# Why Should we Choose between Time and Frequency?

Olivier Maurice

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Olivier Maurice. Why Should we Choose between Time and Frequency?. 2024. hal-04545030

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Submitted on 13 Apr 2024

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<figure><img src="image_1.png" /></figure>
# Why Should we Choose between Time and Frequency?

Olivier Maurice

► To cite this version:

Olivier Maurice. Why Should we Choose between Time and Frequency?. 2024. hal-04545030

HAL Id: hal-04545030

https://hal.science/hal-04545030v1

Submitted on 13 Apr 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_1.png" /></figure>
# Why Should we Choose between Time and Frequency?

Olivier MAURICE*

18/03/2016

## Résumé

Classically, going from the time domain to the frequency one goes through Fourier's transforms. Once a computation is made under the frequency domain, inverse Fourier transform is used for going back to the time domain. But these kinds of operations give good results for linear behaviors and for problems where phase evolving remains sufficiently slow. The direct link between both domains is Parseval's relation. With benefits of Laplace's transform which can use functions defined on finite domains and Parseval's relation applied to local time functions, we can work simultaneously in both domains and solve non linear problems using the frequency representation in various time subdomains. We give an illustration of this technique.

## 1 Introduction

Laplace's transform adds to Fourier's transform the causality and exponential order. These two added values are very important for making robust bridges between both domains of frequency and time. We first recall what means for a function to be of exponential order. Then We recall the initial and final conditions applied in electrical circuits. Particular time domain functions are fast rectangular pulses or Dirac's quasi-pulses. We well know the spectrum and frequency characteristics of these functions. We can also easily compute the Parseval's root mean square criterion in potential to quantified these functions in both time and frequency domains. In case of linear transfer functions, the response to a sequence of quasi-pulses becomes the summation of response to each pulse in a time-driven

<footnote>*AFSCET. olivier.maurice@protonmail.com</footnote>
# Why Should we Choose between Time and Frequency?

Olivier MAURICE*

18/03/2016

## Résumé

Classically, going from the time domain to the frequency one goes through Fourier's transforms. Once a computation is made under the frequency domain, inverse Fourier transform is used for going back to the time domain. But these kinds of operations give good results for linear behaviors and for problems where phase evolving remains sufficiently slow. The direct link between both domains is Parseval's relation. With benefits of Laplace's transform which can use functions defined on finite domains and Parseval's relation applied to local time functions, we can work simultaneously in both domains and solve non linear problems using the frequency representation in various time subdomains. We give an illustration of this technique.

## 1 Introduction

Laplace's transform adds to Fourier's transform the causality and exponential order. These two added values are very important for making robust bridges between both domains of frequency and time. We first recall what means for a function to be of exponential order. Then We recall the initial and final conditions applied in electrical circuits. Particular time domain functions are fast rectangular pulses or Dirac's quasi-pulses. We well know the spectrum and frequency characteristics of these functions. We can also easily compute the Parseval's root mean square criterion in potential to quantified these functions in both time and frequency domains. In case of linear transfer functions, the response to a sequence of quasi-pulses becomes the summation of response to each pulse in a time-driven

<footnote>*AFSCET. olivier.maurice@protonmail.com</footnote>
loop. By using the same approach but where some functions are setted by a previous analysis, we can studied non linear circuits. These studies are leaded under the formalism of tensor analysis of networks.

## 2 Exponential order functions

A time dependent functions $x(t)$ can be separate in a sequence of subdomains. Each subdomain $x_i(t)$ starts at $t = t_i$ and ends at $t = t_j$. On each subdomain, if $\exists (M > 0, \gamma) \in \mathbb{R}^2$ so that, $\forall t > N$:

$$|e^{-\gamma t}F(t)| < M \qquad (1)$$

Then, under these assumptions, a function $exp(-\alpha t)$ can frame $x_i(t)$ and $F(t)$ is an exponential order function $\gamma$ when $t \to \infty$. Calculating locally Fourier's transform :

$$\int_t dt (x_i(t)e^{-\alpha t}) e^{-i\omega t} \qquad (2)$$

Leads to compute Laplace's transform :

$$\int_t dtx_i(t)e^{-kt} \qquad (3)$$

where $k$ is Laplace's opérator. On a sequence of functions, Laplace's transform $\mathcal{L}$ is applied on each subdomain :

$$\int_t dt [x_1(t) + x_2(t) + \dots] e^{-kt} = \mathcal{L}_1 + \mathcal{L}_2 + \dots \qquad (4)$$

Each final condition on a subdomain $x_i$ is postponed as the initial condition of the next subdomain $x_{i+1}$. This use Laplace's functions properties and is possible because Laplace function is of exponential order, $exp(-\alpha t)$ framing $x_i(t)$ and assuring its finite value.

## 3 Laplace's transform of a pulse function

Laplace's transform of a pulse function is a well known expression given by :

$$\mathcal{L}[\delta, \tau] = (1 - e^{-k\tau}) \qquad (5)$$

This spectrum has an amplitude equal to $\tau/T$ where T is the problem time duration. The first zero of the spectrum is at $1/\tau$ frequency. Noting $X[k] = \mathcal{L}[\delta, \tau]$
loop. By using the same approach but where some functions are setted by a previous analysis, we can studied non linear circuits. These studies are leaded under the formalism of tensor analysis of networks.

## 2 Exponential order functions

A time dependent functions $x(t)$ can be separate in a sequence of subdomains. Each subdomain $x_i(t)$ starts at $t = t_i$ and ends at $t = t_j$. On each subdomain, if $\exists (M > 0, \gamma) \in \mathbb{R}^2$ so that, $\forall t > N$:

$$|e^{-\gamma t}F(t)| < M \quad (1)$$

Then, under these assumptions, a function $exp(-\alpha t)$ can frame $x_i(t)$ and $F(t)$ is an exponential order function $\gamma$ when $t \to \infty$. Calculating locally Fourier's transform :

$$\int_t dt (x_i(t)e^{-\alpha t}) e^{-i\omega t} \quad (2)$$

Leads to compute Laplace's transform :

$$\int_t dt x_i(t) e^{-kt} \quad (3)$$

where $k$ is Laplace's opérator. On a sequence of functions, Laplace's transform $\mathcal{L}$ is applied on each subdomain :

$$\int_t dt [x_1(t) + x_2(t) + \dots] e^{-kt} = \mathcal{L}_1 + \mathcal{L}_2 + \dots \quad (4)$$

Each final condition on a subdomain $x_i$ is postponed as the initial condition of the next subdomain $x_{i+1}$. This use Laplace's functions properties and is possible because Laplace function is of exponential order, $exp(-\alpha t)$ framing $x_i(t)$ and assuring its finite value.

