Piecewise linear diode with charge dynamics and junction capacitance

Semiconductors / Fundamental Components

The Commutation Diode block augments the Diode block with a model of charge dynamics. For a description of the piecewise linear diode operation that the Commutation Diode block uses, see Diode.

Use the Commutation Diode block in place of the Diode block when you want to specify precisely the charge dynamics of the device as it operates in reverse mode. For example, suppose that your model uses the diode to divert inductive currents from a motor drive or inverter. In this case, precise reverse-mode operation is important and an appropriate time to use the Commutation Diode block.

The Commutation Diode uses a charge model proposed by Lauritzen and Ma [1]. The defining expressions for this charge model are:

$i=\frac{{q}_{E}-{q}_{M}}{{T}_{M}}$ | (1-1) |

$\frac{d{q}_{M}}{dt}+\frac{{q}_{M}}{\tau}-\frac{{q}_{E}-{q}_{M}}{{T}_{M}}=0$ | (1-2) |

$$\begin{array}{cc}{q}_{E}=\left(\tau +{T}_{M}\right)\left({v}_{D}-{v}_{F}(1-RG))/R\right)& if{v}_{D}{v}_{F}\\ {q}_{E}=\left(\tau +{T}_{M}\right)G{v}_{D}& if{v}_{D}\le {v}_{F}\end{array}$$ | (1-3) |

*i*is the diode current.*q*is the junction charge._{E}*q*is the total stored charge._{M}*T*is the transit time._{M}*τ*is the carrier lifetime.*v*is the voltage across the diode._{D}*v*is the diode forward voltage._{F}*R*is the diode on resistance.*G*is the diode off conductance.

This graphic shows a typical reverse-mode current characteristic for a diode device.

where:

*i*is the peak reverse current._{RM}*i*is the starting forward current when measuring_{F}*i*._{RM}*a*is the rate of change of current when measuring*i*._{RM}*t*is the reverse recovery time._{rr}

On the Charge Dynamics tab of the block, you specify characteristics of your diode device. The block uses these values to calculate the diode charge dynamics expressed in equations 1–1, 1–2, and 1–3.

Data sheets for diodes quote values for peak reverse current for an initial forward current and a steady rate of change of current. The data sheet might also provide values for reverse recovery time and total recovery charge.

The block calculates transit time *T _{M}* and
carrier lifetime

During initial current drop in reverse mode, the diode is still on, and the rate of change of current is determined by an external test circuit.

Using Equation 1–1,

${i}_{F}+at=\frac{{q}_{E}-{q}_{M}}{{T}_{M}}.$ | (1-4) |

Substituting Equation 1–4 into Equation 1–2,

$\frac{d{q}_{M}}{dt}+\frac{{q}_{M}}{\tau}={i}_{F}+at.$ | (1-5) |

Solving Equation 1–5 for *q _{M}*,

${q}_{M}={i}_{F}\tau -a{\tau}^{2}+\frac{k}{\mathrm{exp}\left(\frac{t}{\tau}\right)}+a\tau t,$ | (1-6) |

When *t* is zero, *i* = *i _{F}* and

Substituting these relationships into Equation 1–6 and
solving the equation gives *k* = *aτ ^{2}*.

Therefore,

${q}_{M}={i}_{F}\tau +a{\tau}^{2}\left(\frac{1}{\mathrm{exp}\left(\frac{t}{\tau}\right)}-1\right)+a\tau t.$ | (1-7) |

Substituting these values into Equation 1–1,

${i}_{RM}=\frac{-{q}_{M}}{{T}_{M}}.$ | (1-8) |

$-{T}_{M}{i}_{RM}={i}_{F}\tau +a{\tau}^{2}\left(\frac{1}{\mathrm{exp}\left(\frac{{t}_{s}}{\tau}\right)}-1\right)+a\tau {t}_{s}.$ | (1-9) |

Expressing time *t _{s}* in
terms of

${t}_{s}=\frac{{i}_{RM}-{i}_{F}}{a}.$ | (1-10) |

Consider the diode recovery, that is, when *t* > *t _{s}*.
The diode is reverse biased, and current and junction charge are effectively
zero.

The current is defined by

$i={i}_{RM}\text{exp[}\frac{-(t-{t}_{s})}{{\tau}_{rr}}],$ | (1-11) |

where

$\frac{1}{{\tau}_{rr}}=\frac{1}{\tau}+\frac{1}{{T}_{M}}.$ | (1-12) |

The block now relates the expression in Equation 1–12
to the reverse recovery time *t _{rr}*.

When $t=\frac{{i}_{RM}}{a}+{t}_{rr},$ the current is $\frac{{i}_{RM}}{10}.$

Therefore,

$\mathrm{exp}\left(-\frac{t-{t}_{s}}{{\tau}_{rr}}\right)=0.1$ | (1-13) |

${t}_{rr}={\tau}_{rr}\mathrm{log}\left(10\right)+\frac{{i}_{RM}}{a}.$ | (1-14) |

The block uses equations 1–9 and 1–14 to calculate
values for *T _{M}* and

In addition to allowing you to specify reverse recovery time *t _{rr}* directly,
the block supports two alternative parameterizations. The block can
derive

Reverse recovery time stretch factor

*λ*Reverse recovery charge

*Q*, when the data sheet specifies this value instead of the reverse recovery time._{rr}

The relationship between reverse recovery time stretch factor *λ* and *t _{rr}* is
expressed by the equation

$\lambda =\frac{{t}_{rr}a}{{i}_{RM}}.$

Therefore, a typical value for *λ* is
3. *λ* must be greater than 1.

