# Passive Guaranteed Simulation of Analog Audio Circuits: A Port-Hamiltonian Approach

^{*}

## Abstract

**:**

## 1. Introduction

- Analog circuits combine energy-storing components, dissipative components, and sources.
- Storage components do not produce energy, and dissipative components decrease it.

## 2. Port-Hamiltonian Systems

#### 2.1. Formalism and Property

#### 2.1.1. Components

- ${n}_{S}$
- internal components that store energy $\mathtt{E}\ge 0$ (capacitors or inductors),
- ${n}_{D}$
- internal components that dissipate power $\mathtt{D}\ge 0$ (resistors, diodes, transistors, etc.),
- ${n}_{P}$
- external ports that convey power $\mathtt{S}\phantom{\rule{0.277778em}{0ex}}(\in \mathbb{R})$ from sources (voltage or current generators) or any external system (active, dissipative, or mixed).

#### 2.1.2. Conservative Interconnection

**Theorem**

**1 (Tellegen).**

- ${\mathbf{J}}_{\mathbf{x}}\in {\mathbb{R}}^{{n}_{S}\times {n}_{S}}$
- expresses the conservative power exchanges between storage components (this corresponds to the so-called $\mathbf{J}$ matrix in classical Hamiltonian systems);
- ${\mathbf{J}}_{\mathbf{w}}\in {\mathbb{R}}^{{n}_{D}\times {n}_{D}}$
- expresses the conservative power exchanges between dissipative components;
- ${\mathbf{J}}_{\mathbf{y}}\in {\mathbb{R}}^{{n}_{P}\times {n}_{P}}$
- expresses the conservative power exchanges between ports (direct connections of inputs to outputs);
- $\mathbf{K}\in {\mathbb{R}}^{{n}_{S}\times {n}_{D}}$
- expresses the conservative power exchanges between the storage components and the dissipative components;
- ${\mathbf{G}}_{\mathbf{x}}\in {\mathbb{R}}^{{n}_{S}\times {n}_{P}}$
- expresses the conservative power exchanges between ports and storage components (input gain matrix);
- ${\mathbf{G}}_{\mathbf{w}}\in {\mathbb{R}}^{{n}_{D}\times {n}_{P}}$
- expresses the conservative power exchanges between ports and dissipative components (input gain matrix).

**Property**

**1 (Power Balance).**

**Proof.**

**Remark**

**1 (Power variables).**

#### 2.2. Example

**Remark**

**2 (Reduction).**

## 3. Generation of Equations

**Step 1**:- from a netlist ($\mathcal{L}$) to a graph ($\mathcal{G}$) that represents the Kirchhoff’s laws for a chosen orientation (convention);
**Step 2**:

#### 3.1. Graph Encoding

#### 3.1.1. Netlists

#### 3.1.2. Graph

- Dipoles are made of two nodes and a single branch, defining a single couple of state x and storage function $h\left(x\right)$ (storage component), or dissipative variable w and scalar relation $z\left(w\right)$ (dissipative components).
- More generally, n-ports multipole are made of n nodes and at least $n-1$ branches, defining $n-1$ couples of variables and functions. Typically, the graph for the bipolar junction is made of two branches (base-emitter and base-collector).

- build the internal graph by connecting the elementary graph of the components from the first block of the netlist,
- introduce a reference node ${N}_{0}$ (or datum, see [27] §10) to define the external branches from the second block.

#### 3.1.3. Kirchhoff’s Laws on Graphs

#### 3.2. Realizability Analysis

**i**,

**v**). For storage and sources components, step (i) is straightforward with the constraints given in Table 2. For dissipative components, this step is achieved by selecting each component as voltage-controlled or current-controlled in order to satisfy a criterion on the matrix description of the interconnection scheme. This realizability criterion is given in Section 3.2.1, assuming the control type of every edge is known. A method of choosing the control type of dissipative edges so as to satisfy the realizability criterion is addressed in Section 3.2.2. This leads to Algorithm 1, which solves (i) and (ii).

#### 3.2.1. A Criterion for Realizability

**Proposition**

**1 (Realizability).**

**Proof.**

_{2}in Proposition 1, we state the following remark, which is used in the sequel to derive the realizability analysis algorithm.

