# On the Virtualization of Audio Transducers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nullor-Based Inversion of Circuital Systems

#### 2.1. Nullors

#### 2.2. Inversion Theorem

**Theorem**

**1.**

**Proof.**

#### 2.2.1. Voltage Input Voltage Output (VIVO)

#### 2.2.2. Voltage Input Current Output (VICO)

#### 2.2.3. Current Input Voltage Output (CIVO)

#### 2.2.4. Current Input Current Output (CICO)

#### 2.3. Adjoint Networks

- Passive elements are kept without any changes.
- Nullators are replaced with norators, while norators with nullators.
- The input voltage is replaced with a short circuit (i.e., a current sink). The output of the adjoint circuit will be then the current flowing through such a short, where the positive direction follows the element convention, i.e., from the positive to the negative terminal.
- A current source is connected to the output port. This will be the input of the adjoint circuit. In this case, the direction of the current follows the source convention, i.e., from the negative to the positive terminal.
- Controlled sources are replaced with their dual (e.g., VCVS are replaced with CCCS).

## 3. Direct–Inverse–Direct Chains

#### 3.1. Target-Inverse-Physical Chain (TIPC)

#### 3.2. Physical-Inverse-Target Chain (PITC)

## 4. Sensor Virtualization: Application to Capacitive Microphones

#### 4.1. Inverse Model Validation

#### 4.2. Sensor Virtualization Test

## 5. Actuator Virtualization: Application to Compression Drivers

#### 5.1. Inverse Model Validation

#### 5.2. Actuator Linearization Test

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Nullor circuit symbol. The nullator (port 1) is represented with an ellipse, while the norator (port 2) with two circles.

**Figure 3.**(

**a**) Direct System containing one nullor; (

**b**) Inverse System; (

**c**) circuit employed for the Proof of Theorem 1.

**Figure 4.**Augmenting a circuit with a series connection of nullator and norator. (

**a**) Circuit presenting no nullor; (

**b**) circuit with nullor equivalent to the circuit in (

**a**); (

**c**) inverse of the circuit in (

**b**) and, in turn, of the circuit in (

**a**).

**Figure 5.**Augmenting a circuit with a parallel connection of nullator and norator. (

**a**) Circuit presenting no nullor; (

**b**) circuit with nullor equivalent to the circuit in (

**a**); (

**c**) inverse of the circuit in (

**b**) and, in turn, of the circuit in (

**a**).

**Figure 11.**Possible WD implementation of the circuit shown in Figure 10a.

**Figure 12.**Comparison between the DFT of the impulse responses. The blue curves represent the WD implementation of the circuit in Figure 10a, while the red curves the Mathworks Simscape (SSC) implementation of the same circuit. (

**a**) DFT of the impulse response for microphone BK4134; (

**b**) DFT of the impulse response for microphone BK4146.

**Figure 14.**Validation of the Inverse System for microphone BK4134. (

**a**) Output voltage of the Direct System; (

**b**) comparison between ${P}_{\mathrm{in}}$ (dashed red curve) and ${\widehat{P}}_{\mathrm{in}}$ (continuous blue curve) taking into account the processing chain shown in Figure 13.

**Figure 15.**PITC-based virtualization algorithm. Output voltage signals of: the Direct System, i.e., BK4134, when no virtualization algorithm is present (“No Post-processing”), the Target Direct System, i.e., BK4146 (“Target”), and the PITC (“Virtualized”).

**Figure 16.**(

**a**) Direct circuital model of the compression driver; (

**b**) Inverse circuital model of the compression driver.

**Figure 18.**Possible WD implementation of the circuit shown in Figure 16a.

**Figure 19.**Validation of the Inverse System for loudspeaker SEAS. (

**a**) Output voltage of the Direct System; (

**b**) comparison between ${V}_{\mathrm{in}}$ (dashed red curve) and ${\widehat{V}}_{\mathrm{in}}$ (continuous blue curve) taking into account a processing chain similar to that shown in Figure 13.

