# Modeling non linearities in the frequency domain

Olivier Maurice, Naomi De Mejanes, Philippe Meton

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Olivier Maurice, Naomi De Mejanes, Philippe Meton. Modeling non linearities in the frequency domain. 2024. hal-04527374

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# Modeling non linearities in the frequency domain

Olivier Maurice, Naomi De Mejanes, Philippe Meton

► To cite this version:

Olivier Maurice, Naomi De Mejanes, Philippe Meton. Modeling non linearities in the frequency domain. 2024. hal-04527374

HAL Id: hal-04527374

https://hal.science/hal-04527374v1

Preprint submitted on 30 Mar 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Modeling non linearities in the frequency domain

Olivier Maurice, Philippe Meton, Naomi de Mejanes*

March 26, 2024

## Abstract

The frequency domain remains the fundamental domain for physics, but non linear behaviors constitute a difficulty for solving circuits in this single domain. Nevertheless, it's possible to benefit of both domain advantages under some assumptions. We present in this article conditions leading to this possibility.

We consider an approach that we may call criterion-based steering for time changing impedances. In the criterion-based steering, an invariant which is the root mean square value, eventually weigthed, leads a non linear component impedance behavior.

## 1 Introduction

The frequency domain gives an easier way for computing complex behaviors like the skin effect, losses in lines, etc. But it supposes usually that the electronics circuit is invariant in time. In a first approach, intrinsically, the frequency domain do not take into account the time axis. By the fact it cannot model things that change in time. The purpose of this article is to submit a technique for benefiting of the advantages of both domains. It implies to accept a frequency domain setted by time states and this is intrinsic to Laplace's functions. Then after having defined how an impedance can be parametrized on domains, we use this model for applying our approach on various cases.

## 2 Laplace's function as an exponential order function

A function defined by pieces can be written under Laplace's formalism[1]. The condition for these functions for being writable under Laplace's formalism is that considering one piece $f(t)$, $\exists \epsilon$ so that $f(t)\exp(-\alpha t) < \epsilon$. As $f(t)$ are physical function with finite initial and final conditions, they respect the exponential order function assumption.

<footnote>*AFSCET, afscet.asso.fr, France. contact: olivier.maurice@protonmail.com</footnote>
# Modeling non linearities in the frequency domain

Olivier Maurice, Philippe Meton, Naomi de Mejanes*

March 26, 2024

## Abstract

The frequency domain remains the fundamental domain for physics, but non linear behaviors constitute a difficulty for solving circuits in this single domain. Nevertheless, it's possible to benefit of both domain advantages under some assumptions. We present in this article conditions leading to this possibility.

We consider an approach that we may call criterion-based steering for time changing impedances. In the criterion-based steering, an invariant which is the root mean square value, eventually weigthed, leads a non linear component impedance behavior.

## 1 Introduction

The frequency domain gives an easier way for computing complex behaviors like the skin effect, losses in lines, etc. But it supposes usually that the electronics circuit is invariant in time. In a first approach, intrinsically, the frequency domain do not take into account the time axis. By the fact it cannot model things that change in time. The purpose of this article is to submit a technique for benefiting of the advantages of both domains. It implies to accept a frequency domain setted by time states and this is intrinsic to Laplace's functions. Then after having defined how an impedance can be parametrized on domains, we use this model for applying our approach on various cases.

## 2 Laplace's function as an exponential order function

A function defined by pieces can be written under Laplace's formalism[1]. The condition for these functions for being writable under Laplace's formalism is that considering one piece $f(t)$, $\exists \epsilon$ so that $f(t)\exp(-\alpha t) < \epsilon$. As $f(t)$ are physical function with finite initial and final conditions, they respect the exponential order function assumption.

<footnote>*AFSCET, afscet.asso.fr, France. contact: olivier.maurice@protonmail.com</footnote>
### 3 Equivalent model for a non linear component

We put the assumption that the non linear (NL) electronic component can be modeled using parametrized impedances. The impedances values can be drastically changed playing on some parameter attached with the impedance definition[3].

A well known non linear component is a diode. Its impedance changes drastically depending on its polarisation. For a positive one, the diode behaves like a short-circuit, and for a negative one, it behaves like a open-circuit (this simple model is a good representation of a perfect diode. Obviously, real diodes are more complex in their responses).

This kind of model can be always used in both cases of filters bank or criterion-based steering. The impedance is defined on domains activated by parameters when they belong to an interval attached with the domain. For example, let's define a domain $\tilde{D}_{[p_1...p_2]}$.

p is the parameter that drives the domain value. If $p \in [p_1...p_2]$ then $\tilde{D}_{[p_1...p_2]} = 1$. In other cases, $\tilde{D}_{[p_1...p_2]} = 0$. $\tilde{D}_{[p_1...p_2]}$ can be obtained using sigmoid function like:

$$ \tilde{D}_{[p_1...p_2]} = [1 + \exp(-\alpha(p - p_1))]^{-1} - [1 + \exp(-\alpha(p - p_2))]^{-1} \quad (1) $$

Such domain function drive an impedance value[4]. The same impedance can be so defined by various domain functions, each of them giving the impedance formulation depending on one or more parameters. We write for an impedance $z$:

$$ z = \tilde{D}_{[p_1...p_2]}z_1 + \tilde{D}_{[p_2...p_3]}z_2 + \dots \quad (2) $$

More complex dependencies can be elaborated for components using products of domain functions, translating a non linear behavior projected on two or more parameters. For example, a component can depends (it's often the case) of both current amplitude and temperature. For an exceeding constraint in current, the component can burn its temperature going over a threshold value. In this case, its impedance goes from an original value to a open-circuit.