## 3 Laplace's transform of a pulse function

Laplace's transform of a pulse function is a well known expression given by :

$$\mathcal{L}[\delta, \tau] = (1 - e^{-k\tau}) \quad (5)$$

This spectrum has an amplitude equal to $\tau/T$ where T is the problem time duration. The first zero of the spectrum is at $1/\tau$ frequency. Noting $X[k] = \mathcal{L}[\delta, \tau]$
and $x[n] = \delta$, a fundamental relation between $X$ and $x$ goes through Parseval's theorem. In the principle we approximate our signal writing :

$$x[n] \approx \sum_{n} \mathcal{L}[x[n]\delta, n\tau] \subseteq x[n]\mathcal{L}[\delta_{\tau}]e^{-kn\tau} \quad (6)$$

The spectrum of $\mathcal{L}[\delta_{\tau}]$ being given by (5). The exponential order of the Laplace's transform is used on each pulse, seen as a bounded local time domain.

At the end of a subdomain, potential energy symbolized across capacitors ca, reach some value $V_f$. This final difference of potentials (dop) must be postponed as initial condition in the next subdomain $V_f \to V_i/k$, as a series dop with the concerned capacitor added to the source vector of the problem. The same process must be applied for the kinetic energy, as magnetic flux becoming an initial condition in current for each inductance $L$ symbolizing this storage. The initial condition is $LJ_i$ must be postponed also in the source vector of the problem studied.

## 4 Parseval's theorem applied to pulses

Parseval's theorem gives the equivalence in energy between both representations in both domains frequency and time :

$$\sqrt{\frac{1}{N}x[n]x[n]} = \sqrt{X[k]*X[k]} \quad (7)$$

Parseval's theorem gives a bridge between the time description and the frequency one. Here the temporal domain duration $T$ is major. Once the time step $\tau$ choosen, it defines the number of samples $N$. Being careful that the temporal signal return to zero in $T$, the pulse fundamental amplitude is then defined by $T/\tau$ with $N$ spectrum rays[1]. Equation 7 is called the root mean square value of the signal.

## 5 A time function seen as a pulses sequence

Following 6, the signal is seen as a sequence of pulses with amplitudes equal to the instantaneous time signal value. Computing the rms value 7 for a pulse of amplitude $x[n_0]$ and duration $\tau$ should give again the $x[n_0]$ amplitude at the same moment $n_0$ if we multiply $x[n_0]X[k]exp(-kn_0\tau)$ by $T/\tau$. In fact for various losses, partly due to the non infinite spectrum allocated to $\delta(n_0)$, it remains a difference that must be corrected. For doing that we define the form factor $\mathcal{F}$ with :

$$\mathcal{F} = \alpha \max \left( n_0, |x[n_0]| \left[ \sqrt{\sum_k x[n_0]X^*[k]e^{kn_0\tau}x[n_0]X[k]e^{-kn_0\tau}} \right]^{-1} \right) \quad (8)$$
and $x[n] = \delta$, a fundamental relation between $X$ and $x$ goes through Parseval's theorem. In the principle we approximate our signal writing :

$$x[n] \approx \sum_{n} \mathcal{L}[x[n]\delta, n\tau] \subseteq x[n]\mathcal{L}[\delta_{\tau}] e^{-kn\tau} \quad (6)$$

The spectrum of $\mathcal{L}[\delta_{\tau}]$ being given by (5). The exponential order of the Laplace's transform is used on each pulse, seen as a bounded local time domain.

At the end of a subdomain, potential energy symbolized across capacitors ca, reach some value $V_f$. This final difference of potentials (dop) must be postponed as initial condition in the next subdomain $V_f \to V_i/k$, as a series dop with the concerned capacitor added to the source vector of the problem. The same process must be applied for the kinetic energy, as magnetic flux becoming an initial condition in current for each inductance $L$ symbolizing this storage. The initial condition is $LJ_i$ must be postponed also in the source vector of the problem studied.

## 4 Parseval's theorem applied to pulses

Parseval's theorem gives the equivalence in energy between both representations in both domains frequency and time :

$$\sqrt{\frac{1}{N}x[n]x[n]} = \sqrt{X[k]*X[k]} \quad (7)$$

Parseval's theorem gives a bridge between the time description and the frequency one. Here the temporal domain duration $T$ is major. Once the time step $\tau$ choosen, it defines the number of samples $N$. Being careful that the temporal signal return to zero in $T$, the pulse fundamental amplitude is then defined by $T/\tau$ with $N$ spectrum rays[1]. Equation 7 is called the root mean square value of the signal.

## 5 A time function seen as a pulses sequence

Following 6, the signal is seen as a sequence of pulses with amplitudes equal to the instantaneous time signal value. Computing the rms value 7 for a pulse of amplitude $x[n_0]$ and duration $\tau$ should give again the $x[n_0]$ amplitude at the same moment $n_0$ if we multiply $x[n_0]X[k]exp(-kn_0\tau)$ by $T/\tau$. In fact for various losses, partly due to the non infinite spectrum allocated to $\delta(n_0)$, it remains a difference that must be corrected. For doing that we define the form factor $\mathcal{F}$ with :

$$\mathcal{F} = \alpha \max \left( n_0, |x[n_0]| \left[ \sqrt{\sum_k x[n_0]X^*[k]e^{kn_0\tau}x[n_0]X[k]e^{-kn_0\tau}} \right]^{-1} \right) \quad (8)$$
$\alpha$ is a coefficient to adjust the ratio and obtain a perfect correspondence between the time curve and the curve coming from the weighted rms values.

# 6 The special root mean square function of time

Knowing $\mathcal{F}$ we define a special root mean square value $\chi_{rms}$ by :

$$ \chi_{rms}[n] = \frac{x[n]}{|x[n]|} \sqrt{\sum_{k} x[n] X^*[k] e^{kn\tau} \mathcal{F}x[n] X[k] e^{-kn\tau}} \quad (9) $$

Drawing $\chi_{rmx}[n]$ gives the same curve as $x[n]$. It's a way for verifying if the form factor $\mathcal{F}$ is well dimensioned. If yes, $\chi_{rms}$ is a fast, stable and robust way for going from the Laplace definition of a signal to its time one, locally. The computation of $\chi_{rms}$ should be seen as a local measurement of $x[n]$. The operation (9) make think in the concept of measurement in quantum mechanics, and is in fact quite similar.

