitude [db] Proc. of the 18 th Int. Conference on Digita Audio Effects (DAFx-15), Trondheim, Norway, Nov 3 - Dec 3, Time [s] Frequency [Hz] THD(%) THD+N(%) MS fd Err(%) Fs: 48 ref f: 15 appr Figure 7: Peaks of the spectrum of a 15Hz sinusoid processed through the high pass fiter circuit, samped at 48kHz Time [s] #1-3 Figure 5: Time domain waveforms of the fux density for the signa in Fig. 4. Large time scae (top) and detai (bottom) Frequency [Hz] THD(%) THD+N(%) MS fd Err(%) Fs: 384 ref f: 15 appr Figure 8: Peaks of the spectrum of a 15Hz sinusoid processed through the high pass fiter circuit, samped at 384kHz Frequency [Hz] THD(%) THD+N(%) MS fd Err(%) Fs: 48 ref f: 15 appr Figure 6: Peaks of the spectrum of the signa in Fig. 4 sighty deayed with respect to the. The MS error between the two time-domain votage signas is 2.45%. This rather arge error figure is justified by the fact that the discrepancy between the two waveforms occurs around a rapid votage drop. As human perception of an audio signa is inked more cosey to its frequency content than to its time-domain representation, we find it more reevant to the THD, THD+N and frequency domain MS error figures for the purposes of this evauation. Fig. 6 dispays the and signa in the frequency domain. They are very simiar, with a tota MS error in the frequency domain as sma as.71%. Whie the match is amost perfect for the ower harmonics, a sma discrepancy arises from the 9th harmonic (135Hz and above). No noise or spurious frequencies have been introduced by the mode, as denoted by the fact that the THD and the THD+N vaues are exacty the same. Performing the same anaysis on a 15Hz signa gives a timedomain MS error of 8.63% and a frequency-domain MS error of 1.69%. The spectrum of the signa is shown in Fig. 7. The discrepancies in the spectra ampitudes begin to arise from about 495Hz (33rd harmonic). Again, no spurious frequencies or noise are added. By increasing the samping frequency to 384kHz (8 times oversamping), for the 15Hz sinewave we obtain that the time-domain MS error is cut down to 1.1% and the frequency-domain MS error is.24%. The increase in the samping rate reduced the effect of the unit deay used in the approximation. The frequency response of this oversamped signa is dispayed in Fig. 8. At every time step the vaue of the inductance L[n] changes according to an estimated vaue of the current through the inductor, as expained in Section 4.1. Tabes 1 to 3 show the frequency domain error figures for each of the test signas. Tabe 1 shows that the increase in the ampitude of the input signa causes arger errors. This is expected as to a arger ampitude corresponds a faster saturation DAFX-6

7 Proc. of the 18 th Int. Conference on Digita Audio Effects (DAFx-15), Trondheim, Norway, Nov 3 - Dec 3, 215 of the core and therefore a arger distortion of the output, with a steeper votage drop in the time domain. Tabe 2 shows that resuts are greaty improved by increasing the samping frequency. The error reduction is proportiona to 1/F s. The α parameter, as given by Eq. (34), determines the baance between the weight given to the vaue I[n 1] of the current measured at the previous time step and the estimated vaue I [n] of the current through the inductor at the current time instant, when computing the vaue L[n] of the inductance at the current time instant. When α =.5 the estimated vaue used for the current through the inductor is the average vaue of the current over the time interva between n 1 and n and this is, in theory, the best choice to be used in the computation of L[n], as ong as the estimated vaue I [n] is reasonaby cose to the actua vaue I[n]. Test resuts in Tabe 3 show that most of the times α =.75 performs better than α =.5 and aso that both these vaues perform poory when the signa contains significant amounts of energy at higher frequencies. This can be expained by the fact that as the frequency of the signa increases, the inear extrapoation becomes ess accurate and α =.75 performs better than α =.5 because the overshoot caused by the estimate is mitigated by giving ess weight to it. The vaue of α = 1, corresponding to no inear extrapoation being used, was found to be the one that gives best resuts for a signa of arbitrary frequency content. This effectivey corresponds to the introduction of a 1-sampe deay so that the fiter coefficients are entirey based on the system state at the previous sampe instant. 