## 4 Criterion-based steering approach

### 4.1 Introduction

In this approach the main idea is to be able to create a link between the criterion that drives the non linear impedance behavior and the root mean square value of the constraint presented across the non linear impedance. In a first step we compute an observable value across a high impedance replacing the non linear impedance (NLI). Then this observable is compared with the criterion to drive the impedance function through the domains.
### 3 Equivalent model for a non linear component

We put the assumption that the non linear (NL) electronic component can be modeled using parametrized impedances. The impedances values can be drastically changed playing on some parameter attached with the impedance definition[3].

A well known non linear component is a diode. Its impedance changes drastically depending on its polarisation. For a positive one, the diode behaves like a short-circuit, and for a negative one, it behaves like a open-circuit (this simple model is a good representation of a perfect diode. Obviously, real diodes are more complex in their responses).

This kind of model can be always used in both cases of filters bank or criterion-based steering. The impedance is defined on domains activated by parameters when they belong to an interval attached with the domain. For example, let's define a domain $\mathcal{D}_{[p_1...p_2]}^p$.

$p$ is the parameter that drives the domain value. If $p \in [p_1...p_2]$ then $\mathcal{D}_{[p_1...p_2]}^p = 1$. In other cases, $\mathcal{D}_{[p_1...p_2]}^p = 0$. $\mathcal{D}_{[p_1...p_2]}^p$ can be obtained using sigmoid function like:

$$ \mathcal{D}_{[p_1...p_2]}^p = [1 + \exp(-\alpha(p - p_1))]^{-1} - [1 + \exp(-\alpha(p - p_2))]^{-1} \quad (1) $$

Such domain function drive an impedance value[4]. The same impedance can be so defined by various domain functions, each of them giving the impedance formulation depending on one or more parameters. We write for an impedance $z$:

$$ z = \mathcal{D}_{[p_1...p_2]}^p z_1 + \mathcal{D}_{[p_2...p_3]}^p z_2 + \dots \quad (2) $$

More complex dependencies can be elaborated for components using products of domain functions, translating a non linear behavior projected on two or more parameters. For example, a component can depends (it's often the case) of both current amplitude and temperature. For an exceeding constraint in current, the component can burn its temperature going over a threshold value. In this case, its impedance goes from an original value to a open-circuit.

## 4 Criterion-based steering approach

### 4.1 Introduction

In this approach the main idea is to be able to create a link between the criterion that drives the non linear impedance behavior and the root mean square value of the constraint presented across the non linear impedance. In a first step we compute an observable value across a high impedance replacing the non linear impedance (NLI). Then this observable is compared with the criterion to drive the impedance function through the domains.
## 4.2 Illustration

For illustrating this approach we consider a simple diode power supplied by an impulse signal $x(t)$. The first work consists in determining the Laplace's transform $X(k)$ of $x(n)$, the discrete signal deduced from $x(t)$. In a second step we replace the diode by a high resistance value for determining the incident voltage created across the diode. The voltage obtained $V_d$ is the parameter value $p = V_d$ that determines the diode impedance. In a last step, we compute the voltage and current with the diode, replacing it by the equivalent impedance determined previously and fixed by the domains through the parameter $p$. For making as evidence the non linear behavior we use a source impulse made of two part of two different levels. The first part is of low level : 0.3 volt, while the second part has an amplitude of 3 V. This signal is applied to the diode through a self impedance of the generator of 50 ohms. As long as the voltage across the diode is inferior to 0.6 V, the diode remains blocked. We create two circuits, one with an impedance that remains to be defined and one with the diode.

We define the signal $x(t)$ through its discrete function:

$$x[n] = 0,3 \left[ (1 + e^{-\alpha(n \cdot dt - 50 \mu s)})^{-1} - (1 + e^{-\alpha(n \cdot dt - 250 \mu s)})^{-1} \right] + 2,7 \left[ (1 + e^{-\alpha(n \cdot dt - 250 \mu s)})^{-1} - (1 + e^{-\alpha(n \cdot dt - 500 \mu s)})^{-1} \right] \quad (3)$$

The signal $x[n]$ is drawn figure 1.

The Laplace's discrete transform of this signal is given by (figure 2):

$$X[k] = \frac{1}{N} \sum_{0}^{N-1} x[n]e^{-j2\pi k \frac{n}{N}} \quad (4)$$

We can easily verify that the inverse Laplace's transform of $X[k]$ gives $x[n]$ with an error of $2.2 \cdot 10^{-10}$ (this is obtained just making the difference between the original temporal signal and the one coming from the inverse Laplace's transform.