# 7 Description of a non linear fundamental impedance tensor

Noting $\chi_{rms}(t)$ a time function deduced from a spectral one using 9. A non linear impedance can be set by domain functions parametrized. For example we may have a non linear impedance $z$ setted by the voltage accross it[2]. The technique uses domain functions $\tilde{D}_{I_1(p)}$. This function provoke the change of $z$ depending on the parameter $p$. If $p$ belongs to the intervalle $I_1(p)$ then $z$ has some particular equation. So $z$ can be developed in general following an expression like :

$$ z = \tilde{D}_{I_1(p)}z_1 + \tilde{D}_{I_2(p)}z_2 + \dots + \tilde{D}_{I_i(p)}z_i \quad (10) $$

Through this development, $z$ will automatically change of expression when $p$ changes. But $p$ is defined in the time domain. It means that $p \equiv p_{rms}$. So, a frequency loop makes the computation of the circuit inside a time loop where some parameters are updated. Both techniques of the special rms function and domain functions give the facility for modeling non linear and complex circuits. The non linear behavior is determined in two steps. In a first step we determine the parameter value without the non linear component. In a second step we set the domain function and compute the whole circuit with the non linear component.
$\alpha$ is a coefficient to adjust the ratio and obtain a perfect correspondence between the time curve and the curve coming from the weighted rms values.

# 6 The special root mean square function of time

Knowing $\mathcal{F}$ we define a special root mean square value $\chi_{rms}$ by :

$$ \chi_{rms}[n] = \frac{x[n]}{|x[n]|} \sqrt{\sum_{k} x[n] X^*[k] e^{kn\tau} \mathcal{F}x[n] X[k] e^{-kn\tau}} \quad (9) $$

Drawing $\chi_{rmx}[n]$ gives the same curve as $x[n]$. It's a way for verifying if the form factor $\mathcal{F}$ is well dimensioned. If yes, $\chi_{rms}$ is a fast, stable and robust way for going from the Laplace definition of a signal to its time one, locally. The computation of $\chi_{rms}$ should be seen as a local measurement of $x[n]$. The operation (9) make think in the concept of measurement in quantum mechanics, and is in fact quite similar.

# 7 Description of a non linear fundamental impedance tensor

Noting $\chi_{rms}(t)$ a time function deduced from a spectral one using 9. A non linear impedance can be set by domain functions parametrized. For example we may have a non linear impedance $z$ setted by the voltage accross it[2]. The technique uses domain functions $\tilde{D}_{I_1(p)}$. This function provoke the change of $z$ depending on the parameter $p$. If $p$ belongs to the intervalle $I_1(p)$ then $z$ has some particular equation. So $z$ can be developed in general following an expression like :

$$ z = \tilde{D}_{I_1(p)}z_1 + \tilde{D}_{I_2(p)}z_2 + \dots + \tilde{D}_{I_i(p)}z_i \quad (10) $$

Through this development, $z$ will automatically change of expression when $p$ changes. But $p$ is defined in the time domain. It means that $p \equiv p_{rms}$. So, a frequency loop makes the computation of the circuit inside a time loop where some parameters are updated. Both techniques of the special rms function and domain functions give the facility for modeling non linear and complex circuits. The non linear behavior is determined in two steps. In a first step we determine the parameter value without the non linear component. In a second step we set the domain function and compute the whole circuit with the non linear component.
<figure><img src="image_2.png" /><figcaption>FIGURE 1 – Source function</figcaption></figure>

# 8 Illustration on a spark gap modeling

For illustrating our techniques we choose to model a spark gap. This is a non linear component activated by a high voltage while it is stoped by a low current.

## 8.1 Imaging a circuit equiped with a spark gap

We imagine a circuit made of an electromotive force in series with a resistor. This generator power supplies a spark gap and we measure the voltage across this load.

## 8.2 Source function

The electromotive force $E$ generates a double exponential function defined by :

$$E(t) = E_0 (e^{-t/t_d} - e^{-t/t_m}) \qquad (11)$$

We careful that the curve (figure 1) return to zero before the end of the time domain.

The software used for tracing the curves is :

```python
import numpy as np
import plab as plt
#
```
<figure><img src="image_2.png" /><figcaption>FIGURE 1 – Source function</figcaption></figure>

# 8 Illustration on a spark gap modeling

For illustrating our techniques we choose to model a spark gap. This is a non linear component activated by a high voltage while it is stoped by a low current.

## 8.1 Imaging a circuit equiped with a spark gap

We imagine a circuit made of an electromotive force in series with a resistor. This generator power supplies a spark gap and we measure the voltage across this load.

## 8.2 Source function

The electromotive force $E$ generates a double exponential function defined by :

$$E(t) = E_0 (e^{-t/t_d} - e^{-t/t_m}) \qquad (11)$$

We careful that the curve (figure 1) return to zero before the end of the time domain.

The software used for tracing the curves is :

```python
import numpy as np
import pylab as plt
```
```python
T=1E-3 # time domain
tau=1E-6 # time step
td=150E-6 # fall time
tm=10E-6 # time rise
Eo=1E5 # source amplitude
N=int(T/tau)
#
E= np(zeros(N,dtype= float))
abst= np(zeros(N,dtype= float))
for n in range(N):
    t=n*tau
    abst[n]=t*1E6 # time in us
    E[n]=Eo*(np.exp(-t/td)-np.exp(-t/tm))
    plt plot(abst,E, label='Source signal')
    plt grid(True)
    plt xlabel('Time [µs]')
    plt legend()
    plt show()
```

## 8.3 Form factor determination

Once the source is defined, we develop a pulse function $\delta$ of duration $\tau$ and compute its Laplace's function. For normalizing it we multiply this Laplace's transform by $T/\tau$. It is shown on figure 2.

After these operations, we can compute the sequence of pulses weighted by $E[n]$, to set the form factor $F$ in order that both curves are equals. Once $F$ is known we can construct the curve $\chi_{rms}[n]$. We obtain the comparison shown figure 3.

