- Theoretical model of the
**supercapacitor model**

The basic principle of electric double-layer capacitors is established based on the capacitance characteristics between the solid-liquid interface. This property was discovered by Helmholtz in 1853. Electric double-layer capacitors store energy through space charges generated on the solid-liquid surface under electrostatic action. This space charge is called the electric double layer, and its thickness is below the nanometer scale. Based on the Helmholtz double-layer structure, various theoretical models have been proposed to describe the properties of capacitors.

1.1 Helmholtz two-layer model

Helmholtz was the first to study the capacitive properties of the solid-liquid interface. Helmholtz believes that the charges are evenly distributed on both sides of the solid-liquid interface, as shown in Figure 2. In this case, the surface capacitance (Surfacic Capacitance) of the electric double layer can be calculated by formula (1):

where ε is the dielectric constant of the electrolyte;

d – the spacing of the bilayer structure, in the Helmholtz model, this value should be equal to the molecular diameter of the electrolyte. For example, theoretically, the surface capacitance of the aqueous electrolyte can be calculated by formula (5-4) to obtain C*=340uF/cm2 (ε=78, d=0.2nm).

The Helmholtz electric double layer model is quite simple. It proposes that the capacitance is proportional to the size of the contact surface and inversely proportional to the spacing, which points out the direction for future generations to improve the performance of the capacitor. The disadvantage of this model is that the charge is assumed to be uniformly distributed (this is not possible on the electrolyte side because of the poor conductivity of the electrolyte), and the influence of the conductive properties of the electrolyte is not considered. In applications, the actual surface capacitance of capacitors is often more than an order of magnitude larger than the experimental results (usually 10~30μF/cm2).

1.2 The double-layer structure model of Gouy and Chapman

In order to make up for the shortcomings of the Helmholtz model and describe the relationship between the electric double layer capacitance and voltage, Gouy introduced the random thermal motion model in 1910, which considered the spatial distribution of ionic charges in the electrolyte, which is now also called the diffusion layer model, as shown in Figure 3. Show.

Gouy’s mathematical formula for the diffusion layer was proposed by Chapman in 1913. This formula is based on the Poisson (Poisson) equation and the Boltzmann (Botzmann) distribution function. Under the action of a single electric field, the surface capacitance between the electrode and the electrolyte can be expressed by the following formula (4):

In the formula, z is the ion valence in the electrolyte; n0 is the concentration of anions and cations after thermodynamic equilibrium of the electrolyte; ε is the dielectric constant of the electrolyte; q is the basic charge; uT is the thermodynamic potential unit, uT=kT/q; k is the Boltsmann constant; T is the absolute temperature; ψ0 is the surface electromotive force.

At 25 °C, the relationship between the surface capacitance of the aqueous electrolyte and the surface potential is shown in Figure 5. It can be seen that the Gouy and Chapman models exaggerate the capacitance related to the electric double layer except in the case of dilute electrolyte and low potential. This is because some ions can be infinitely close to the interface after taking ions as point charges.

1.3 Stern’s double-layer structure model

In 1924, Stern improved the Gouy and Chapman model. Stern introduced the size of ions and solvent molecules on the basis of the previous model, and divided the charging area into two parts: one part is the diffusion layer of the previous model, and the surface capacitance is Cd* ; The other part is the dense layer, which is composed of ions adsorbed on the electrode surface, and the surface capacitance is Cc*.

The surface capacitance C” of the electric double layer is composed of the two capacitors mentioned above in series, and is calculated by the following formula（6）:

In the formula, Cd* can be calculated by formula (4), just replace ψ0 with ψd. It can be seen that the influence of Cd* on the total capacitance of the electric double layer is greater at low potential, but when ψdSSSSSSS is greater than a certain thermodynamic unit, especially when the electrolyte is relatively concentrated, the influence of Cd* can be ignored.

- Supercapacitor equivalent circuit model

The theoretical model of supercapacitors mainly focuses on the study of the characteristics of the electric double layer, which has guiding significance for the study of improving the energy and capacity level of capacitors. However, this kind of theoretical model that focuses on microscopic conditions and describes a single interface cannot accurately describe the supercapacitor. The external characteristics of supercapacitors cannot meet the requirements for the use of supercapacitors and control research. In order to accurately describe the external characteristics of capacitors, a variety of equivalent models have been designed to study the external characteristics of supercapacitors by using simple electrical components to form a network, and to study the physical and chemical meanings of each component separately. These models have their own characteristics and limitations, and are suitable for certain situations. Here are some of the most representative equivalent models.