But $x[n]$ can be cutted out in two signals:

$$x[n] = x_1[n] + x_2[n] \quad (5)$$

By the fact, its Laplace's transform can also be separated in two parts:

$$L[x(n)] = X_1[k] + X_2[k] \quad (6)$$

## 4.3 Methodology

The non linear circuit can be studied in two steps, one step per Laplace's transform. Computing both Laplace's transform of $x_1$ and $x_2$ on the same temporal domain gives the way for controling the root mean square value of each signal.
## 4.2 Illustration

For illustrating this approach we consider a simple diode power supplied by an impulse signal $x(t)$. The first work consists in determining the Laplace's transform $X(k)$ of $x(n)$, the discrete signal deduced from $x(t)$. In a second step we replace the diode by a high resistance value for determining the incident voltage created across the diode. The voltage obtained $V_d$ is the parameter value $p = V_d$ that determines the diode impedance. In a last step, we compute the voltage and current with the diode, replacing it by the equivalent impedance determined previously and fixed by the domains through the parameter $p$. For making as evidence the non linear behavior we use a source impulse made of two part of two different levels. The first part is of low level : 0.3 volt, while the second part has an amplitude of 3 V. This signal is applied to the diode through a self impedance of the generator of 50 ohms. As long as the voltage across the diode is inferior to 0.6 V, the diode remains blocked. We create two circuits, one with an impedance that remains to be defined and one with the diode.

We define the signal $x(t)$ through its discrete function:

$$x[n] = 0,3 \left[ (1 + e^{-\alpha(n \cdot dt - 50 \mu s)})^{-1} - (1 + e^{-\alpha(n \cdot dt - 250 \mu s)})^{-1} \right] + 2,7 \left[ (1 + e^{-\alpha(n \cdot dt - 250 \mu s)})^{-1} - (1 + e^{-\alpha(n \cdot dt - 500 \mu s)})^{-1} \right] \quad (3)$$

The signal $x[n]$ is drawn figure 1.

The Laplace's discrete transform of this signal is given by (figure 2):

$$X[k] = \frac{1}{N} \sum_{0}^{N-1} x[n]e^{-j2\pi k \frac{n}{N}} \quad (4)$$

We can easily verify that the inverse Laplace's transform of $X[k]$ gives $x[n]$ with an error of $2.2 \cdot 10^{-10}$ (this is obtained just making the difference between the original temporal signal and the one coming from the inverse Laplace's transform.

But $x[n]$ can be cutted out in two signals:

$$x[n] = x_1[n] + x_2[n] \quad (5)$$

By the fact, its Laplace's transform can also be separated in two parts:

$$L[x(n)] = X_1[k] + X_2[k] \quad (6)$$

## 4.3 Methodology

The non linear circuit can be studied in two steps, one step per Laplace's transform. Computing both Laplace's transform of $x_1$ and $x_2$ on the same temporal domain gives the way for controling the root mean square value of each signal.
<figure><img src="image_2.png" /><figcaption>Figure 1: Source signal</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Laplace transform module</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 1: Source signal</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2: Laplace transform module</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: Time signal separated in two parts</figcaption></figure>

So if we can make a link with the NLI state and the root mean square (rms) value of each signal, it becomes possible to study the non linear circuit under the Laplace's domain.

For recall, the rms value is obtained making:

$$rms = \sqrt{\frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2} = \sqrt{\sum_{k=0}^{N-1} |X[k]|^2} \quad (7)$$

Now we make the same exercise cutting $x[n]$ in two parts:

$$x_1[n] = 0,3 \left[ (1 + e^{-\alpha(n \cdot dt - 50 \mu s)})^{-1} - (1 + e^{-\alpha(n \cdot dt - 250 \mu s)})^{-1} \right] \quad (8)$$

$$x_2[n] = 2,7 \left[ (1 + e^{-\alpha(n \cdot dt - 250 \mu s)})^{-1} - (1 + e^{-\alpha(n \cdot dt - 500 \mu s)})^{-1} \right]$$

After what we can compute $X_1[k]$ and $X_2[k]$ with the same performances as before. This leads to the curves figure 3 and 4. Inverse Laplace's transform gives deviations less than $3 \cdot 10^{-10}$ and still less at the rms level.

$(X_h$ is the spectrum for the diode blocked while $X_l$ is the spectrum for the diode activated).

## 4.4 Diode case

Studying the response of a simple diode circuit is made in two steps:
<figure><img src="image_5.png" /><figcaption>Figure 3: Time signal separated in two parts</figcaption></figure>

So if we can make a link with the NLI state and the root mean square (rms) value of each signal, it becomes possible to study the non linear circuit under the Laplace's domain.

For recall, the rms value is obtained making:

$$rms = \sqrt{\frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2} = \sqrt{\sum_{k=0}^{N-1} |X[k]|^2} \quad (7)$$

Now we make the same exercise cutting $x[n]$ in two parts:

$$x_1[n] = 0,3 \left[ (1 + e^{-\alpha(n \cdot dt - 50 \mu s)})^{-1} - (1 + e^{-\alpha(n \cdot dt - 250 \mu s)})^{-1} \right] \quad (8)$$

$$x_2[n] = 2,7 \left[ (1 + e^{-\alpha(n \cdot dt - 250 \mu s)})^{-1} - (1 + e^{-\alpha(n \cdot dt - 500 \mu s)})^{-1} \right]$$

After what we can compute $X_1[k]$ and $X_2[k]$ with the same performances as before. This leads to the curves figure 3 and 4. Inverse Laplace's transform gives deviations less than $3 \cdot 10^{-10}$ and still less at the rms level.