**Remark**

**3 (Necessary condition for realizability).**

#### 3.2.2. Algorithm

**(C1)**- The potential on each node $n\in [1,\cdots ,{n}_{N}]$ is uniquely defined so that ${\sum}_{b=1}^{{n}_{B}}{[\Lambda ]}_{n,b}=1$.
**(C2)**- Each Current-controlled edge $b\in [{n}_{1}+1,\cdots ,{n}_{N}]$ propagates the knowledge of the potential on one node to the other, so that ${\sum}_{n=1}^{{n}_{N}}{[\Lambda ]}_{n,b}=1$.
**(C3)**- No edge imposes the reference potential ${e}_{0}$ so that ${\lambda}_{0}={\mathbb{0}}_{1\times {n}_{B}}$.
**(C4)**- No voltage-controlled edge $b\in [1,\cdots ,{n}_{1}]$ imposes any potential so that ${\lambda}_{1}={\mathbb{0}}_{{n}_{N}\times {n}_{2}}$.

Algorithm 1: Analysis of realizability. If successfully complete, the resulting PHS structure is given by the procedure in the proof of Proposition 1 |

#### 3.2.3. Example

## 4. Guaranteed-Passive Simulation

#### 4.1. Numerical Scheme

**Remark**

**4 (Multi-variate components).**

#### 4.2. Solving the Implicit Equations

**Remark**

**5 (Explicit mapping).**

Algorithm 2: Simulation, with ${n}_{t}$ the number of time-steps and ${n}_{NR}$ the (fixed) number of Newton–Raphson iterations. |

#### 4.3. Comparison with Standard Methods

## 5. Applications

#### 5.1. Diode Clipper

#### 5.2. Common-Emitter BJT Audio Amplifier

#### 5.3. Wah Pedal

`C++`code is automatically generated; Second, this code is encapsulated in a Juce template to compile the audio plugin (see [44]). The sample rate ${f}_{s}$ is imposed by the host digital audio workstation (here Ableton Live!), and we force five Newton–Raphson iterations. The simulation performed well (audio examples are available at the url [45]). The CPU load on a laptop (Macbook 2.9 GHz Intel Core i7 with 8Go RAM) is 37% for ${f}_{s}=96$ kHz, and 20% for ${f}_{s}=48$ kHz.

**Remark**

**6 (Time-varying stability).**

## 6. Conclusions

- the graph theory to describe the interconnection network of a given circuit’s schematic,
- a dictionary of elementary components which are conformable with PHS formalism.

`C++`simulation code to be used in the core of a real-time VST audio plug-in simulating the Dunlop Cry-Baby wah pedal.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Reduction

**Remark**

**A1 (Reduced explicit mapping).**

## Appendix B. Dictionary of Elementary Components

2-Ports | |||||

Storage | Diagram | $\mathit{x}$ | Stored Energy ${E}=\mathit{h}(\mathit{x})$ | Voltage $\mathit{v}$ | Current $\mathit{i}$ |

Inductance | ϕ | $\frac{{\varphi}^{2}}{2L}$ | $\frac{d\varphi}{dt}$ | $\frac{dh}{d\varphi}$ | |

Capacitance | q | $\frac{{q}^{2}}{2C}$ | $\frac{dh}{dq}$ | $\frac{dq}{dt}$ | |

Dissipative | Diagram | $\mathit{w}$ | Dissipated Power ${D}(\mathit{w})=\mathit{w}.\mathit{z}(\mathit{w})$ | Voltage $\mathit{v}$ | Current $\mathit{i}$ |

Resistance | i | $R.{i}^{2}$ | $z\left(w\right)$ | w | |

Conductance | v | ${v}^{2}/R$ | w | $z\left(w\right)$ | |

PN Diode | v | $v.{I}_{S}\left(exp\left(\frac{v}{\mu {v}_{0}}\right)-1\right)+{v}^{2}.{G}_{min}$ | w | $z\left(w\right)$ | |