The program for reaching this result is given below.

```python
# making E as a sequence of pulses without F
Wrms = np zeros(N, type = float)
WX = np zeros(N, type = complex)

for n in range(N):
    t = n * tau
    for s in range(N):
        k = 1 * J * 2 * np pi * s / T
        WX[s] = E[n] * X[s] * np exp(-k * t)
        Wrms[n] = np sqrt(abs(np dot(np conj(WX), WX)))

# we determine F
```
```python
T=1E-3 # time domain
tau=1E-6 # time step
td=150E-6 # fall time
tm=10E-6 # time rise
Eo=1E5 # source amplitude
N=int(T/tau)
#
E= np(zeros(N,dtype= float))
abst= np(zeros(N,dtype= float))
for n in range(N):
    t=n*tau
    abst[n]=t*1E6 # time in us
    E[n] = Eo*(np.exp(-t/td)-np.exp(-t/tm))
    plt plot(abst,E, label='Source signal')
    plt grid(True)
    plt xlabel('Time [µs]')
    plt legend()
    plt show()
```

## 8.3 Form factor determination

Once the source is defined, we develop a pulse function $\delta$ of duration $\tau$ and compute its Laplace's function. For normalizing it we multiply this Laplace's transform by $T/\tau$. It is shown on figure 2.

After these operations, we can compute the sequence of pulses weighted by $E[n]$, to set the form factor $F$ in order that both curves are equals. Once $F$ is known we can construct the curve $\chi_{rms}[n]$. We obtain the comparison shown figure 3.

The program for reaching this result is given below.

```python
# making E as a sequence of pulses without F
Wrms = np zeros(N, type = float)
WX = np zeros(N, type = complex)

for n in range(N):
    t = n * tau
    for s in range(N):
        k = 1 * J * 2 * np pi * s / T
        WX[s] = E[n] * X[s] * np exp(-k * t)
        Wrms[n] = np sqrt(abs(np dot(np conj(WX), WX)))

# we determine F
```
<figure><img src="image_3.png" /><figcaption>FIGURE 2 – Pulse normalized spectrum</figcaption></figure>

```python
F=2.*np:sqrt(max(abs(E))/max(abs(WX)))*8.6/6.
print(F)

for n in range(N):
    t=n*tau

    for s in range(N):
        k=1J*2.*np.pi*s/T
        WX[s]=E[n]*X[s]*np.exp(-k*t)
        Wrms[n]=np:sqrt(abs(np.dot(np.conj(WX),F*WX)))

plt plot(abst, Wrms, 'x', label='Signal Wrms')
plt grid(True)
plt xlabel(u'Time [µs]')
plt legend()

plt show()
```

The form factor *F* being known, we are now able to use it in any Laplace to time bridge.
<figure><img src="image_3.png" /><figcaption>FIGURE 2 – Pulse normalized spectrum</figcaption></figure>

```python
F=2.*np:sqrt(max(abs(E))/max(abs(WX)))*8.6/6.
print(F)

for n in range(N):
    t=n*tau

    for s in range(N):
        k=1J*2.*np.pi*s/T
        WX[s]=E[n]*X[s]*np.exp(-k*t)
        Wrms[n]=np:sqrt(abs(np.dot(np.conj(WX),F*WX)))

plt plot(abst, Wrms, 'x', label='Signal Wrms')
plt grid(True)
plt xlabel(u'Time [µs]')
plt legend()

plt show()
```

The form factor *F* being known, we are now able to use it in any Laplace to time bridge.
<figure><img src="image_4.png" /><figcaption>FIGURE 3 – Comparison $E_0$, $Wrms$</figcaption></figure>

## 8.4 Spark gap model

The spark gap is a capacitor when not activated, and an arc when activated. The activation is linked with a voltage value going over a threshold value. Once the spark gap is activated, it returns to a capacitor behavior when the current goes under another threshold value. For modeling this behavior we need two functions of domains. One function controls the voltage response of the spark gap and another function controls the current response of the spark gap. If the voltage is under the threshold $V_b$ whatever the current value, the spark gap impedance $Z_{sp}$ must be a capacitor: $Z_{sp} = \dot{D}_{<V_b} (kC)^{-1}$. If the voltage goes over $V_b$, the spark gap impedance must be a low resistance $R_a$, also if the current is higher than the minimum value $I_m$ required for maintaining the spark gap activated. The impedance becomes:

$$Z_{sp} = \left( \dot{D}_{<V_b} + \dot{D}_{<I_m} \right) (kC)^{-1} + \dot{D}_{>V_b} \dot{D}_{>I_m} R_a \quad (12)$$

The fact to have an OR function in neutral state characterize through the capacitor the dielectric strength of the spark gap is. We will have to make the computation in two steps: a first step for determining the domain function parameter values, with only the spark gap capacitor in the circuit. And a second step applying the voltage obtained in the previous step with the complete spark gap model. As the spark gap behavior depends on two domain functions, the current in presence of the arc must be determined for setting the current domain functions once the spark
<figure><img src="image_4.png" /><figcaption>FIGURE 3 – Comparison $E_0$, $Wrms$</figcaption></figure>

## 8.4 Spark gap model

The spark gap is a capacitor when not activated, and an arc when activated. The activation is linked with a voltage value going over a threshold value. Once the spark gap is activated, it returns to a capacitor behavior when the current goes under another threshold value. For modeling this behavior we need two functions of domains. One function controls the voltage response of the spark gap and another function controls the current response of the spark gap. If the voltage is under the threshold $V_b$ whatever the current value, the spark gap impedance $Z_{sp}$ must be a capacitor: $Z_{sp} = \dot{D}_{<V_b} (kC)^{-1}$. If the voltage goes over $V_b$, the spark gap impedance must be a low resistance $R_a$, also if the current is higher than the minimum value $I_m$ required for maintaining the spark gap activated. The impedance becomes:

$$Z_{sp} = \left( \dot{D}_{<V_b} + \dot{D}_{<I_m} \right) (kC)^{-1} + \dot{D}_{>V_b} \dot{D}_{>I_m} R_a \quad (12)$$

The fact to have an OR function in neutral state characterize through the capacitor the dielectric strength of the spark gap is. We will have to make the computation in two steps: a first step for determining the domain function parameter values, with only the spark gap capacitor in the circuit. And a second step applying the voltage obtained in the previous step with the complete spark gap model. As the spark gap behavior depends on two domain functions, the current in presence of the arc must be determined for setting the current domain functions once the spark
<figure><img src="image_5.png" /><figcaption>FIGURE 4 – Domain function in volts</figcaption></figure>

gap activated. It means that each time both open circuit voltage and short circuit current must be computed before setting the spark gap model.