Reverse recovery charge *Q _{rr}* is
the integral over time of the reverse current from the point where
the current goes negative until it decays back to zero.

The initial charge, to time t_{s} (as shown
in the figure), is expressed by the equation

$${Q}_{s}=\frac{1}{2}\left(-{i}_{RM}\right)\frac{{i}_{RM}}{a}.$$ | (1-15) |

Integrating Equation 1–11 gives the charge between times
t_{s} and inf. This charge is equal to

${\tau}_{rr}{i}_{RM}.$

Therefore, total reverse recovery charge is given by the equation

${Q}_{rr}=-\frac{{i}_{RM}^{2}}{2a}+{\tau}_{rr}{i}_{RM}.$ | (1-16) |

Rearranging Equation 1–16 to solve for *τ _{rr}* and
substituting the result into Equation 1–14 gives an equation
that expresses

${t}_{rr}=\left(\frac{{Q}_{rr}}{{i}_{RM}}+\frac{{i}_{RM}}{2a}\right)\mathrm{log}\left(10\right)+\frac{{i}_{RM}}{a}.$

The block provides a thermal modeling variant. To select a variant,
right-click the block in your model. From the context menu, select **Simscape** > **Block
choices**, and then one of these variants:

**No thermal port**— This variant does not simulate heat generation in the device. This variant is the default.**Show thermal port**— This variant contains a thermal port that allows you to model the heat that conduction losses generate. For numerical efficiency, the thermal state does not affect the electrical behavior of the block. The thermal port is hidden by default. When you select a thermal variant of the block, the thermal port appears.

`+`

Electrical conserving port associated with the diode positive terminal

`-`

Electrical conserving port associated with the diode negative terminal

`H`

Thermal conserving port. The thermal port is optional and is hidden by default. To enable this port, select a variant that includes a thermal port.

**Forward voltage**Minimum voltage required across the

`+`

and`-`

block ports for the gradient of the diode I-V characteristic to be 1/R_{on}, where R_{on}is the value of**On resistance**. The default value is`0.8`

`V`

.**On resistance**Rate of change of voltage versus current above the

**Forward voltage**. The default value is`0.001`

`Ohm`

.**Off conductance**Conductance of the reverse-biased diode. The default value is

`1e-5`

`1/Ohm`

.**Number of series diodes**The number of diodes connected in series between the

`+`

and`-`

block ports. Each diode has the**Forward voltage**,**On resistance**, and**Off conductance**that you specify. The default value is`1`

.**Number of parallel diodes**The number of parallel diodes, or number of parallel paths formed by series-connected diodes, between the

`+`

and`-`

block ports. Each diode has the**Forward voltage**,**On resistance**, and**Off conductance**that you specify. The default value is`1`

.

**Junction capacitance**Diode junction capacitance. The default value is

`50`

`nF`

.**Peak reverse current, iRM**Peak reverse current measured by an external test circuit. This value must be less than zero. The default value is

`-235`

`A`

.**Initial forward current when measuring iRM**Initial forward current when measuring peak reverse current. This value must be greater than zero. The default value is

`300`

`A`

.**Rate of change of current when measuring iRM**Rate of change of current when measuring peak reverse current. This value must be less than zero. The default value is

`-50`

`A/μs`

.**Reverse recovery time parameterization**Determines how you specify reverse recovery time in the block. The default value is

`Specify reverse recovery time directly`

.If you select

`Specify stretch factor`

or`Specify reverse recovery charge`

, you specify a value that the block uses to derive the reverse recovery time. For more information on these options, see Alternatives to Specifying trr Directly.**Reverse recovery time, trr**Interval between the time when the current initially goes to zero (when the diode turns off) and the time when the current falls to less than 10% of the peak reverse current. The default value is

`15`

`μs`

.This parameter is visible only if you set

**Reverse recovery time parameterization**to`Specify reverse recovery time directly`

.The value of the

**Reverse recovery time, trr**parameter must be greater than the value of the**Peak reverse current, iRM**parameter divided by the value of the**Rate of change of current when measuring iRM**parameter.**Reverse recovery time stretch factor**Value that the block uses to calculate

**Reverse recovery time, trr**. This value must be greater than`1`

. The default value is`3`

.This parameter is visible only if you set

**Reverse recovery time parameterization**to`Specify stretch factor`

.Specifying the stretch factor is an easier way to parameterize the reverse recovery time than specifying the reverse recovery charge. The larger the value of the stretch factor, the longer it takes for the reverse recovery current to dissipate.

**Reverse recovery charge, Qrr**Value that the block uses to calculate

**Reverse recovery time, trr**. Use this parameter if the data sheet for your diode device specifies a value for the reverse recovery charge instead of a value for the reverse recovery time.The reverse recovery charge is the total charge that continues to dissipate when the diode turns off. The value must be less than $$-\frac{{i}^{2}{}_{RM}}{2a},$$

where:

*i*is the value specified for_{RM}**Peak reverse current, iRM**.*a*is the value specified for**Rate of change of current when measuring iRM**.

The default value is

`1500`

`μAs`

.The parameter is visible only if you set

**Reverse recovery time parameterization**to`Specify reverse recovery charge`

.

[1] Lauritzen, P.O. & C.L. Ma, “A Simple Diode
Model with Reverse Recovery.” *IEEE ^{®} Transactions
on Power Electronics*. Vol. 6, No. 2, 1991, pp. 188–191.

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