$(X_h$ is the spectrum for the diode blocked while $X_l$ is the spectrum for the diode activated).

## 4.4 Diode case

Studying the response of a simple diode circuit is made in two steps:
<figure><img src="image_5.png" /><figcaption>Figure 4: Spectre signal separated in two parts</figcaption></figure>

1. computing the open voltage across all passive components of the diode equivalent circuit;

2. replacing the diode by its NLI.

For making a link between the signal rms value and its peak one, we use a form factor $F$ defined by (remember that the rms value is the only common observable for both domains, frequency and time):

$$F = \frac{T}{D} \qquad (9)$$

where $T$ is the temporal domain duration and $D$ the signal duration. We understand here the interest to keep the same period for both parts of the original signal: it gives a direct correspondence between the time step $T/N$ and the frequency $k/T$. We define the diode impedance by:

$$z_D = \frac{\frac{u}{D_{<1}}R_{\infty} + \frac{u}{D_{>1}}R_0}{1 + kC \left( \frac{u}{D_{<1}}R_{\infty} + \frac{u}{D_{>1}}R_0 \right)} \qquad (10)$$

Where 1 is the diode threshold (that's why the domains are separated by this value allocated to $u$). This impedance results from a non linear resistance in parallel with a capacitance $C$. $R_{\infty}$ is the high resistance when the diode is blocked and $R_0$ the low resistance when the diode is passer-by. The non
<figure><img src="image_6.png" /><figcaption>Figure 4: Spectre signal separated in two parts</figcaption></figure>

1. computing the open voltage across all passive components of the diode equivalent circuit;

2. replacing the diode by its NLI.

For making a link between the signal rms value and its peak one, we use a form factor $\mathcal{F}$ defined by (remember that the rms value is the only common observable for both domains, frequency and time):

$$ \mathcal{F} = \frac{T}{D} \qquad (9) $$

where $T$ is the temporal domain duration and $D$ the signal duration. We understand here the interest to keep the same period for both parts of the original signal: it gives a direct correspondence between the time step $T/N$ and the frequency $k/T$. We define the diode impedance by:

$$ z_D = \frac{\frac{u}{D_{<1}} R_{\infty} + \frac{u}{D_{>1}} R_0}{1 + kC \left( \frac{u}{D_{<1}} R_{\infty} + \frac{u}{D_{>1}} R_0 \right)} \qquad (10) $$

Where 1 is the diode threshold (that's why the domains are separated by this value allocated to $u$). This impedance results from a non linear resistance in parallel with a capacitance $C$. $R_{\infty}$ is the high resistance when the diode is blocked and $R_0$ the low resistance when the diode is passer-by. The non
linear circuit story is described in two states: a first state corresponding to the $x_1$ excitation, and another state corresponding to the $x_2$ excitation. The computation follows the sequence of these two states story.

Once the NLI defined, we compute what may be called a polarisation circuit where the NLI are replaced by high impedance in order to determine the open circuit voltage ($V_{CO}$) values across them. These $V_{CO}$ values are the criteria setting the NLI values through their parametrized domains. In the software developed for computing the circuit the NLI is simply represented by the function:

$$NLI(V_{CO}) = \left\{ 1 - \left[ 1 + \exp(-\alpha(V_{CO} - V_t)) \right]^{-1} \right\} R_{\infty} \quad (11)$$

The threshold $V_t$ is the value for which the NLI changes of state. The $V_{CO}$ voltage is obtained using:

$$V_{CO} = \sqrt{\mathcal{F} \sum_{k=0}^{N-1} |X[k]|^2} \quad (12)$$

After the first computation where $V_{CO}$ is computed, a second computation takes into account the complete NL impedance with (11) and all the circuits currents are known for each state, one for $x_1$ and the other for $x_2$. It's important to understand that in the first run, we consider all the NL passive impedances components, including the high frequency capacitance of the NL that comes in parallel with the impedance $R_{\infty}$ in the blocked state.

For the polarisation step and the first state, the impedance is $\zeta_{01} = R_{50} + R_t/(1+kR_tC)$, $R_{50}$ being the source self impedance, $R_t$ the test impedance for determining the $V_{CO}$ (a quite high resistance value) and $C$ the diode capacitance. The $V_{CO}$ is obtained by calculating:

$$V_{CO} = R_t \left( \frac{X_1[k]}{\zeta_{01}} \right) \quad (13)$$

For the second state this becomes:

$$V_{CO} = R_t \left( \frac{X_2[k]}{\zeta_{01}} \right) \quad (14)$$

In both cases we calculate the corrected rms value using (12). Then 11 gives us the NLI expression and allow us to compute the NL circuit in a second step, and this for the two possible states. Writing this computation we obtain in our case the curves given figure 5. For the first state, a rms value of 0.53 volt is measured across the NLI, and for the second state where it behaves nearly a short circuit, a rms value of $2.4 \ 10^{-11}$ is computed.

All the method declined on a simple diode example can be applied to any complex circuit. Each state is defined on the temporal base and has its own dating. This is very important: it says that the results obtained in the Laplace's domain follow a defined sequence fixed by the temporal base. In the case where many NL belong to the same circuit, their changes of states must be studied respecting the causality fixed by the sequence in which they are addressed.
linear circuit story is described in two states: a first state corresponding to the $x_1$ excitation, and another state corresponding to the $x_2$ excitation. The computation follows the sequence of these two states story.