Did this model works? We can test it on our temporal source signal replacing the capacitor by a high resistance value to avoid finite difference time domain computation. For programming the domain function we use sigmoid functions. We write :

$$ \dot{D}_{V_b}^u = 1 - [1 + \exp(-\sigma(V - V_b))]^{-1} \quad (13) $$

Figure 4 we trace the component $\dot{D}_{>V_b}^u R_a$ with $R_{a0} = 100k \Omega$, blocking value.

We see that with our sets, the impedance becomes low when the voltage goes over 20 kV.

Having verified the domain function, we can trace the four functions, each having a complementary one giving an opposite behavior. Noting $\dot{D}_V^u = 1 - \dot{D}_{V_b}^u$ and $\dot{D}_{cV}^u = \dot{D}_{V_b}^u$, also $\dot{D}_I^i = 1 - \dot{D}_{I_m}^i$ and $\dot{D}_{cI}^i = \dot{D}_{I_m}^i$. In that case, the spark gap impedance becomes :

$$ Z_{sp} = (\dot{D}_V^u + \dot{D}_I^i) (kC)^{-1} + \dot{D}_{cV}^u \dot{D}_{cI}^i R_a \quad (14) $$

Figure 5 shows the four domains. The listing is given below.
<figure><img src="image_5.png" /><figcaption>FIGURE 4 – Domain function in volts</figcaption></figure>

gap activated. It means that each time both open circuit voltage and short circuit current must be computed before setting the spark gap model.

Did this model works? We can test it on our temporal source signal replacing the capacitor by a high resistance value to avoid finite difference time domain computation. For programming the domain function we use sigmoid functions. We write :

$$ \dot{D}_{V_b}^u = 1 - [1 + \exp(-\sigma(V - V_b))]^{-1} \quad (13) $$

Figure 4 we trace the component $\dot{D}_{>V_b}^u R_a$ with $R_{a0} = 100k\Omega$, blocking value.

We see that with our sets, the impedance becomes low when the voltage goes over 20 kV.

Having verified the domain function, we can trace the four functions, each having a complementary one giving an opposite behavior. Noting $\dot{D}_V^u = 1 - \dot{D}_{V_b}^u$ and $\dot{D}_{cV}^u = \dot{D}_{V_b}^u$, also $\dot{D}_I^i = 1 - \dot{D}_{I_m}^i$ and $\dot{D}_{cI}^i = \dot{D}_{I_m}^i$. In that case, the spark gap impedance becomes :

$$ Z_{sp} = (\dot{D}_V^u + \dot{D}_I^i) (kC)^{-1} + \dot{D}_{cV}^u \dot{D}_{cI}^i R_a \quad (14) $$

Figure 5 shows the four domains. The listing is given below.
<figure><img src="image_6.png" /><figcaption>FIGURE 5 – Domain functions</figcaption></figure>

```python
# domain functions test
Rao=1E5 # resistance off high value
Vb = 20E3 # spark gap voltage threshold
Im = 1E-3 # spark gap current threshold
D4V(np(zeros(100, dtype.float)) # domain function in volt
D4I(np(zeros(100, type.float)) # domain function in current
Dc4V(np(zeros(100, type.float)) # complementary function
Dc4I(np(zeros(100, type.float)) # complementary function

for ii in range(100):
    V=ii*1E3
    D4V[kkV] = (1.-1./ (1.+np.exp(-1E-3*(V-Vb))))
    Dc4V[kkV] = 1.-D4V[kkV]

pltsubplot(1,2,1)
plot(D4V label='D4V')
plot(Dc4V label='Dc4V')

grid(True)
legend()
xlabel('kV')

for ii in range(100):
    I=ii*1E-4
    D4I[kkV] = 1.-1./ (1.+np.exp(-1E4*(I-Im)))
```
<figure><img src="image_6.png" /><figcaption>FIGURE 5 – Domain functions</figcaption></figure>

```python
# domain functions test
Rao=1E5 # resistance off high value
Vb = 20E3 # spark gap voltage threshold
Im = 1E-3 # spark gap current threshold
D4V(np(zeros(100, type=float)) # domain function in volt
D4I(np(zeros(100, type=float)) # domain function in current
Dc4V(np(zeros(100, type=float)) # complementary function
Dc4I(np(zeros(100, type=float)) # complementary function

for ii in range(100):
    V=ii*1E3
    D4V[kkV] = (1.-1./ (1.+np.exp(-1E-3*(V-Vb))))
    Dc4V[kkV] = 1.-D4V[kkV]

pltsubplot(1,2,1)
plot(D4V label='D4V')
plot(Dc4V label='Dc4V')
grid(True)
legend()
xlabel('kV')

for ii in range(100):
    I=ii*1E-4
    D4I[kkV] = 1.-1./ (1.+np.exp(-1E-4*(I-Im)))
```
<figure><img src="image_7.png" /><figcaption>FIGURE 6 – Spark gap non linear impedance testing</figcaption></figure>

```python
Dc4I(ii)=1.-D4I(ii)
pltsubplot(1,2,2)
pltplot(D4I label='D4I')
pltplot(Dc4I label='Dc4I')
pltgrid(True)
pltlegend()
pltxlabel(u'mA')

pltshow()
```

In a first step we can test our spark gap impedance model getting out its capacitance. This allows to compute its response directly in the time domain without having to program time integration. In that case, the capacitor is replaced by a high resistance when the spark gap is not activated. We obtain the curves shown in figure 6.

We verify that once the open voltage goes over the voltage threshold, the spark gap impedance falls down fastly. Then, as the current is low enough, the spark gap returns to an inactivated mode when the voltage becomes inferior to the voltage threshold. The test program is given here after.

```matlab
# spark gap impedance testing
Rhigh=1E5
Ro=1E2
```
<figure><img src="image_7.png" /><figcaption>FIGURE 6 – Spark gap non linear impedance testing</figcaption></figure>

```python
Dc4I(ii)=1.-D4I(ii)
pltsubplot(1,2,2)
plt plot(D4I label='D4I')
plt plot(Dc4I, label='Dc4I')
plt gridTrue)
plt legend()
plt xlabel(u'mA')

plt show()
```

In a first step we can test our spark gap impedance model getting out its capacitance. This allows to compute its response directly in the time domain without having to program time integration. In that case, the capacitor is replaced by a high resistance when the spark gap is not activated. We obtain the curves shown figure 6.