3-Ports | |||||

Dissipative | Diagram | $\mathbf{w}$ | $\mathbf{z}(\mathbf{w})$ | ||

NPN Transistor | $\left(\begin{array}{c}{v}_{BC}\\ {v}_{BE}\end{array}\right)$ | $\left(\begin{array}{c}{i}_{BC}\\ {i}_{BE}\end{array}\right)=\left(\begin{array}{cc}{\alpha}_{R}& -1\\ -1& {\alpha}_{F}\end{array}\right).\left(\begin{array}{c}{I}_{S}\left({e}^{{v}_{BC}/{v}_{t}}-1\right)+{v}_{BC}.{G}_{min}\\ {I}_{S}\left({e}^{{v}_{BE}/{v}_{t}}-1\right)+{v}_{BE}.{G}_{min}\end{array}\right)$ | |||

Potentiometer | $\left(\begin{array}{c}{v}_{p1}\\ {i}_{p2}\end{array}\right)$ | $\left(\begin{array}{c}{i}_{p1}\\ {v}_{p2}\end{array}\right)=\left(\begin{array}{c}{v}_{p1}/(1+\alpha .{R}_{p})\\ {i}_{p2}.(1+(1-\alpha ).{R}_{p})\end{array}\right)$ |

#### Appendix B.1. Storage Components

#### Appendix B.2. Linear Dissipative Components

#### Appendix B.3. Nonlinear Dissipative Components

#### Appendix B.4. Incidence Matrices Γ

## Appendix C. Discrete Gradient for Multi-Variate Hamiltonian

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**Figure 2.**Schematic and corresponding graph of a simple bipolar-junction transistor (BJT) amplifier with feedback. The grey part corresponds to the components, and the outer elements correspond to the external ports, or sources (as in Table 1).

**Figure 3.**Definitions and orientations for a single current-controlled edge b from node i to node j, with nodes potentials ${e}_{i}$ and ${e}_{j}$, respectively. The knowledge of the potential ${e}_{j}$ is transferred to node i with ${e}_{i}={v}_{b}+{e}_{j}$.

**Figure 4.**Simulation results and comparison of the methods in Table 3, for a nonlinear conservative system $\frac{\mathtt{d}\mathbf{x}}{\mathtt{d}t}={\mathbf{J}}_{\mathbf{x}}\xb7\nabla \mathcal{H}\left(\mathbf{x}\right)$ with $\mathcal{H}\left(\mathbf{x}\right)$ given in (30): (

**a**) Trapezoidal rule; (

**b**) Midpoint rule; (

**c**) PHS combined with discrete gradient; (

**d**) Relative error on energy balance.The comparison measure is the relative error on the power balance defined by $\u03f5\left(k\right)=\frac{\left|\mathcal{H}\left(\mathbf{x}(k+1)\right)-\mathcal{H}\left(\mathbf{x}\left(k\right)\right)\right|}{\mathcal{H}\left(\mathbf{x}\left(0\right)\right)}$. We see from Figure 4d that the error associated with the proposed method (PHS approach combined with the discrete gradient method) is low compared to the two other methods (with machine precision ≃10${}^{-16}$). The accumulation of these errors is responsible for the apparently unstable behavior of the trapezoidal rule.

**Figure 5.**Simulation (Figure 5b) of a dissipative diode clipper (Figure 5a) at the sample rate ${f}_{s}=96$ kHz, with three Newton–Raphson iterations, for a 10 ms sinusoidal excitation at 1 kHz with linearly increasing amplitude between 0 V and 2 V. (

**a**) Diode clipper schematic; (

**b**) Simulation of the diode clipper of Figure 5a.

**Figure 6.**Simulation (Figure 6b) of the common-emitter bipolar-junction transistor (BJT) amplifier with feedback (Figure 6a) at the sample rate ${f}_{s}=384$ kHz, with 10 Newton–Raphson iterations, for a 10 ms sinusoidal excitation at 1 kHz with linearly increasing amplitude between 0 V and $0.2$ V. (

**a**) Schematic of a common-emitter BJT amplifier with feedback; (

**b**) Simulation of the BJT amplifier in Figure 6a.

**Figure 7.**Results for the common-emitter BJT amplifier with feedback (Figure 6a). The input voltage signal is a 4 s exponential chirp on the audio range (20 Hz–20 kHz) with amplitude $0.05$ V (logarithmic frequency scale). Simulation starts at 0.3 s (after the switching transient). (