Once the NLI defined, we compute what may be called a polarisation circuit where the NLI are replaced by high impedance in order to determine the open circuit voltage ($V_{CO}$) values across them. These $V_{CO}$ values are the criteria setting the NLI values through their parametrized domains. In the software developed for computing the circuit the NLI is simply represented by the function:

$$NLI(V_{CO}) = \left\{ 1 - \left[ 1 + \exp(-\alpha(V_{CO} - V_t)) \right]^{-1} \right\} R_{\infty} \quad (11)$$

The threshold $V_t$ is the value for which the NLI changes of state. The $V_{CO}$ voltage is obtained using:

$$V_{CO} = \sqrt{\mathcal{F} \sum_{k=0}^{N-1} |X[k]|^2} \quad (12)$$

After the first computation where $V_{CO}$ is computed, a second computation takes into account the complete NL impedance with (11) and all the circuits currents are known for each state, one for $x_1$ and the other for $x_2$. It's important to understand that in the first run, we consider all the NL passive impedances components, including the high frequency capacitance of the NL that comes in parallel with the impedance $R_{\infty}$ in the blocked state.

For the polarisation step and the first state, the impedance is $\zeta_{01} = R_{50} + R_t/(1+kR_tC)$, $R_{50}$ being the source self impedance, $R_t$ the test impedance for determining the $V_{CO}$ (a quite high resistance value) and $C$ the diode capacitance. The $V_{CO}$ is obtained by calculating:

$$V_{CO} = R_t \left( \frac{X_1[k]}{\zeta_{01}} \right) \quad (13)$$

For the second state this becomes:

$$V_{CO} = R_t \left( \frac{X_2[k]}{\zeta_{01}} \right) \quad (14)$$

In both cases we calculate the corrected rms value using (12). Then 11 gives us the NLI expression and allow us to compute the NL circuit in a second step, and this for the two possible states. Writing this computation we obtain in our case the curves given figure 5. For the first state, a rms value of 0.53 volt is measured across the NLI, and for the second state where it behaves nearly a short circuit, a rms value of 2.4 $10^{-11}$ is computed.

All the method declined on a simple diode example can be applied to any complex circuit. Each state is defined on the temporal base and has its own dating. This is very important: it says that the results obtained in the Laplace's domain follow a defined sequence fixed by the temporal base. In the case where many NL belong to the same circuit, their changes of states must be studied respecting the causality fixed by the sequence in which they are addressed.
<figure><img src="image_6.png" /><figcaption>Figure 5: Measured voltage for the two states</figcaption></figure>

# 5 Generalization

Let's consider a temporal and discretized signal $x(n)$ defined on the temporal domain $N$. If we can decompose it approximatively in a sequence of rectangular impulse signal $x_σ(n)$ of amplitude $A_σ$ ($ndt = nT/N$):

$$x(n) = \sum_{\sigma} A_{\sigma} \pi_{\sigma}(n) \delta_{\sigma} \qquad (15)$$

Then, its spectrum is immediately obtained knowing the one of $A_σx_σ$:

$$X_{\sigma}(k) = \sum_{n} A_{\sigma} \pi_{\sigma}(n) \exp \left( -i2\pi \frac{k}{T} \tau_{\sigma} \right) \exp \left( -i2\pi k \frac{n}{N} \right) \quad (16)$$

We can compute the transfer function $H_{co}$ between the source and the non linear component in its absence, making a measurement *Obs* across the component edge:

$$H_{co}(k) = Obs(X_{\sigma}) \qquad (17)$$

The form factor of the elementary excitation $x_σ(n)$ is well known: $F = T/\delta t$. Then we can compute the "weighted rms" $w$ value at the same observation point:

$$w = \left( \sum_{k} H_{co}^{*}(k^{*}) F H_{co}(k) \right)^{1/2} \qquad (18)$$
<figure><img src="image_7.png" /><figcaption>Figure 5: Measured voltage for the two states</figcaption></figure>

# 5 Generalization

Let's consider a temporal and discretized signal $x(n)$ defined on the temporal domain $N$. If we can decompose it approximatively in a sequence of rectangular impulse signal $x_σ(n)$ of amplitude $A_σ$ ($ndt = nT/N$):

$$x(n) = \sum_{\sigma} A_{\sigma} \pi_{\sigma}(n) \delta_{\sigma} \qquad (15)$$

Then, its spectrum is immediately obtained knowing the one of $A_σx_σ$:

$$X_{\sigma}(k) = \sum_{n} A_{\sigma} \pi_{\sigma}(n) \exp \left( -i2\pi \frac{k}{T} \tau_{\sigma} \right) \exp \left( -i2\pi k \frac{n}{N} \right) \qquad (16)$$

