Resistor model including velocity saturation, tolerance, operational limits, fault behavior, and noise
Simscape / Electronics / Passive Devices
The Diffusion Resistor block represents a resistor with velocity saturation, while letting you model the following effects:
You can turn these modeling options on and off independently of each other.
In its simplest form, the resistance of the Diffusion Resistor block is:
$$R={R}_{0}\left(1-{p}_{2}-{p}_{3}+{p}_{2}\sqrt{1+{\left({\theta}_{2}{v}_{pn}\right)}^{2}}+{p}_{3}\sqrt[3]{1+{\left|{\theta}_{3}{v}_{pn}\right|}^{3}}\right)$$
where:
R_{0} is zero-bias resistance.
p_{2} and p_{3} are the quadratic and linear voltage coefficients, respectively.
θ_{2} and θ_{3} are inverse voltages for quadratic and linear voltage activation, respectively.
v_{pn} is applied voltage across the resistor.
At low bias,
$$R\approx {R}_{0}\left(1+\frac{{p}_{2}{\theta}_{2}^{2}{v}_{pn}^{2}}{2}\right)$$
and therefore p_{2} and θ_{2} determine the low-bias quadratic behavior of the resistor.
At high bias,
$$R\approx {R}_{0}\left(1-{p}_{2}-{p}_{3}+\left|{v}_{pn}\right|\left({p}_{2}{\theta}_{2}+{p}_{3}{\theta}_{3}\right)\right)$$
and therefore p_{3} and θ_{3} impact only the high-bias linear behavior of the resistor.
You can use the voltage-dependence of the resistance to model velocity saturation in a diffused resistor. For sufficiently high voltage,
$${i}_{sat}=\frac{1}{{R}_{0}\left({p}_{2}{\theta}_{2}+{p}_{3}{\theta}_{3}\right)}$$
where i_{sat} is saturation current.
The simplified parameterization model assumes that the quadratic and linear coefficients are the same. This is one of the recommended assumptions for the r2_cmc model, as a reasonable initial guess when performing parameter extraction (see the r2_cmc documentation at https://projects.si2.org/cmc_index.php). With this assumption, it is possible to define two new parameters, Critical voltage and Corner voltage, which provide a simpler means for parameterizing models:
$$\begin{array}{l}{p}_{2}={p}_{3}=\frac{{v}_{co}}{2{v}_{crit}}\\ {\theta}_{2}={\theta}_{3}=\frac{1}{2{v}_{co}}\end{array}$$
where:
v_{crit} is critical voltage.
v_{co} is corner voltage.
At high voltage,
$$\frac{dR}{d{v}_{pn}}\approx \frac{{R}_{0}}{{v}_{crit}}$$
and therefore, critical voltage is the reciprocal of the slope of the increase of R/R_{0} with voltage.
With this parameterization, the saturation current is
$${i}_{sat}=\frac{{v}_{crit}}{{R}_{0}}$$
You can apply tolerances to the nominal value you provide for the Resistance parameter. Datasheets typically provide a tolerance percentage for a given resistor type. The table shows how the block applies tolerances and calculates resistance based on the selected Tolerance application option.
Option | Resistance Value |
---|---|
| R_{0} |
| Uniform distribution: R_{0} ·
(1 – tol + 2· tol· Gaussian
distribution: R_{0} ·
(1 + tol · |
| R_{0} · (1 + tol ) |
| R_{0} · (1 – tol ) |
In the table,
R_{0} is the Resistance parameter value, nominal zero-bias resistance.
tol is fractional tolerance, Tolerance (%) /100.
nSigma is the value you provide for the Number of standard deviations for quoted tolerance parameter.
rand
and randn
are standard MATLAB^{®} functions
for generating uniform and normal distribution random numbers.
You can specify operating limits in terms of power and maximum working voltage. For the thermal variant of the block (see Thermal Port), you can also specify operating limits in terms of temperature.
When an operating limit is exceeded, the block can either generate a warning or stop the simulation with an error. For more information, see the Operating Limits parameters section.