**a**) Spectrogram of output ${v}_{\mathrm{OUT}}$ obtained with the proposed method; (

**b**) Spectrogram of output ${v}_{\mathrm{OUT}}$ obtained with LT-Spice.

**Figure 8.**Simulations of the Cry-Baby’s circuit of Figure 9, for the potentiometer parameter $\alpha =0$ (

**a**) and $\alpha =1$ (

**b**) in the frequency domain, compared with LT-Spice simulations on the audio range 20 Hz–20 kHz.

**Figure 9.**Schematic of the Cry-Baby wah pedal. Note the IN/OUT terminals and the 9 V supply. The potentiometer $\mathtt{P}$ controls the effect.

Line | Label | Node List | Type | Parameters |
---|---|---|---|---|

${\ell}_{1}$ | $C1$ | ${N}_{1},{N}_{2}$ | CapaLin | $20{e}^{-9}$ |

${\ell}_{2}$ | $R1$ | ${N}_{3},{N}_{4}$ | Resistor | $1.5{e}^{3}$ |

${\ell}_{3}$ | $Q1$ | ${N}_{2},{N}_{3},{N}_{5}$ | NPN_ Type1 | List of parameters |

${\ell}_{4}$ | Vcc | ${N}_{4}$ | Voltage | 9 |

${\ell}_{5}$ | IN | ${N}_{1}$ | Voltage | ∼ |

${\ell}_{6}$ | OUT | ${N}_{3}$ | Current | 0 |

${\ell}_{7}$ | GRD | ${N}_{5}$ | Voltage | 0 |

Component type | Current-Controlled | Voltage-Controlled |
---|---|---|

${[\mathbf{a}]}_{\mathit{b}}={\mathit{i}}_{\mathit{b}}$ | ${[\mathbf{a}]}_{\mathit{b}}={\mathit{v}}_{\mathit{b}}$ | |

${[\mathbf{b}]}_{\mathit{b}}={\mathit{v}}_{\mathit{b}}$ | ${[\mathbf{b}]}_{\mathit{b}}={\mathit{i}}_{\mathit{b}}$ | |

storages | capacitor | inductor |

resistors | resistance | conductance |

nonlinear | diodes, transistors | |

sources | voltage source | current source |

Method | Update |
---|---|

Trapezoidal rule | $\mathbf{x}(k+1)=\mathbf{x}\left(k\right)+T\xb7{\mathbf{J}}_{\mathbf{x}}\xb7\frac{\nabla \mathcal{H}\left(\mathbf{x}\left(k\right)\right)+\nabla \mathcal{H}\left(\mathbf{x}(k+1)\right)}{2}$ |

Midpoint rule | $\mathbf{x}(k+1)=\mathbf{x}\left(k\right)+T\xb7{\mathbf{J}}_{\mathbf{x}}\xb7\nabla \mathcal{H}\left(\frac{\mathbf{x}\left(k\right)+\mathbf{x}(k+1)}{2}\right)$ |

PHS with discrete gradient | $\mathbf{x}(k+1)=\mathbf{x}\left(k\right)+T\xb7{\mathbf{J}}_{\mathbf{x}}\xb7{\nabla}_{d}\mathcal{H}\left(\mathbf{x}\left(k\right),\mathbf{x}(k+1)\right)$ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Falaize, A.; Hélie, T. Passive Guaranteed Simulation of Analog Audio Circuits: A Port-Hamiltonian Approach. *Appl. Sci.* **2016**, *6*, 273.
https://doi.org/10.3390/app6100273

**AMA Style**

Falaize A, Hélie T. Passive Guaranteed Simulation of Analog Audio Circuits: A Port-Hamiltonian Approach. *Applied Sciences*. 2016; 6(10):273.
https://doi.org/10.3390/app6100273

**Chicago/Turabian Style**

Falaize, Antoine, and Thomas Hélie. 2016. "Passive Guaranteed Simulation of Analog Audio Circuits: A Port-Hamiltonian Approach" *Applied Sciences* 6, no. 10: 273.
https://doi.org/10.3390/app6100273