We verify that once the open voltage goes over the voltage threshold, the spark gap impedance falls down fastly. Then, as the current is low enough, the spark gap return to an inactivated mode when the voltage becomes inferior to the voltage threshold. The test program is given here after.

```matlab
# spark gap impedance testing
Rhigh=1E5
Ro=1E2
```
```
Ra=1.
RH=1E5
Usg= np.zeros(N, type = float)
Jt= np zeros(N, type = float)
for n in range(N):
    t = n * tau
    # open circuit voltage and short circuit current values
    VCO = E[n] * Rhigh / (Rhigh + Ro)
    Icc = E[n] / Ro
    # complete circuit computation
    zsg = (D4V[int(VCO/1000.)] + D4I[int(Icc/10.)])*RH\ + Dc4V[int(VCO/1000.)]*Dc4I[int(Icc/10.])*Ra
    Usg[n] = zsg * E[n] / (Ro + zsg)
    Jt[n] = E[n] / (Ro + zsg)

plt. figure (1)
plt. plot(abst, Usg, label = 'Usg [V]')
plt. grid(True)
plt. legend()
plt. xlabel('time [µs]')

plt. figure (2)
plt. plot(abst, Jt, label = 'Jt [A]')
plt. grid(True)
plt. legend()
plt. xlabel('time [µs]')

plt. show()
```

## 8.5 Final computation

We can now replace the high resistance by the capacitor and include the frequency loop using the pulse spectrum at each time step. This leads to the curve figure 7. The listing is given here after.

```
# complete spark gap impedance model computing
Rhigh=1E5
Ro=1E2
Ra=1.
RH=1E5
```
<figure><img src="image_8.png" /><figcaption>FIGURE 7 – Final computation</figcaption></figure>

```python
Usgt= np(zeros(N, type = float))
Usgf= np(zeros(N, type = complex))
Jt= np(zeros(N, type = float))
C=10E-12
Qf= np(zeros(N, type = complex))  # final to initial condition
# for the capacitor
Qt= np(zeros(N, type = float))
# microvalue for getting up the E[n] limit singularity
epsilon=1E-6
for n in range(N):
    t=n*tau
    # open circuit voltage and short circuit current values
    VCO=E[n]*Rhigh/(Rhigh+Ro)
    ICC=E[n]/Ro
    for s in range(1,N):
        k=1J*2.*np.pi*s/T
        # complete circuit computation
        WX[s] = (E[n] - Qt[n])*X[s]*np.exp(-k*t)
        zsg = (D4V[int(VCO/1000.)]+D4I[int(ICC/10.)])/(k*C)\ +D4V[int(VCO/1000.)]*D4I[int(ICC/10.)]*Ra
        Jsg = WX[s] / (Ro + zsg)
        Qf[s] = (1/(k*C)*Jsg)\
```
<figure><img src="image_8.png" /><figcaption>FIGURE 7 – Final computation</figcaption></figure>

```python
Usgt= np(zeros(N, type = float))
Usgf= np(zeros(N, type = complex))
Jt= np(zeros(N, type = float))
C=10E-12
Qf= np(zeros(N, type = complex))  # final to initial condition
# for the capacitor
Qt= np(zeros(N, type = float))
# microvalue for getting up the E[n] limit singularity
epsilon=1E-6
for n in range(N):
    t=n*tau
    # open circuit voltage and short circuit current values
    VCO=E[n]*Rhigh/(Rhigh+Ro)
    Icc=E[n]/Ro
    for s in range(1,N):
        k=1J*2.*np.pi*s/T
        # complete circuit computation
        WX[s] = (E[n] - Qt[n])*X[s]*np.exp(-k*t)
        zsg = (D4V[int(VCO/1000.)]+D4I[int(Icc/10.)])/(k*C)\ +Dc4V[int(VCO/1000.)]*Dc4I[int(Icc/10.)]*Ra
        Jsg=WX[s]/(Ro+zsg)
        Qf[s] =(1./ (k*C)*Jsg*\
```
(Dc4V[int(VCO/1000.))*Dc4I[int(Icc/10.)*Ra]/zsg)/k
Usgf[s]=WX[s]-Ro*Jsg
Usgf[0]=Usgf[1] # DC value supposes to be equal to
# fundamental
Qf[0]=Qf[1]
Usgt[n]=(E[n]+epsilon)/abs((E[n]+epsilon))*\np.sqrt(abs(np.dot(np.conj(Usgf),F*Usgf)))
Qt[n]=Qt[n-1]+(Qt[n]+epsilon)/abs((Qt[n]+epsilon))*\np.sqrt(abs(np.dot(np.conj(Qf),F*Qf)))

plt.plot(abst,E label='E')
plt.plot(abst,Usgt, label='Usgt')
plt.grid(True)
plt.xlabel('u'time [µs]')
plt.legend()
plt.show()

You can note the little difference in the curves coming from the presence of the
capacitor. As there is this capacitor, we need to postpone its final loading voltage
from the previous step time to the next one. This report is taken into account also
in the time domain. It influences the source value which becomes $E[n]$ decreased
by $Qt[n]$. In the weighted rms value, the sign is obtained making the ratio for
example $E[n]/|E[n]|$. But if $E[n] = 0$ this leads to a singularity, that's why we
must consider an infinitesimal scalar $\epsilon$ for making $(E[n] + \epsilon)/|(E[n] + \epsilon)|$. $\epsilon$ must
be small compared to the nominal scale of $E[n]$ in order that it doesn't impact its
amplitude in the other cases.