**Figure 20.**Validation of the Inverse System. (

**a**) Input voltage ${V}_{\mathrm{in}}$ of the Direct System; (

**a**) output velocity ${v}_{\mathrm{out}}$ (i.e., a current) of the Direct System; (

**c**) output voltage ${\widehat{V}}_{\mathrm{in}}$ of the Inverse System.

**Figure 21.**TIPC-based linearization algorithm. The first two plots are obtained considering $A=5$ V: (

**a**) power spectrum of the Physcal Direct System output (“Non Compensated”); (

**b**) power spectrum of the TIPC output (“Compensated”). Instead, the remaining rows are obtained considering $A=9$ V: (

**c**) power spectrum of the Physical Direct System (“Non Compensated”); (

**d**) power spectrum of the TIPC output (“Compensated”).

Input Signal | Output Signal | Direct System | Inverse System |
---|---|---|---|

Voltage | Voltage | ||

Network A. | Network B. | ||

Voltage | Current | ||

Network C. | Network D. | ||

Current | Voltage | ||

Network E. | Network F. | ||

Current | Current | ||

Network G. | Network H. |

Parameter | BK4134 | BK4146 | Parameter | BK4134 | BK4146 |
---|---|---|---|---|---|

${R}_{\mathrm{a}1}$ $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{4}\mathrm{s}}\right]$ | $4.12\times {10}^{6}$ | $1.03\times {10}^{6}$ | ${M}_{\mathrm{ah}}$ $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{4}{\mathrm{s}}^{2}}\right]$ | $278.2$ | $209.52$ |

${R}_{\mathrm{a}2}$ $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{4}\mathrm{s}}\right]$ | $6.54\times {10}^{6}$ | $1.66\times {10}^{6}$ | ${C}_{\mathrm{ab}}$ $\left[\frac{{\mathrm{m}}^{4}{\mathrm{s}}^{2}}{\mathrm{kg}}\right]$ | $0.89\times {10}^{-12}$ | $4.76\times {10}^{-12}$ |

${M}_{\mathrm{a}1}$ $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{4}{\mathrm{s}}^{2}}\right]$ | $54.83$ | $27.44$ | ${M}_{\mathrm{md}}$ $\left[\mathrm{kg}\right]$ | $3.69\times {10}^{-6}$ | $14.73\times {10}^{-6}$ |

${C}_{\mathrm{a}1}$ $\left[\frac{{\mathrm{m}}^{4}{\mathrm{s}}^{2}}{\mathrm{kg}}\right]$ | $1.95\times {10}^{-12}$ | $15.58\times {10}^{-12}$ | ${C}_{\mathrm{md}}$ $\left[\frac{\mathrm{m}}{\mathrm{N}}\right]$ | $12.58\times {10}^{-6}$ | $26.55\times {10}^{-6}$ |

${C}_{\mathrm{ag}}$ $\left[\frac{{\mathrm{m}}^{4}{\mathrm{s}}^{2}}{\mathrm{kg}}\right]$ | $9.12\times {10}^{-15}$ | $46.54\times {10}^{-15}$ | ${C}_{\mathrm{e}0}$ $\left[\mathrm{F}\right]$ | $27.36\times {10}^{-12}$ | $90.72\times {10}^{-12}$ |

${R}_{\mathrm{as}}$ $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{4}\mathrm{s}}\right]$ | $4.13\times {10}^{3}$ | $444.58$ | ${R}_{\mathrm{L}}$ $\left[\Omega \right]$ | $100\times {10}^{6}$ | $100\times {10}^{6}$ |

${M}_{\mathrm{as}}$ $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{4}{\mathrm{s}}^{2}}\right]$ | $18.8$ | $6.24$ | ${S}_{\mathrm{d}}$ $\left[{\mathrm{m}}^{2}\right]$ | $62.2\times {10}^{-6}$ | $248.3\times {10}^{-6}$ |