6. DISCUSSION In this paper we introduced a way to sove a high pass fiter circuit containing a non-inear inductor using a time-varying II fiter, whose bock diagram is shown in Fig. 9. The mode is physicay informed and expoits the fact that the actua inductance vaue for a soenoid changes according to the saturation of its core. As pointed out in 4.1, these changes affect the frequency response of the fiter by moduating its cutoff frequency Performance The time-variant II fiters used here to emuate the behavior of a non-inear inductor produced resuts comparabe to the mode and they did not exhibit any inherent instabiity. The time domain error due to the intrinsic deay in the approximation does not affect negativey the perceived sound, as it produces rather sma error figures in the frequency-domain. For certain combinations of parametes (e.g. arge µ i, arge input votage) ringing and overshoot effects have been observed, due to the sudden change in the fiter coefficients. These effects can be attenuated by hard imiting the sew rate of the fiter coefficients, imposing a maximum-change-per-sampe imit, or otherwise suppressing the transient using one of the techniques proposed in [8]. The use of inear extrapoation to compute an estimate of the inductance vaue at the current time sampe did not improve the resuts. On the other hand, the mode produces good resuts when not using inear extrapoation, with frequency-domain errors beow 1.75% in a the cases under exam and beow 1.2% when tested with rea-word audio signas. Inductors mosty saturate at ow frequencies, therefore the use of oversamping is not a requirement when modeing this type of non-inearity, as the higher partias generated by the distortion are V i[n] V i[n 1] z 1 b [n] L[n]= b 1[n] Compute fiter coefficients Saturating inductor mode ( c+ bn a 1[n] b [n] = 2L[n]/(T + 2L[n]) b 1 [n] = 2L[n]/(T + 2L[n]) a 1 [n] = (T 2L[n])/(T + 2L[n])) L[n] cn 2 S V i [n 1] Vo[n 1] ) 2 z 1 V o[n] V o[n 1] Figure 9: Bock diagram of the digita fiter for the circuit in Fig. 2, when α = 1. ikey to be beow the Nyquist frequency even for a samping rate of 48kHz. Despite this genera consideration, the choice of the oversamping factor has to be evauated on a case-by-case basis, according to the characteristics of the inductor (e.g. saturation fux density), of the circuit (e.g. presence of other non inear eements and fiters) and of the expected frequency content of the input signa. Nevertheess, the accuracy of the mode takes advantage of oversamping, which reduces the effects of the 1-sampe deay and gives better resuts overa. As outined above, Eq. (17) is a particuar case which can be soved with ower computationa cost than most non-inear equations found in DSP systems. This considered, soving the high pass circuit using the non-inear mode and Eq. (17) requires 13 mutipies, 12 additions and 2 square roots per sampe. Soving the same circuit with the time-variant II fiter in Section 4 requires 5 mutipies, 4 additions and 2 divisions per sampe. As on modern CPUs the execution time of square roots is greater or equa than the one for divisions, the time-variant II mode turns out to have a ower computationa cost than the non-inear one. Improvements in speed can become even greater when a simiar approach is used to mode non-inearities which are otherwise soved through computationay-expensive transcendenta functions. The stabiity and performance of time-variant II modes have to be evauated on a case-by-case basis. For instance, modeing of a diode cipper circuit as a time-variant resistor has been attempted by the authors which ed to a conditionay-working mode that requires oversamping and other adjustments to prevent DC drift of the output Appications The idea that is at the base of this research, that is the use of timevariant inear fiters with recursive coefficient computation to impement non-inearities, proved to be not ony achievabe, but aso we suited for the emuation of a rea eectronic component, the inductor. This mode has been successfuy used to extend existing systems, without requiring major re-designs. For instance, it was used to add the non-inearities of the inductor to the wah-wah DAFX-7

