# single diode equivalent circuit models

russische frauen aus deutschland kennenlernen In, diode modelling refers to the mathematical models used to approximate the actual behaviour of real diodes to enable calculations and circuit analysis. A 's - curve is (it is well described by the ). This nonlinearity complicates calculations in circuits involving diodes so simpler models are often required.

This article discusses the modelling of diodes, but the techniques may be generalized to other diodes.

## Large-signal modelling[]

### Shockley diode model[]

The relates the diode current I {\displaystyle I} of a er sucht sie in paderborn markt de diode to the diode voltage V D {\displaystyle V_{D}} . This relationship is the diode I-V characteristic:

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I = I S ( e V D n V T − 1 ) {\displaystyle I=I_{S}\left(e^{\frac {V_{D}}{nV_{\text{T}}}}-1\right)} ,

where I S {\displaystyle I_{S}} is the saturation current or scale current of the diode (the magnitude of the current that flows for negative V D {\displaystyle V_{D}} in excess of a few V T {\displaystyle V_{\text{T}}} , typically 10−12 A). The scale current is proportional to the cross-sectional area of the diode. Continuing with the symbols: V T {\displaystyle V_{\text{T}}} is the ( k T / q {\displaystyle kT/q} , about 26 mV at normal temperatures), and n {\displaystyle n} is known as the diode ideality factor (for silicon diodes n {\displaystyle n} is approximately 1 to 2).

When V D ≫ n V T {\displaystyle V_{D}\gg nV_{\text{T}}} the formula can be simplified to:

I ≈ I S ⋅ e V D n V T {\displaystyle I\approx I_{S}\cdot e^{\frac {V_{D}}{nV_{\text{T}}}}} .

This expression is, however, only an approximation of a more complex I-V characteristic. Its applicability is particularly limited in case of ultrashallow junctions, for which better analytical models exist.

#### Diode-resistor circuit example[]

To illustrate the complications in using this law, consider the problem of finding the voltage across the diode in Figure 1.

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Figure 1: Diode circuit with resistive load.

Because the current flowing through the diode is the same as the current throughout the entire circuit, we can lay down another equation. By, the current flowing in the circuit is

I = V S − V D R {\displaystyle I={\frac {V_{S}-V_{D}}{R}}} .

These two equations determine the diode current and the diode voltage. To solve these two equations, we could substitute the current I {\displaystyle I} from the second equation into the first equation, and then try to rearrange the resulting equation to get V D {\displaystyle V_{D}} in terms of V S {\displaystyle V_{S}} . A difficulty with this method is that the diode law is nonlinear. Nonetheless, a formula expressing I {\displaystyle I} directly in terms of V S {\displaystyle V_{S}} without involving V D {\displaystyle V_{D}} can be obtained using the Lambert W {\displaystyle W} -function, which is the of f ( w ) = w e w {\displaystyle f(w)=we^{w}} , that is, w = W ( f ) {\displaystyle w=W(f)} . This solution is discussed next.

### Explicit solution[]

An explicit expression for the diode current can be obtained in terms of the (also called the Omega function). A guide to these manipulations follows. A new variable w {\displaystyle w} is introduced as

w = I S R n V T ( I I S + 1 ) {\displaystyle w={\frac {I_{S}R}{nV_{\text{T}}}}\left({\frac {I}{I_{S}}}+1\right)} .