${R}_{\mathrm{ah}}$ $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{4}\mathrm{s}}\right]$ | $99.93\times {10}^{3}$ | $86.45\times {10}^{3}$ | $\alpha $ $\left[\frac{N}{V}\right]$ | $121.17$ | 140 |

Parameter | SEAS | Parameter | SEAS |
---|---|---|---|

${R}_{\mathrm{a}1}$ $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{4}\mathrm{s}}\right]$ | $0.72\times {10}^{6}$ | ${R}_{\mathrm{e}}$ $\left[\Omega \right]$ | $4.9$ $\Omega $ |

${R}_{\mathrm{a}2}$ $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{4}\mathrm{s}}\right]$ | $1.64\times {10}^{6}$ | ${L}_{\mathrm{e}}$ $\left[\mathrm{H}\right]$ | $50\times {10}^{-6}$ |

${M}_{\mathrm{a}1}$ $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{4}{\mathrm{s}}^{2}}\right]$ | $36.32$ | $B{l}_{0}$ $\left[\frac{\mathrm{N}}{\mathrm{A}}\right]$ | $3.14$ |

${C}_{\mathrm{a}1}$ $\left[\frac{{\mathrm{m}}^{4}{\mathrm{s}}^{2}}{\mathrm{kg}}\right]$ | $30.11\times {10}^{-12}$ | $B{l}_{1}$ $\left[\frac{\mathrm{N}}{\mathrm{A}\phantom{\rule{4.pt}{0ex}}\mathrm{mm}}\right]$ | $2.7\times {10}^{-2}$ |

${C}_{\mathrm{af}}$ $\left[\frac{{\mathrm{m}}^{4}{\mathrm{s}}^{2}}{\mathrm{kg}}\right]$ | $9.88\times {10}^{-12}$ | $B{l}_{2}$ $\left[\frac{\mathrm{N}}{\mathrm{A}\phantom{\rule{4.pt}{0ex}}{\mathrm{mm}}^{2}}\right]$ | $1\times {10}^{-2}$ |

${R}_{\mathrm{md}}$ $\left[\frac{\mathrm{kg}}{\mathrm{m}}\right]$ | $0.92$ | $B{l}_{3}$ $\left[\frac{\mathrm{N}}{\mathrm{A}\phantom{\rule{4.pt}{0ex}}{\mathrm{mm}}^{3}}\right]$ | $1.2\times {10}^{-3}$ |

${M}_{\mathrm{md}}$ $\left[\mathrm{kg}\right]$ | $298.64\times {10}^{-6}$ | $B{l}_{4}$ $\left[\frac{\mathrm{N}}{\mathrm{A}\phantom{\rule{4.pt}{0ex}}{\mathrm{mm}}^{4}}\right]$ | $2.2\times {10}^{-4}$ |

${C}_{\mathrm{md}}$ $\left[\frac{\mathrm{m}}{\mathrm{N}}\right]$ | $14.1\times {10}^{-6}$ | ${S}_{\mathrm{d}}$ $\left[{\mathrm{m}}^{2}\right]$ | $0.7\times {10}^{-3}$ |

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**MDPI and ACS Style**

Giampiccolo, R.; Bernardini, A.; Massi, O.; Sarti, A.
On the Virtualization of Audio Transducers. *Sensors* **2023**, *23*, 5258.
https://doi.org/10.3390/s23115258

**AMA Style**

Giampiccolo R, Bernardini A, Massi O, Sarti A.
On the Virtualization of Audio Transducers. *Sensors*. 2023; 23(11):5258.
https://doi.org/10.3390/s23115258

**Chicago/Turabian Style**

Giampiccolo, Riccardo, Alberto Bernardini, Oliviero Massi, and Augusto Sarti.
2023. "On the Virtualization of Audio Transducers" *Sensors* 23, no. 11: 5258.
https://doi.org/10.3390/s23115258