We can compute the transfer function $H_{co}$ between the source and the non linear component in its absence, making a measurement *Obs* across the component edge:

$$H_{co}(k) = Obs(X_{\sigma}) \qquad (17)$$

The form factor of the elementary excitation $x_σ(n)$ is well known: $\mathcal{F} = T/\delta t$. Then we can compute the "weighted rms" $w$ value at the same observation point:

$$w = \left( \sum_{k} H_{co}^{*}(k^{*}) \mathcal{F} H_{co}(k) \right)^{1/2} \qquad (18)$$
Using this value allows to set the non linear component behavior through its domain definitions $\mathcal{D}_I$. Then the circuit response can be computed with the setted NL component at the same frequency. As $X_\sigma$ must be calculate only once, this process is simple to implement. The spectrum of $x(n)$ is completely defined through the $A_\sigma$. Once the circuit response obtained, we can go back at this moment in the time domain, computing the weighted rms value of Obs. The sequence of actions is:

1. decompose $x(n)$ in a summation of rectangular pulse of amplitudes $A_\sigma$;

2. compute the open-circuit voltage in response to $X_\sigma$ and its weighted rms value $w_\sigma$;

3. compute the NL circuit response using $w_\sigma$ in the non linear impedance defined on domains;

4. compute the weighted rms value of this response to evaluate the local value of the observable in the time domain.

Taking for the pulse duration the sampling period, the succession of rms values makes a curve $w(n)$ which rebuilds the observable temporal response. As an application we consider a double exponential curve for the temporal source figure 6. We first define the source function as:

$$x(t) = \exp\left(-\frac{t}{150.10^{-6}}\right) - \exp\left(-\frac{t}{10.10^{-6}}\right) \quad (19)$$

This temporal function is discretized with a time step $dt$. This is typically a lightning current function. We can compute on a another way the spectrum of a normalized pulse function having $\tau = ndt$ for duration, $X(k)$ with:

$$X(k) = \frac{T}{\tau k} [1 - \exp(-k\tau)] \quad (20)$$

Then in a temporal loop, we can create an instantaneous spectrum associated with each source time step value:

$$X^i = x(n) \cdot X(k) \quad (21)$$

At the same time event, the weighted rms value can be computed and compared with the corresponding source value. This gives the way for setting the form factor amplitude $\mathcal{F}$. The non linear circuit is then solved. In a first step we determine the non linear setting parameter $p$ as the observable $\Theta$ (it can be an open circuit voltage, a short circuit current, etc.). Having $p$ value, we make the frequency computation once again, with the non linear impedance determined by its domains and $p$ [2]. Having compute the same observable $\Theta$ with the complete circuit, we can compute also its weighted rms value $w(n)$ (equation 18) and trace the time response of this observable as the sequence of $w(t)$.
Using this value allows to set the non linear component behavior through its domain definitions $\mathcal{D}_I$. Then the circuit response can be computed with the seted NL component at the same frequency. As $X_\sigma$ must be calculate only once, this process is simple to implement. The spectrum of $x(n)$ is completely defined through the $A_\sigma$. Once the circuit response obtained, we can go back at this moment in the time domain, computing the weighted rms value of Obs. The sequence of actions is:

1. decompose $x(n)$ in a summation of rectangular pulse of amplitudes $A_\sigma$;

2. compute the open-circuit voltage in response to $X_\sigma$ and its weighted rms value $w_\sigma$;

3. compute the NL circuit response using $w_\sigma$ in the non linear impedance defined on domains;

4. compute the weighted rms value of this response to evaluate the local value of the observable in the time domain.

Taking for the pulse duration the sampling period, the succession of rms values makes a curve $w(n)$ which rebuilds the observable temporal response. As an application we consider a double exponential curve for the temporal source figure 6. We first define the source function as:

$$x(t) = \exp\left(-\frac{t}{150.10^{-6}}\right) - \exp\left(-\frac{t}{10.10^{-6}}\right) \quad (19)$$

This temporal function is discretized with a time step $dt$. This is typically a lightning current function. We can compute on a another way the spectrum of a normalized pulse function having $\tau = ndt$ for duration, $X(k)$ with:

$$X(k) = \frac{T}{\tau k} [1 - \exp(-k\tau)] \quad (20)$$

Then in a temporal loop, we can create an instantaneous spectrum associated with each source time step value:

$$X^i = x(n) \cdot X(k) \quad (21)$$

At the same time event, the weighted rms value can be computed and compared with the corresponding source value. This gives the way for setting the form factor amplitude $\mathcal{F}$. The non linear circuit is then solved. In a first step we determine the non linear setting parameter $p$ as the observable $\Theta$ (it can be an open circuit voltage, a short circuit current, etc.). Having $p$ value, we make the frequency computation once again, with the non linear impedance determined by its domains and $p$ [2]. Having compute the same observable $\Theta$ with the complete circuit, we can compute also its weighted rms value $w(n)$ (equation 18) and trace the time response of this observable as the sequence of $w(t)$.
<figure><img src="image_7.png" /><figcaption>Figure 6: Example developed for generalization</figcaption></figure>

All this process was applied figure 6. The source function is a double exponential one (curve "x"). We verified that the sequence of rms values gives the same curve ("rmst"). Then applying the previous process we obtain the normalized pulse spectrum (curve "pulse spectrum" on the right). The circuit was a simple resistance in series with a clipper diode setted at 0,4 volt. The sequence of rms values obtained accross the clipper gives the curve "rest". In this example there are no memory effect and no final or initial conditions that should be reported.