Following the substitutions I / I S = e V D / n V T − 1 {\displaystyle I/I_{S}=e^{V_{D}/nV_{\text{T}}}-1} :

w e w = I S R n V T e V D n V T e I S R n V T ( I I S + 1 ) {\displaystyle we^{w}={\frac {I_{S}R}{nV_{\text{T}}}}e^{\frac {V_{D}}{nV_{\text{T}}}}e^{{\frac {I_{S}R}{nV_{\text{T}}}}\left({\frac {I}{I_{S}}}+1\right)}}

and V D = V S − I R {\displaystyle V_{D}=V_{S}-IR} :

w e w = I S R n V T e V S n V T e − I R n V T e I R I S n V T I S e I S R n V T {\displaystyle we^{w}={\frac {I_{S}R}{nV_{\text{T}}}}e^{\frac {V_{S}}{nV_{\text{T}}}}e^{\frac {-IR}{nV_{\text{T}}}}e^{\frac {IRI_{S}}{nV_{\text{T}}I_{S}}}e^{\frac {I_{S}R}{nV_{\text{T}}}}}

rearrangement of the diode law in terms of w becomes:

w e w = I S R n V T e V s + I s R n V T {\displaystyle we^{w}={\frac {I_{S}R}{nV_{\text{T}}}}e^{\frac {V_{s}+I_{s}R}{nV_{\text{T}}}}} ,

which using the Lambert W {\displaystyle W} -function becomes

w = W ( I S R n V T e V s + I s R n V T ) {\displaystyle w=W\left({\frac {I_{S}R}{nV_{\text{T}}}}e^{\frac {V_{s}+I_{s}R}{nV_{\text{T}}}}\right)} .

With the approximations (valid for the most common values of the parameters) I s R ≪ V S {\displaystyle I_{s}R\ll V_{S}} and I / I S ≫ 1 {\displaystyle I/I_{S}\gg 1} , this solution becomes

I ≈ n V T R W ( I S R n V T e V s n V T ) {\displaystyle I\approx {\frac {nV_{\text{T}}}{R}}W\left({\frac {I_{S}R}{nV_{\text{T}}}}e^{\frac {V_{s}}{nV_{\text{T}}}}\right)} .

Once the current is determined, the diode voltage can be found using either of the other equations.

For large x, W ( x ) {\displaystyle W(x)} can be approximated by W ( x ) = ln ⁡ x − ln ⁡ ln ⁡ x + o ( 1 ) {\displaystyle W(x)=\ln x-\ln \ln x+o(1)} . For common physical parameters and resistances, I S R n V T e V s n V T {\displaystyle {\frac {I_{S}R}{nV_{\text{T}}}}e^{\frac {V_{s}}{nV_{\text{T}}}}} will be on the order of 1040.

#### Iterative solution[]

The diode voltage V D {\displaystyle V_{D}} can be found in terms of V S {\displaystyle V_{S}} for any particular set of values by an using a calculator or computer. The diode law is rearranged by dividing by I S {\displaystyle I_{S}} , and adding 1. The diode law becomes

e V D n V T = I I S + 1 {\displaystyle e^{\frac {V_{D}}{nV_{\text{T}}}}={\frac {I}{I_{S}}}+1} .

By taking natural logarithms of both sides the exponential is removed, and the equation becomes

V D n V T = ln ⁡ ( I I S + 1 ) {\displaystyle {\frac {V_{D}}{nV_{\text{T}}}}=\ln \left({\frac {I}{I_{S}}}+1\right)} .

For any I {\displaystyle I} , this equation determines V D {\displaystyle V_{D}} . However, I {\displaystyle I} also must satisfy the Kirchhoff's law equation, given above. This expression is substituted for I {\displaystyle I} to obtain

V D n V T = ln ⁡ ( V S − V D R I S + 1 ) {\displaystyle {\frac {V_{D}}{nV_{\text{T}}}}=\ln \left({\frac {V_{S}-V_{D}}{RI_{S}}}+1\right)} ,

or

V D = n V T ln ⁡ ( V S − V D R I S + 1 ) {\displaystyle V_{D}=nV_{\text{T}}\ln \left({\frac {V_{S}-V_{D}}{RI_{S}}}+1\right)} .