9 Projection on a Riemann's sphere and fiber space

We can project the complex spectrum *S* of the observable at a defined location
in time on an imaginary circle *I* = (ℝ, ℂ). The complete revolution of the circle is
associated with the temporal domain duration 2π. As the spectrum changes at each
time step, i.e. at each sphere angle corresponding to the circle rotation, a surface
is drawn on the sphere volume *W*. The sphere being drawn by the circle rotation
is a Riemann's one (with a normalization). The weighted rms formula projects the
surface on the temporal observable curve. This process is similar to this of a fiber
space. Both representations are dual and show that the time curve is a sequence
of measurements in the Laplace's domain. The temporal curve is a manifold *M*
for which each step is in correspondence with a spectrum *S* on *I*(θ). The spectrum
can be seen as a fiber *S* = *V* ∈ *I*(θ) and it exists as many fibers *V* as *M* allows
(Dc4V[int(VCO/1000.))*Dc4I[int(Icc/10.)*Ra]/zsg)/k
Usgf[s]=WX[s]-Ro*Jsg
Usgf[0]=Usgf[1] # DC value supposes to be equal to
# fundamental
Qf[0]=Qf[1]
Usgt[n]=(E[n]+epsilon)/abs((E[n]+epsilon))*\np.sqrt(abs(np.dot(np.conj(Usgf),F*Usgf)))
Qt[n]=Qt[n-1]+(Qt[n]+epsilon)/abs((Qt[n]+epsilon))*\np.sqrt(abs(np.dot(np.conj(Qf),F*Qf)))

plt.plot(abst,E label='E')
plt.plot(abst,Usgt, label='Usgt')
plt.grid(True)
plt.xlabel('u'time [µs]')
plt.legend()
plt.show()

You can note the little difference in the curves coming from the presence of the
capacitor. As there is this capacitor, we need to postpone its final loading voltage
from the previous step time to the next one. This report is taken into account also
in the time domain. It influences the source value which becomes $E[n]$ decreased
by $Qt[n]$. In the weighted rms value, the sign is obtained making the ratio for
example $E[n]/|E[n]|$. But if $E[n] = 0$ this leads to a singularity, that's why we
must consider an infinitesimal scalar $\epsilon$ for making $(E[n] + \epsilon)/|(E[n] + \epsilon)|$. $\epsilon$ must
be small compared to the nominal scale of $E[n]$ in order that it doesn't impact its
amplitude in the other cases.

9 Projection on a Riemann’s sphere and fiber space

We can project the complex spectrum *S* of the observable at a defined location
in time on an imaginary circle *I* = (ℝ, ℂ). The complete revolution of the circle is
associated with the temporal domain duration 2π. As the spectrum changes at each
time step, i.e. at each sphere angle corresponding to the circle rotation, a surface
is drawn on the sphere volume *W*. The sphere being drawn by the circle rotation
is a Riemann’s one (with a normalization). The weighted rms formula projects the
surface on the temporal observable curve. This process is similar to this of a fiber
space. Both representations are dual and show that the time curve is a sequence
of measurements in the Laplace’s domain. The temporal curve is a manifold *M*
for which each step is in correspondence with a spectrum *S* on *I*(θ). The spectrum
can be seen as a fiber *S* = *V* ∈ *I*(θ) and it exists as many fibers *V* as *M* allows
it. In linear cases, the fiber space $\mathcal{W}$ is equal to the product of $\mathcal{M}$ by the fiber $\mathcal{V}$:
$\mathcal{W} = \mathcal{M} \times \mathcal{V}$. But in the case of non linear behaviors with sudden changes of state,
if $\mathcal{W}$ keep the dimension of $\mathcal{M} \times \mathcal{V}$ it can be different from $\mathcal{M} \times \mathcal{V}[3]$. In case of non
linear behaviors, the twisted fiber space can be seen as a sequence of linear ones,
going back to our first assumption using Laplace's exponential order property (and
its final and initial conditions postpone capacities) and the separation of $\mathcal{M}$ in a
sequence of pulse functions and associated time interval domains. So in general $\mathcal{W}$
is a twisted fiber space.

# 10 Space and time or frequency axes

The first experiences give very efficient results, making think that the physical
space is this blend of both time and frequency through the fiber space $\mathcal{W}$. The ap-
proach is no more to choice between time or frequency, but working simultaneously
in the manifold $\mathcal{M}$ of the time perception and in the fiber $\mathcal{V}$ of the underlying La-
place's domains[4]. The time domain is associated with a rhythm of a stationnary
field waves where each beat leads the time metric. If the field exists only under
these stationnary waves, the wave beat gives the time and the wave succession of
maxima gives the space metric. The link and transfer of energy between $\mathcal{V}$ and
$\mathcal{M}$ can be modeled using both wave and cycle spaces[4]. Our perceptible space is
the cycle one and the topology in which we practice electromagnetic compatibility
and all physical realizations. The particle side exists in $\mathcal{M}$ while the wave function
side exists in $\mathcal{V}$ through $S$. Once more, our perceptible reality is described by $\mathcal{W}$.

Energy creates gravitation. Mass is the added property making a field going from $\mathcal{V}$ to $\mathcal{M}$, but $\mathcal{V}$ energy contributes also to the gravitation. It means that the common field that belongs to $\mathcal{W}$ is the gravitational field which defines the space-rhythm metric $\mathcal{W}$.

In this context, particles duality may be lifted following de Broglie's concept of "pilot wave". As we cannot make measurements on any loaded particle without passing through $\mathcal{V}$, measurements made on electrons for example will show the particle behavior through $\mathcal{M}$ and the wave one through $\mathcal{V}[5]$. It seems that such bridge arrives when the field self mode goes over $10^{19}$ Hz. We know that field modes drive the physical world, and it exists scale effects making this world systemic rather than cartesian. Various emergences come from cross talks between systems when going from one scale to another. Mass acquisition can perhaps be seen as such an emergence, as if the quantum volume in relation with the particle wavelength involved jumps from some states set to a new category of sets. With the submit formalism, this kind of step can be modeled using a domain function $\mathcal{D}_{[I \in \mathcal{W}]}^m$ driving
it. In linear cases, the fiber space $\mathcal{W}$ is equal to the product of $\mathcal{M}$ by the fiber $\mathcal{V}$ :
$\mathcal{W} = \mathcal{M} \times \mathcal{V}$. But in the case of non linear behaviors with sudden changes of state,
if $\mathcal{W}$ keep the dimension of $\mathcal{M} \times \mathcal{V}$ it can be different from $\mathcal{M} \times \mathcal{V}[3]$. In case of non
linear behaviors, the twisted fiber space can be seen as a sequence of linear ones,
going back to our first assumption using Laplace's exponential order property (and
its final and initial conditions postpone capacities) and the separation of $\mathcal{M}$ in a
sequence of pulse functions and associated time interval domains. So in general $\mathcal{W}$
is a twisted fiber space.