Another case were conducted replacing the diode by a simple capacitance.
The result is shown figure 7, the capacitance making a low frequency pass filter.
In that case, the past charging value reached accross the capacitor at each time
step was taken into account and reported on the next time step.

6 Interpretation

Various criteria can be used to set the NLI states, here we use a pseudo-peak
amplitude defined by rms(*F*) (the *F* factor intervenes under the root). But
other criterion may be defined; the objective being to be able to set the NLI
values, based on these criteria. The approach is similar to a piano player. At
each rhythm time, he plays a chord more or less strongly. His action can be seen
as a Dirac's attack and the piano may be not a succession of piano keys, but an
instantaneous filter that determines each time the chord played. This filter is
the non linear component and player's force of attack is first characterized on a
linear filter.

Mathematically, we consider an attack *e* rising up for example. A circuit response depends on *e* amplitude, i.e. it exists a set of impulse responses {H(|*e*|<sub>*i*</sub>)}
<figure><img src="image_8.png" /><figcaption>Figure 6: Example developed for generalization</figcaption></figure>

All this process was applied figure 6. The source function is a double exponential one (curve "x"). We verified that the sequence of rms values gives the same curve ("rmst"). Then applying the previous process we obtain the normalized pulse spectrum (curve "pulse spectrum" on the right). The circuit was a simple resistance in series with a clipper diode setted at 0,4 volt. The sequence of rms values obtained accross the clipper gives the curve "rest". In this example there are no memory effect and no final or initial conditions that should be reported.

Another case were conducted replacing the diode by a simple capacitance.
The result is shown figure 7, the capacitance making a low frequency pass filter.
In that case, the past charging value reached accross the capacitor at each time
step was taken into account and reported on the next time step.

6 Interpretation

Various criteria can be used to set the NLI states, here we use a pseudo-peak
amplitude defined by rms(*F*) (the *F* factor intervenes under the root). But
other criterion may be defined; the objective being to be able to set the NLI
values, based on these criteria. The approach is similar to a piano player. At
each rhythm time, he plays a chord more or less strongly. His action can be seen
as a Dirac's attack and the piano may be not a succession of piano keys, but an
instantaneous filter that determines each time the chord played. This filter is
the non linear component and player's force of attack is first characterized on a
linear filter.

Mathematically, we consider an attack *e* rising up for example. A circuit re-
sponse depends on *e* amplitude, i.e. it exists a set of impulse responses {H(|*e*|<sub>*i*</sub>)}
<figure><img src="image_8.png" /><figcaption>Figure 7: Example with a low frequency pass filter</figcaption></figure>

leading to the circuit response $s$. The instantaneous impulse reponse $H(|e_i|$) is chosen depending on the local $|e|$ amplitude and may change when $e$ is rising up in this example. Making the classical convolution product:

$$s(n) = e_0 H_n + e_1 H_{n-1} + \dots \qquad (22)$$

becomes:

$$s(n) = e_0 H_n(|e_0|) + e_1 H_{n-1}(|e_1|) + \dots \qquad (23)$$

If we can obtain in energy locally ($H_n(|e_q|) = H_\beta$):

$$h_{\alpha\beta} = e_{\alpha} H_{\beta} + h_{y} \qquad (24)$$

Where $h_y$ is a value coming from the past product in time. then:

$$s(\beta) \approx \sqrt{h_{\alpha\beta}^* \mathcal{F} h_{\alpha\beta}} \qquad (25)$$

That's what we have done.

# 7 Synthesis

The technique enriches the original spectrum. This increasing of spectrum is a typical signature of non linear electronics.

Through this technique we can benefit of both advantages of the frequency domain for its adequation in taking into account complex models and of the time domain for taking into account brutal change of states.
<figure><img src="image_9.png" /><figcaption>Figure 7: Example with a low frequency pass filter</figcaption></figure>

leading to the circuit response s. The instantaneous impulse reponse $H(|e_i|)$ is choosen depending on the local $|e|$ amplitude and may change when e is rising up in this example. Making the classical convolution product:

$$s(n) = e_0 H_n + e_1 H_{n-1} + \dots \qquad (22)$$

becomes:

$$s(n) = e_0 H_n(|e_0|) + e_1 H_{n-1}(|e_1|) + \dots \qquad (23)$$

If we can obtain in energy locally ($H_n(|e_q|) = H_\beta$):

$$h_{\alpha\beta} = e_{\alpha} H_{\beta} + h_{y} \qquad (24)$$

Where $h_y$ is a value coming from the past product in time. then:

$$s(\beta) \approx \sqrt{h_{\alpha\beta}^* \mathcal{F} h_{\alpha\beta}} \qquad (25)$$

That's what we have done.

# 7 Synthesis

The technique enriches the original spectrum. This increasing of spectrum is a typical signature of non linear electronics.

Through this technique we can benefit of both advantages of the frequency domain for its adequation in taking into account complex models and of the time domain for taking into account brutal change of states.
## 8 Going further

In [5] we submit to consider the 4-space:

$$ \mathbb{R}C : (\mathbb{C}, \mathbb{R}^3) \leftrightarrow (\lambda, x^u) \qquad (26) $$

Previous developments suggest that if we see the time as a set of appointments where measurements are done, the cone of light becomes drawn in a 3D projection quite difficult to imagine, where each time point is projected into Laplace's domain. A value of an electrical observable in space and time results from the weighted root mean square operation and comes in fact from $\mathbb{R}C$.