The voltage of the source V S {\displaystyle V_{S}} is a known given value, but V D {\displaystyle V_{D}} is on both sides of the equation, which forces an iterative solution: a starting value for V D {\displaystyle V_{D}} is guessed and put into the right side of the equation. Carrying out the various operations on the right side, we come up with a new value for V D {\displaystyle V_{D}} . This new value now is substituted on the right side, and so forth. If this iteration converges the values of V D {\displaystyle V_{D}} become closer and closer together as the process continues, and we can stop iteration when the accuracy is sufficient. Once V D {\displaystyle V_{D}} is found, I {\displaystyle I} can be found from the Kirchhoff's law equation.

Sometimes an iterative procedure depends critically on the first guess. In this example, almost any first guess will do, say V D = 600 mV {\displaystyle V_{D}=600\,{\text{mV}}} . Sometimes an iterative procedure does not converge at all: in this problem an iteration based on the exponential function does not converge, and that is why the equations were rearranged to use a logarithm. Finding a convergent iterative formulation is an art, and every problem is different.

#### Graphical solution[]

Graphical determination of the operating point through the intersection of the diode characteristic with the resistive load line.

Graphical analysis is a simple way to derive a numerical solution to the equations describing the diode. As with most graphical methods, it has the advantage of easy visualization. By plotting the I-V curves, it is possible to obtain an approximate solution to any arbitrary degree of accuracy. This process is the graphical equivalent of the two previous approaches, which are more amenable to computer implementation.

This method plots the two current-voltage equations on a graph and the point of intersection of the two curves satisfies both equations, giving the value of the current flowing through the circuit and the voltage across the diode. The figure illustrates such method.

### Piecewise linear model[]

A piecewise linear approximation of the diode characteristic.

In practice, the graphical method is complicated and impractical for complex circuits. Another method of modelling a diode is called (PWL) modelling. In mathematics, this means taking a function and breaking it down into several linear segments. This method is used to approximate the diode characteristic curve as a series of linear segments. The real diode is modelled as 3 components in series: an ideal diode, a voltage source and a.

The figure shows a real diode I-V curve being approximated by a two-segment piecewise linear model. Typically the sloped line segment would be chosen tangent to the diode curve at the. Then the slope of this line is given by the reciprocal of the resistance of the diode at the Q-point.

#### Mathematically idealized diode[]

I-V characteristic of an ideal diode.

Firstly, let us consider a mathematically idealized diode. In such an ideal diode, if the diode is reverse biased, the current flowing through it is zero. This ideal diode starts conducting at 0 V and for any positive voltage an infinite current flows and the diode acts like a short circuit. The I-V characteristics of an ideal diode are shown below:

#### Ideal diode in series with voltage source[]

Now let us consider the case when we add a voltage source in series with the diode in the form shown below:

Ideal diode with a series voltage source.

When forward biased, the ideal diode is simply a short circuit and when reverse biased, an open circuit.

If the of the diode is connected to 0 V, the voltage at the will be at Vt and so the potential at the cathode will be greater than the potential at the anode and the diode will be reverse biased. In order to get the diode to conduct, the voltage at the anode will need to be taken to Vt. This circuit approximates the cut-in voltage present in real diodes. The combined I-V characteristic of this circuit is shown below:

I-V characteristic of an ideal diode with a series voltage source.

The Shockley diode model can be used to predict the approximate value of V t {\displaystyle V_{t}} .

I = I S ( e V D n ⋅ V T − 1 ) ⇔ ln ⁡ ( 1 + I I S ) = V D n ⋅ V T ⇔ V D = n ⋅ V T ln ⁡ ( 1 + I I S ) ≈ n ⋅ V T ln ⁡ ( I I S ) ⇔ V D ≈ n ⋅ V T ⋅ ln ⁡ 10 ⋅ log 10 ⁡ ( I I S ) {\displaystyle {\begin{aligned}&I=I_{S}\left(e^{\frac {V_{D}}{n\cdot V_{\text{T}}}}-1\right)\\\Leftrightarrow {}&\ln \left(1+{\frac {I}{I_{S}}}\right)={\frac {V_{D}}{n\cdot V_{\text{T}}}}\\\Leftrightarrow {}&V_{D}=n\cdot V_{\text{T}}\ln \left(1+{\frac {I}{I_{S}}}\right)\approx n\cdot V_{\text{T}}\ln \left({\frac {I}{I_{S}}}\right)\\\Leftrightarrow {}&V_{D}\approx n\cdot V_{\text{T}}\cdot \ln {10}\cdot \log _{10}{\left({\frac {I}{I_{S}}}\right)}\end{aligned}}}