The Diffusion Resistor block allows you to model an electrical fault as an instantaneous change in resistance. The block can trigger fault events:
At a specific time
When a current limit is exceeded for longer than a specific time interval
You can enable or disable these trigger mechanisms separately, or use them together if more than one trigger mechanism is required in a simulation. When more than one mechanism is enabled, the first mechanism to trigger the fault takes precedence. In other words, component fails no more than once per simulation.
When the resistor fails, its resistance is changed to the value you specify for the Faulted zero-voltage resistance parameter. You can also choose whether to issue an assertion when a fault occurs, by using the Reporting when a fault occurs parameter. The assertion can take the form of a warning or an error. By default, the block does not issue an assertion.
The Diffusion Resistor block can generate thermal noise
current. If you set the Noise mode parameter to
Enabled
, then the block includes a noise current
source connected in parallel to the diffusion resistor.
If the sampling time is h, then the thermal noise is given by:
$${i}_{N}=\sqrt{2kT/R}\frac{N\left(0,1\right)}{\sqrt{h}}$$
where:
k is the Boltzmann constant, 1.3806504e-23 J/K.
T is temperature.
R is resistance.
N is a Gaussian random number with zero mean and standard deviation of one.
2kT/R is the double-sided thermal noise power distribution (the single-sided equivalent is 4kT/R).
The block generates Gaussian noise by using the Random Number source in the Simscape™ Foundation library. You can control the random number seed by setting the Repeatability parameter:
Not repeatable
—
Every time you simulate your model, the block resets the random seed
using the MATLAB random number generator:
seed = randi(2^32-1);
Repeatable
— The block automatically generates a seed value
and stores it inside the block, to always start the simulation with the same
random number. This auto-generated seed value is set when you add a
Diffusion Resistor block from the block
library to the model. When you make a new copy of the
Diffusion Resistor block from an existing
one in a model, a new seed value is generated. The block sets the value
using the MATLAB random number generator command shown above.
Specify seed
— If
you select this option, the additional Seed parameter
lets you directly specify the random number seed value.
The block has an optional thermal port, hidden by default. To expose the thermal port, right-click the block in your model, and then from the context menu select Simscape > Block choices > Show thermal port. This action displays the thermal port H on the block icon, and adds the Thermal tab and the Variables tab to the block dialog box.
Use the Thermal tab to specify how the resistance value changes with temperature and to set the thermal mass. Use the Variables tab to set the initial temperature target.
For the thermal variant, the defining equation for the resistance is augmented with additional temperature scaling:
$$R={R}_{0}\left(1+{T}_{C1}^{eff}\Delta T+{T}_{C2}^{eff}{\left(\Delta T\right)}^{2}\right)\left(1-{p}_{2}-{p}_{3}+{p}_{2}\sqrt{1+{\left({\theta}_{2}{v}_{pn}\right)}^{2}}+{p}_{3}\sqrt[3]{1+{\left|{\theta}_{3}{v}_{pn}\right|}^{3}}\right)$$
where $${T}_{C1}^{eff}$$ and $${T}_{C2}^{eff}$$ are the linear and quadratic temperature scaling coefficients, respectively.
$$\Delta T={T}_{sim}-{T}_{meas}$$
where:
T_{sim} is simulation temperature.
T_{meas} is measurement temperature.
With the thermal port exposed, the generated noise uses the temperature at the thermal port when determining the instantaneous noise value. Exposing the thermal port also extends the options on the Operating Limits tab as follows:
The Power rating parameter becomes temperature dependent. You define a temperature up to which the full power rating is available, plus a higher temperature for which the power rating is reduced to zero. It is assumed that the power rating decreases linearly with temperature between these two values.
An additional parameter, Operating temperature range, [Tmin Tmax], lets you define the valid temperature range for block operation.
Use the Variables section of the block interface to set the priority and initial target values for the block variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables (Simscape).
This section appears only for the blocks with exposed thermal port. The Temperature variable lets you specify a high-priority target for the temperature at the start of simulation.
Simulating with noise enabled slows down simulation. Choose the sample time (h) so that noise is generated only at frequencies of interest, and not higher.