8 Proc. of the 18 th Int. Conference on Digita Audio Effects (DAFx-15), Trondheim, Norway, Nov 3 - Dec 3, 215 peda mode based on the DK-method presented in [26] by simpy repacing the static inductance in the circuit with a time-varying one. What emerged from the simuation of the wah-wah peda is that the current through the inductor was too sma to cause audibe saturation, when using for the inductor parameters simiar to the ones of a rea wah-wah inductor. By introducing fictitious physica parameters for the inductor, we aowed the input signa to drive it into saturation. As a resut we obtained increased harmonic distortion and a shift of the cutoff frequency of the fiter. The inductor mode presented is not compete yet, as a fu mode of the inductor woud require at east to add the hysteresis of the magnetic core. From what we have seen so far, it is reasonabe to think that this additiona step wi not add much to the compexity of the mode. 7. EFEENCES [1] J. Macak, J. Schimme, and V. Väimäki, ea-time guitar preamp simuation using modified bockwise method and approximations, EUASIP Journa on Advances in Signa Processing, vo. 211, pp. 2, 211. [2] D. T. M. Yeh, Automated physica modeing of noninear audio circuits for rea-time audio effects;part II: BJT and vacuum tube exampes, Audio, Speech, and Language Processing, IEEE Transactions on, vo. 2, no. 4, pp , May 212. [3] K. Meerkötter and. Schoz, Digita simuation of noninear circuits by wave digita fiter principes, in Circuits and Systems, 1989., IEEE Internationa Symposium on. IEEE, 1989, pp [4] S. Petrausch and. abenstein, Wave digita fiters with mutipe noninearities, in Signa Processing Conference, 24 12th European, Sept 24, pp [5] J. O. Smith, Introduction to Digita Fiters with Audio Appications, W3K Pubishing, [6] G. Stoyanov and M. Kawamata, Variabe digita fiters, J. Signa Processing, vo. 1, no. 4, pp , [7] J. Laroche, On the stabiity of time-varying recursive fiters, Journa of the Audio Engineering Society, vo. 55, no. 6, pp , 27. [8] V. Väimäki and T.I. Laakso, Suppression of transients in time-varying recursive fiters for audio signas, in Proceedings of the 1998 IEEE Internationa Conference on Acoustics, Speech and Signa Processing, 1998, May 1998, vo. 6, pp vo.6. [9] A. Wishnick, Time-varying fiters for musica appications, in Proceedings of the 17th Internationa Conference on Digita Audio Effects (DAFx-14), 214, pp [1] D ossum, Making digita fiters sound anaog, pp. 3 33, [11] V. Väimäki, S. Bibao, J. O. Smith, J. S. Abe, J. Pakarinen, and D. Berners, Virtua anaog effects, DAFX: Digita Audio Effects, Second Edition, pp , 211. [12] J. Keimoa, V. Lazzarini, V. Vaimaki, and J. Timoney, Feedback ampitude moduation synthesis, EUASIP Journa on Advances in Signa Processing,, no , 211. [13] J. Pakarinen, V. Väimäki, and M. Karjaainen, Physicsbased methods for modeing noninear vibrating strings, Acta Acustica united with Acustica, vo. 91, no. 2, pp , 25. [14] D. C. Jies, J. B. Thoeke, and M. K. Devine, Numerica determination of hysteresis parameters for the modeing of magnetic properties using the theory of ferromagnetic hysteresis, Magnetics, IEEE Transactions on, vo. 28, no. 1, pp , [15] J. Macak, Noninear audio transformer simuation using approximation of differentia equations, Eektrorevue, vo. 2, no. 4, December 211. [16]. C. D. de Paiva, J. Pakarinen, V. Väimäki, and M. Tikander, ea-time audio transformer emuation for virtua tube ampifiers, EUASIP Journa on Advances in Signa Processing, vo. 211, no. 1, pp , 211. [17] T. Stison and J. Smith, Anayzing the moog vcf with considerations for digita impementation, in Proceedings of the 1996 Internationa Computer Music Conference, Hong Kong, Computer Music Association, [18] A. Huoviainen, Noninear digita impementation of the moog adder fiter, in Proc. Int. Conf. on Digita Audio Effects (Napes, Itay, October 24), 24, pp [19] S. D Angeo and V. Väimäki, Generaized moog adder fiter: Part ii expicit noninear mode through a nove deayfree oop impementation method, Audio, Speech, and Language Processing, IEEE/ACM Transactions on, vo. 22, no. 12, pp , 214. [2] A. Härmä, Impementation of recursive fiters having deay free oops, in Acoustics, Speech and Signa Processing, Proceedings of the 1998 IEEE Internationa Conference on. IEEE, 1998, vo. 3, pp [21] D. J. Griffiths, Introduction to eectrodynamics, vo. 3, Prentice ha Upper Sadde iver, NJ, [22] D. C. Jies, Introduction to Magnetism and Magnetic Materias, Second Edition, Tayor & Francis, [23] D. T. M. Yeh, Digita impementation of musica distortion circuits by anaysis and simuation, Ph.D. thesis, Stanford University, 29. [24] Incrementa magnetic permeabiity, in Computer Science and Communications Dictionary, pp Springer US, 21. [25] G. Borin, G. De Poi, and D. occhesso, Eimination of deay-free oops in discrete-time modes of noninear acoustic systems, Speech and Audio Processing, IEEE Transactions on, vo. 8, no. 5, pp , 2. [26] M. Hoters and U. Zözer, Physica modeing of a wah-wah effect peda as a case study for appication of the noda DK method to circuits with variabe parts, in Proceedings of the 14th Internationa Conference on Digita Audio Effects DAFx11, 211, pp DAFX-8