Using n = 1 {\displaystyle n=1} and T = 25 °C {\displaystyle T=25\,{\text{°C}}} :

V D ≈ 0.05916 ⋅ log 10 ⁡ ( I I S ) {\displaystyle V_{D}\approx 0.05916\cdot \log _{10}{\left({\frac {I}{I_{S}}}\right)}}

Typical values of the at room temperature are:

• I S = 10 − 12 {\displaystyle I_{S}=10^{-12}} for silicon diodes;
• I S = 10 − 6 {\displaystyle I_{S}=10^{-6}} for germanium diodes.

As the variation of V D {\displaystyle V_{D}} goes with the logarithm of the ratio I I S {\displaystyle {\frac {I}{I_{S}}}} , its value varies very little for a big variation of the ratio. The use of base 10 logarithms makes it easier to think in orders of magnitude.

For a current of 1.0 mA:

• V D ≈ 0.53 V {\displaystyle V_{D}\approx 0.53\,{\text{V}}} for silicon diodes (9 orders of magnitude);
• V D ≈ 0.18 V {\displaystyle V_{D}\approx 0.18\,{\text{V}}} for germanium diodes (3 orders of magnitude).

For a current of 100 mA:

• V D ≈ 0.65 V {\displaystyle V_{D}\approx 0.65\,{\text{V}}} for silicon diodes (11 orders of magnitude);
• V D ≈ 0.30 V {\displaystyle V_{D}\approx 0.30\,{\text{V}}} for germanium diodes (5 orders of magnitude).

Values of 0.6 or 0.7 volts are commonly used for silicon diodes.

#### Diode with voltage source and current-limiting resistor[]

The last thing needed is a resistor to limit the current, as shown below:

Ideal diode with a series voltage source and resistor.

The I-V characteristic of the final circuit looks like this:

I-V characteristic of an ideal diode with a series voltage source and resistor.

The real diode now can be replaced with the combined ideal diode, voltage source and resistor and the circuit then is modelled using just linear elements. If the sloped-line segment is tangent to the real diode curve at the, this approximate circuit has the same circuit at the Q-point as the real diode.

#### Dual PWL-diodes or 3-Line PWL model[]

I-V characteristic of the standard PWL model (marked by red-triangles), as described above. Shown for reference is the standard Shockley-diode model (marked by blue-diamonds). The Shockley parameters are Is = 1e - 12 A, Vt = 0.0258 V

When more accuracy is desired in modelling the diode's turn-on characteristic, the model can be enhanced by doubling-up the standard PWL-model. This model uses two piecewise-linear diodes in parallel, as a way to model a single diode more accurately.

PWL Diode model with 2 branches. The top branch has a lower forward-voltage and a higher resistance. This allows the diode to switch on more gradually, and in this regard more accurately models a real diode. The bottom branch has a higher forward voltage and a lower resistance, thus allowing high current at high voltage
Plot of the I-V Characteristic of this model (marked by red-triangles), as compared to the standard Shockley-diode model (marked by blue-diamonds). The Shockley parameters are Is = 1e - 12 A, Vt = 0.0258 V