2.1 Simple charge-discharge model

In this simple model, the supercapacitor is equivalent to a simple RC loop, as shown in Figure 9.

When discharging there are (7):

While charging there are (8):

In the formula, RESR is the equivalent internal resistance of the capacitor; C is the ideal capacitance; R0 is the load resistance; USCAP is the working voltage of the supercapacitor; UC0 is the voltage of the supercapacitor C at t=0; UL is the charging voltage. This model follows the equivalent DC circuit model of ordinary capacitors, with simple structure, clear physical meaning of each parameter, and easy to obtain directly in the test. Therefore, this model has been widely used.

2.2 Two-branch model

The two-branch model was proposed by R.Bonert and L.Zubieta, which divides electrostatic energy into two parts, namely the fast stored or transmitted energy and the slow stored or transmitted energy.

Figure 10 shows a double-branch circuit. This model is considered from the transition process of super capacitors. C0 and R0 are the main circuits, which are mainly used to represent the energy change during charging and discharging. Cr and Rr become auxiliary circuits. The loop mainly describes the internal energy redistribution process after the charging and discharging process.

In order to further improve the accuracy, an RC loop can also be added to achieve the purpose of improving the accuracy of the model. A 5-time constant model was proposed in the literature [4], which is realized by adding an RC loop on the basis of this model. However, the equivalent components of this type of model are considered from the perspective of improving the simulation accuracy, and their physical meaning is not obvious, and the model is complicated due to too many parameters, which is inconvenient to use and cannot be directly measured in the test. Class models are not widely used.

2.3 First-order AC equivalent circuit model

If supercapacitors are used in AC working conditions, the capacitors will show many AC characteristics different from their DC characteristics, mainly including AC impedance characteristics, sound characteristics, and efficiency and temperature characteristics under AC working conditions. A first-order AC equivalent circuit model is given in the literature [. In this model, four ideal electrical components are included. Rs is the equivalent series resistance (ESR), which mainly affects the charging and discharging of the supercapacitor; the inductance L is mainly used to simulate the inherent inductance of the capacitor, and its inductance value is very small. It can be ignored in many applications, especially at high frequencies; capacitor C and resistor Rp are used to simulate the resistance affected by the self-discharge of the super capacitor, where Rp is used to simulate the self-discharge loss of the capacitor, also known as leakage current resistance. In practice, the resistance Rp is always much larger than Rs. During high power discharge, Rp can also be ignored.

The capacitor itself will have a certain inherent inductance, which is determined by its structure and characteristics. The inductance of the capacitor includes the inductance caused by the internal core, the inductance caused by the lead wire and the inductance of the metal casing. The internal electrode of the super capacitor used in the test adopts a laminated foil structure, a plastic casing, and the inductance is small, and the capacitor is used in the electric drive. In the system, it mainly experiences large-value unsteady direct current instead of small-amplitude alternating current, so the actual impact of inductance L on the performance of supercapacitors can be ignored. The research on the performance of supercapacitors here mainly focuses on the study of DC characteristics.

2.4 Improved dynamic equivalent circuit model of supercapacitor

The above models are all based on the research methods of ordinary capacitors, and can make good predictions on the performance of supercapacitors under steady-state working conditions (constant current charge and discharge). However, these models are not completely suitable for the power-type supercapacitors used in electric vehicle drive systems. The main problem is that the above models all assume that the equivalent ideal capacitor capacity and equivalent series resistance are fixed, which is inconsistent with reality.

The working conditions in the electric vehicle drive system are very complex. The supercapacitor used as an energy storage element actually works under the condition of high current pulsating DC, and the current, voltage and ambient temperature fluctuate greatly. A large number of test results show that the working state of supercapacitors has obvious changes with the change of the working environment, and the above-mentioned model with fixed parameters is not completely suitable for the study of supercapacitors for electric vehicles.

To this end, the above model needs to be modified. As analyzed above, in order to adapt to the research of electric vehicle drive system, a variable-parameter supercapacitor equivalent circuit model that adapts to high-current DC conditions should be established. In the model, C is the ideal capacitor of the supercapacitor, which is the energy storage element in the supercapacitor. RESR is the equivalent series internal resistance of the supercapacitor. This parameter has an important influence on the working voltage, efficiency and temperature rise of the supercapacitor.

Read more: What is an AC Permanent Magnet Motor Drive System?