Electronics is perhaps the physic job that involves the most of physics. It implies quantum mechanics and chemical, electronics and electromagnetism and finally mechanics. Multiphysic problems can be quite easily solved when they concern local electronics system, while at the scale of the universe, it becomes very difficult. Perhaps we may begin by modelling theoretically what's happen locally before to take a look further.

When we think of our method for nonlinear components using a sequence of pulse function, the generalization has shown that the time behavior can be seen as the response of a succession of Laplace's responses measured through the local rms value. If we consider one pulse only, it interacts in a transfer function of a given circuit with various elements. If These interactions occurs in a short time compared to the pulse duration, all seems as if both elements in interaction was in the same referential. There are no propagation, no phase shift. If the distance is high enough so that the element response comes after the end of the pulse duration, then the interaction concerns two objects that do not belong to the same referential. Using clocks with a rhythm $T$, both cases can be evaluated looking if the interaction response comes before the end of the half period $T/2$. Then drawing a cone of light with the wavelength, it exists a zone of $\lambda/2 = cT/2$ in which past as future doesn't exist, the interaction being instantaneous. In another zone, propagation exists between two objects and in this case, rigorously, each actor cannot know what's append in the other referential. He can only perceived the other referential through some transformation laws (figure 8).
## 8 Going further

In [5] we submit to consider the 4-space:

$$ \mathbb{R}C : (\mathbb{C}, \mathbb{R}^3) \leftrightarrow (\lambda, x^u) \qquad (26) $$

Previous developments suggest that if we see the time as a set of appointments where measurements are done, the cone of light becomes drawn in a 3D projection quite difficult to imagine, where each time point is projected into Laplace's domain. A value of an electrical observable in space and time results from the weighted root mean square operation and comes in fact from $\mathbb{R}C$.

Electronics is perhaps the physic job that involves the most of physics. It implies quantum mechanics and chemical, electronics and electromagnetism and finally mechanics. Multiphysic problems can be quite easily solved when they concern local electronics system, while at the scale of the universe, it becomes very difficult. Perhaps we may begin by modelling theoretically what's happen locally before to take a look further.

When we think of our method for nonlinear components using a sequence of pulse function, the generalization has shown that the time behavior can be seen as the response of a succession of Laplace's responses measured through the local rms value. If we consider one pulse only, it interacts in a transfer function of a given circuit with various elements. If These interactions occurs in a short time compared to the pulse duration, all seems as if both elements in interaction was in the same referential. There are no propagation, no phase shift. If the distance is high enough so that the element response comes after the end of the pulse duration, then the interaction concerns two objects that do not belong to the same referential. Using clocks with a rhythm $T$, both cases can be evaluated looking if the interaction response comes before the end of the half period $T/2$. Then drawing a cone of light with the wavelength, it exists a zone of $\lambda/2 = cT/2$ in which past as future doesn't exist, the interaction being instantaneous. In another zone, propagation exists between two objects and in this case, rigorously, each actor cannot know what's append in the other referential. He can only perceived the other referential through some transformation laws (figure 8).
<figure><img src="image_9.png" /><figcaption>Figure 8: Another view of the light of cone</figcaption></figure>

References

[1] Murray R.Spiegel, "Schaum's outline of theory and problems of Laplace transforms. Mc-Graw Hill editor. 1965. ISBN : 0-07-060231-X ; 978-0-07-060231-1.

[2] Victor, J., & Shapley, R. (1980). A method of nonlinear analysis in the frequency domain. Biophysical Journal, 29(3), 459-483.

[3] Tymerski, R., Vorpérian, V., Lee, F. C., & Baumann, W. T. (1989). Nonlinear modeling of the PWM switch. IEEE Transactions on Power Electronics, 4(2), 225-233.

[4] Moreno P., Gomez P., Naredo J.L., & Guardado J.L., Frequency domain transient analysis of electrical networks including non-linear conditions. (2005) International Journal of Electrical Power & Energy Systems, 27(2) 139-146.

[5] Olivier Maurice. FROM R4 CURVILINEAR SPACE TO THE CYCLE SPACE. 2024. (hal-04448080)
<figure><img src="image_10.png" /><figcaption>Figure 8: Another view of the light of cone</figcaption></figure>

References

[1] Murray R.Spiegel, "Schaum's outline of theory and problems of Laplace transforms. Mc-Graw Hill editor. 1965. ISBN : 0-07-060231-X ; 978-0-07-060231-1.

[2] Victor, J., & Shapley, R. (1980). A method of nonlinear analysis in the frequency domain. Biophysical Journal, 29(3), 459-483.

[3] Tymerski, R., Vorpérian, V., Lee, F. C., & Baumann, W. T. (1989). Nonlinear modeling of the PWM switch. IEEE Transactions on Power Electronics, 4(2), 225-233.

[4] Moreno P., Gomez P., Naredo J.L., & Guardado J.L., Frequency domain transient analysis of electrical networks including non-linear conditions. (2005) International Journal of Electrical Power & Energy Systems, 27(2) 139-146.

[5] Olivier Maurice. FROM R4 CURVILINEAR SPACE TO THE CYCLE SPACE. 2024. (hal-04448080)