## Small-signal modelling[]

### Resistance[]

Using the Shockley equation, the small-signal diode resistance r D {\displaystyle r_{D}} of the diode can be derived about some operating point () where the DC bias current is I Q {\displaystyle I_{Q}} and the Q-point applied voltage is V Q {\displaystyle V_{Q}} . To begin, the diode small-signal conductance g D {\displaystyle g_{D}} is found, that is, the change in current in the diode caused by a small change in voltage across the diode, divided by this voltage change, namely:

g D = d I d V | Q = I s n ⋅ V T e V Q n ⋅ V T ≈ I Q n ⋅ V T {\displaystyle g_{D}=\left.{\frac {dI}{dV}}\right|_{Q}={\frac {I_{s}}{n\cdot V_{\text{T}}}}e^{\frac {V_{Q}}{n\cdot V_{\text{T}}}}\approx {\frac {I_{Q}}{n\cdot V_{\text{T}}}}} .

The latter approximation assumes that the bias current I Q {\displaystyle I_{Q}} is large enough so that the factor of 1 in the parentheses of the Shockley diode equation can be ignored. This approximation is accurate even at rather small voltages, because the V T ≈ 25 mV {\displaystyle V_{\text{T}}\approx 25\,{\text{mV}}} at 300 K, so V Q / V T {\displaystyle V_{Q}/V_{\text{T}}} tends to be large, meaning that the exponential is very large.

Noting that the small-signal resistance r D {\displaystyle r_{D}} is the reciprocal of the small-signal conductance just found, the diode resistance is independent of the ac current, but depends on the dc current, and is given as

r D = n ⋅ V T I Q {\displaystyle r_{D}={\frac {n\cdot V_{\text{T}}}{I_{Q}}}} .

### Capacitance[]

The charge in the diode carrying current I Q {\displaystyle I_{Q}} is known to be

Q = I Q τ F + Q J {\displaystyle Q=I_{Q}\tau _{F}+Q_{J}} ,

where τ F {\displaystyle \tau _{F}} is the forward transit time of charge carriers: The first term in the charge is the charge in transit across the diode when the current I Q {\displaystyle I_{Q}} flows. The second term is the charge stored in the junction itself when it is viewed as a simple ; that is, as a pair of electrodes with opposite charges on them. It is the charge stored on the diode by virtue of simply having a voltage across it, regardless of any current it conducts.

In a similar fashion as before, the diode capacitance is the change in diode charge with diode voltage:

C D = d Q d V Q = d I Q d V Q τ F + d Q J d V Q ≈ I Q V T τ F + C J {\displaystyle C_{D}={\frac {dQ}{dV_{Q}}}={\frac {dI_{Q}}{dV_{Q}}}\tau _{F}+{\frac {dQ_{J}}{dV_{Q}}}\approx {\frac {I_{Q}}{V_{\text{T}}}}\tau _{F}+C_{J}} ,

where C J = d Q J d V Q {\displaystyle C_{J}={\frac {dQ_{J}}{dV_{Q}}}} is the junction capacitance and the first term is called the, because it is related to the current diffusing through the junction.

## Variation of forward voltage with temperature[]

The Shockley diode equation has an exponential of V D / ( k T / q ) {\displaystyle V_{D}/(kT/q)} , which would lead one to expect that the forward-voltage increases with temperature. In fact, this is generally not the case: as temperature rises, the saturation current I S {\displaystyle I_{S}} rises, and this effect dominates. So as the diode becomes hotter, the forward-voltage (for a given current) decreases.

Here is some detailed, which shows this for a 1N4005 silicon diode. In fact, some silicon diodes are used as temperature sensors; for example, the CY7 series from OMEGA has a forward voltage of 1.02 V in liquid nitrogen (77 K), 0.54 V at room temperature, and 0.29 V at 100 °C.

In addition, there is a small change of the material parameter bandgap with temperature. For LEDs, this bandgap change also shifts their colour: they move towards the blue end of the spectrum when cooled.

Since the diode forward-voltage drops as its temperature rises, this can lead to in bipolar-transistor circuits (base-emitter junction of a acts as a diode), where a change in bias leads to an increase in power-dissipation, which in turn changes the bias even further.