# 10 Space and time or frequency axes

The first experiences give very efficient results, making think that the physical
space is this blend of both time and frequency through the fiber space W. The ap-
proach is no more to choice between time or frequency, but working simultaneously
in the manifold M of the time perception and in the fiber V of the underlying La-
place's domains[4]. The time domain is associated with a rhythm of a stationnary
field waves where each beat leads the time metric. If the field exists only under
these stationnary waves, the wave beat gives the time and the wave succession of
maxima gives the space metric. The link and transfer of energy between V and
M can be modeled using both wave and cycle spaces[4]. Our perceptible space is
the cycle one and the topology in which we practice electromagnetic compatibility
and all physical realizations. The particle side exists in M while the wave function
side exists in V through S. Once more, our perceptible reality is described by W.
Energy creates gravitation. Mass is the added property making a field going
from V to M, but V energy contributes also to the gravitation. It means that
the common field that belongs to W is the gravitational field which defines the
space-rhythm metric W.

Energy creates gravitation. Mass is the added property making a field going from $\mathcal{V}$ to $\mathcal{M}$, but $\mathcal{V}$ energy contributes also to the gravitation. It means that the common field that belongs to $\mathcal{W}$ is the gravitational field which defines the space-rhythm metric $\mathcal{W}$.

In this context, particles duality may be lifted following de Broglie's concept of "pilot wave". As we cannot make measurements on any loaded particle without passing through $\mathcal{V}$, measurements made on electrons for example will show the particle behavior through $\mathcal{M}$ and the wave one through $\mathcal{V}[5]$. It seems that such bridge arrives when the field self mode goes over $10^{19}$ Hz. We know that field modes drive the physical world, and it exists scale effects making this world systemic rather than cartesian. Various emergences come from cross talks between systems when going from one scale to another. Mass acquisition can perhaps be seen as such an emergence, as if the quantum volume in relation with the particle wavelength involved jumps from some states set to a new category of sets. With the submit formalism, this kind of step can be modeled using a domain function $\mathcal{D}_{[I \in \mathcal{W}]}^m$ driving
the heavy mass property. A state in W may be repered by the variables :

$$ [(n, l, m, s), \mu], x^k \in \mathbb{R}^3, (t \oplus S) \in \mathcal{M}(\mathbb{R}) \times \mathcal{V}(\mathbb{C}) $$

$n, l, m, s$ being the fourth quantum numbers and $\mu$ the energy of heavy mass.

# 11 Conclusion

In electromagnetic compatibility (EMC) we are regularly in front of the di-lemma to work in the time or frequency domains. Generally, standards and mea-surements are realized in the frequency domain. But non linear electronics (a spark gap is a good example) invite to work in the time domain. From many years I have remarked that Laplace's transform offers possibilities to treat sequentially non linear models by pieces. Starting from Parseval's relation, it seemed that some efficient bridge should exist between both domains, avoiding to consider each of them as a single block, as imposed by Fourier's transform. By discovering the trick of the weighted root mean square relation, I've noticed with friends[2] that it seems to give amazing and efficient results, applied in various cases for EMC. This was inspired by the possibility to model all these electronic behaviors in the tensor analysis of networks[6] extended with strings[7]. Today I present this trick of working in W and using $\mathcal{D}_{[I \in W]}^p$ all under the tensor analysis of networks, for benefiting of external critics and tests in order to discover its limits or confirm its universality for electromagnetic compatibility and perhaps further.

# Références

[1] Bellanger, M. (2022). Traitement numérique du signal-10e éd. Dunod.

[2] Maurice, O., De Mejanes, N., & Meton, P. (2024). Modeling non linearities in the frequency domain.

[3] Penrose, R. (2006). The road to reality. Random house, chap. 15 in the French version.

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

[5] Louis de BROGLIE, "Mécanique ondulatoire du photon - théorie quantique des champs". Gauthier-Villars editor, Paris, 1967.

[6] Kron, G. (1939). Tensor analysis of networks. New York : J. Wiley & Sons.

[7] Maurice, O., Reineix, A., Durand, P., & Dubois, F. (2014). Kron's method and cell complexes for magnetomotive and electromotive forces. arXiv preprint arXiv :1412.0187.
the heavy mass property. A state in W may be repered by the variables :

$$ [(n, l, m, s), \mu], x^k \in \mathbb{R}^3, (t \oplus S) \in \mathcal{M}(\mathbb{R}) \times \mathcal{V}(\mathbb{C}) $$

$n, l, m, s$ being the fourth quantum numbers and $\mu$ the energy of heavy mass.

# 11 Conclusion

In electromagnetic compatibility (EMC) we are regularly in front of the di-lemma to work in the time or frequency domains. Generally, standards and mea-surements are realized in the frequency domain. But non linear electronics (a spark gap is a good example) invite to work in the time domain. From many years I have remarked that Laplace's transform offers possibilities to treat sequentially non linear models by pieces. Starting from Parseval's relation, it seemed that some efficient bridge should exist between both domains, avoiding to consider each of them as a single block, as imposed by Fourier's transform. By discovering the trick of the weighted root mean square relation, I've noticed with friends[2] that it seems to give amazing and efficient results, applied in various cases for EMC. This was inspired by the possibility to model all these electronic behaviors in the tensor analysis of networks[6] extended with strings[7]. Today I present this trick of working in W and using $\mathcal{D}_{[I \in W]}^p$ all under the tensor analysis of networks, for benefiting of external critics and tests in order to discover its limits or confirm its universality for electromagnetic compatibility and perhaps further.

# Références

[1] Bellanger, M. (2022). Traitement numérique du signal-10e éd. Dunod.

[2] Maurice, O., De Mejanes, N., & Meton, P. (2024). Modeling non linearities in the frequency domain.

[3] Penrose, R. (2006). The road to reality. Random house, chap. 15 in the French version.

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

[5] Louis de BROGLIE, "Mécanique ondulatoire du photon - théorie quantique des champs". Gauthier-Villars editor, Paris, 1967.

[6] Kron, G. (1939). Tensor analysis of networks. New York : J. Wiley & Sons.

[7] Maurice, O., Reineix, A., Durand, P., & Dubois, F. (2014). Kron's method and cell complexes for magnetomotive and electromotive forces. arXiv preprint arXiv :1412.0187.