## References[]

1. . Popadic, Miloš; Lorito, Gianpaolo; Nanver, Lis K. (2009).. IEEE Transactions on Electron Devices. 56: 116–125. :.
2. Banwell, T.C.; Jayakumar, A.. Electronics Letters. 36 (4). :.
3. . A.S. Sedra and K.C. Smith (2004). (Fifth ed.). New York: Oxford. Example 3.4 p. 154.  .
4. Kal, Santiram (2004). "Chapter 2". Basic Electronics: Devices, Circuits and IT Fundamentals (Section 2.5: Circuit Model of a P-N Junction Diode ed.). Prentice-Hall of India Pvt.Ltd.  .
5. ^ R.C. Jaeger and T.N. Blalock (2004). (second ed.). McGraw-Hill.  .
6. datasheet

To describe the operating of a PV module, we use the Shockley's simple "one diode" model (primarily designed for a single cell), described, for example, in. This model is based on the following equivalent circuit for decribing a PV cvell:

The model was primarily developed for a single cell. Its generalization to the whole module implies that all cells are considered as rigorously identical.

A more sophisticated model, implying 2 different diodes, is sometimes proposed for the very accurate modelling of a single cell. But in PVsyst, we think that small discrepancies in the cell parameters, inducing internal mismatch, as well as the moderate accuracy of our basic input parameters (usually from manufacturer), make no sense to use it. In the one-diode model the two diodes are considered identical, and the Gamma factor - ranging theoretically from 1 to 2 - defines the mix between them.

This model is well-suited for the description of the Si-crystalline modules, but needs some adaptations for reproducing the. We observed that the CIS technology obeys quite well to this standard model.

The main expression describing the general "one-diode" model is written as:

I  =  Iph  -   Io  [ exp  (q · (V+I·Rs) / ( Ncs·Gamma·k·Tc) ) - 1 ]    -    (V + I·Rs) / Rsh

with :

I        =        Current supplied by the module  [A].

V        =        Voltage at the terminals of the module  [V].

Iph        =        Photocurrent [A], proportional to the irradiance G,  with a correction as function of  Tc  (see below).

ID        =        Diode current, is the product   Io  ·  [exp(     ) -1].

Io        =        inverse saturation current, depending on the temperature [A]  (see expression below).

Rs        =        Series resistance [ohm].

Rsh        =        Shunt resistance [ohm].

q        =        Charge of the electron  =  1.602·E-19 Coulomb

k        =        Bolzmann's constant =  1.381 E-23  J/K.

Gamma=          Diode quality factor, normally between 1 and 2

Ncs        =        Number of cells in series.

Tc        =        Effective temperature of the cells [Kelvin]

The photocurrent varies with irradiance and temperature. The one-diode model assumes that it is perfectly proportional to the irradiance.

Its variation with temperature is low and positive (muISC parameter, of the order of  +0.05 % / °C).

Therefore for any condition, the Iph will be determined with respect to the values given for reference conditions  (Gref, Tref):

Iph =   ( G / Gref ) ·   [  Iph ref  +   muISC    (Tc   -  Tc ref) ]

where        G and Gref          =         effective and reference irradiance [W/m²].

Tc and Tc ref        =        effective and reference cell's temperature [°K].

muISC                   =        temperature coefficient of the photocurrent (or short-circuit current).

The diode's reverse saturation current is supposed to vary with the temperature according to the expression:

Io  =  Io ref    ( Tc / Tc ref   )3  ·   exp  [ ( q · EGap / Gamma · k)  ·  ( 1 / Tc ref  -   1 / Tc ) ]

 where EGap = Gap's energy of the material (1.12  eV  for cristalline Si, 1.03 eV  for CIS, 1.7 eV for amorphous silicon, 1.5 eV for CdTe).
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