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N-Channel metal oxide semiconductor field effect transistor using either Shichman-Hodges equation or surface-potential-based model

**Library:**Simscape / Electrical / Semiconductors & Converters

The N-Channel MOSFET block provides two main modeling variants:

Based on threshold voltage — Uses the Shichman-Hodges equation to represent the device. This modeling approach, based on threshold voltage, has the benefits of simple parameterization and simple current-voltage expressions. However, these models have difficulty in accurately capturing transitions across the threshold voltage and lack some important effects, such as velocity saturation. For details, see Threshold-Based Model.

Based on surface potential — Uses the surface-potential equation to represent the device. This modeling approach provides a greater level of model fidelity than the simple square-law (threshold-voltage-based) models can provide. The trade-off is that there are more parameters that require extraction. For details, see Surface-Potential-Based Model.

Together with the thermal port variants (see Thermal Port),
the block therefore provides you with four choices. To select the desired variant, right-click
the block in your model. From the context menu, select **Simscape** >
**Block choices**, and then one of the following options:

**Threshold-based**— Basic model, which represents the device using the Shichman-Hodges equation (based on threshold voltage) and does not simulate thermal effects. This is the default.**Threshold-based with thermal**— Model based on threshold voltage and with exposed thermal port.**Surface-potential-based**— Model based on surface potential. This model does not simulate thermal effects.**Surface-potential-based with thermal**— Thermal variant of the model based on surface potential.

The threshold-based variant of the block uses the Shichman and Hodges equations [1] for an insulated-gate field-effect transistor to represent an N-Channel MOSFET.

The drain-source current, *I _{DS}*, depends on the
region of operation:

In the off region (

*V*<_{GS}*V*), the drain-source current is:_{th}$${I}_{DS}=0$$

In the linear region (0 <

*V*<_{DS}*V*–_{GS}*V*), the drain-source current is:_{th}$${I}_{DS}=K\left(({V}_{GS}-{V}_{th}){V}_{DS}-{V}_{DS}{}^{2}/2\right)\left(1+\lambda \left|{V}_{DS}\right|\right)$$

In the saturated region (0 <

*V*–_{GS}*V*<_{th}*V*), the drain-source current is:_{DS}$${I}_{DS}=(K/2){({V}_{GS}-{V}_{th})}^{2}\left(1+\lambda \left|{V}_{DS}\right|\right)$$

In the preceding equations:

*K*is the transistor gain.*V*is the positive drain-source voltage._{DS}*V*is the gate-source voltage._{GS}*V*is the threshold voltage. For the four terminal parameterization,_{th}*V*is obtained using these equations:_{th}*V*Range_{BS}*V*Equation_{th}$${V}_{BS}\le 0$$ $${V}_{th}={V}_{T0}+\gamma \left(-\sqrt{2{\varphi}_{B}}\right)+\gamma \left(\sqrt{2{\varphi}_{B}-{V}_{BS}}\right)$$ $$0<{V}_{BS}\le 4{\varphi}_{B}$$ $${V}_{th}={V}_{T0}-\frac{\gamma {V}_{BS}}{\sqrt{2{\varphi}_{B}}}$$ $${V}_{BS}>4{\varphi}_{B}$$ $${V}_{th}={V}_{T0}+\gamma \left(-\sqrt{2{\varphi}_{B}}\right)$$ *λ*is the channel modulation.

The block models junction capacitances either by fixed capacitance values, or by tabulated
values as a function of the drain-source voltage. In either case, you can either directly
specify the gate-source and gate-drain junction capacitance values, or let the block derive them
from the input and reverse transfer capacitance values. Therefore, the
**Parameterization** options for charge model on the **Junction
Capacitance** tab are:

`Specify fixed input, reverse transfer and output capacitance`

— Provide fixed parameter values from datasheet and let the block convert the input and reverse transfer capacitance values to junction capacitance values, as described below. This is the default method.`Specify fixed gate-source, gate-drain and drain-source capacitance`

— Provide fixed values for junction capacitance parameters directly.`Specify tabulated input, reverse transfer and output capacitance`

— Provide tabulated capacitance and drain-source voltage values based on datasheet plots. The block converts the input and reverse transfer capacitance values to junction capacitance values, as described below.`Specify tabulated gate-source, gate-drain and drain-source capacitance`

— Provide tabulated values for junction capacitances and drain-source voltage.

Use one of the tabulated capacitance options (```
Specify tabulated input, reverse
transfer and output capacitance
```

or ```
Specify tabulated gate-source,
gate-drain and drain-source capacitance
```

) when the datasheet provides a plot of
junction capacitances as a function of drain-source voltage. Using tabulated capacitance values
gives more accurate dynamic characteristics and avoids the need for interactive tuning of
parameters to fit the dynamics.

If you use the ```
Specify fixed gate-source, gate-drain and drain-source
capacitance
```

or ```
Specify tabulated gate-source, gate-drain and
drain-source capacitance
```

option, the **Junction Capacitance**
tab lets you specify the **Gate-drain junction capacitance**,
**Gate-source junction capacitance**, and **Drain-source junction
capacitance** parameter values (fixed or tabulated) directly. Otherwise, the block
derives them from the **Input capacitance, Ciss**, **Reverse transfer
capacitance, Crss**, and **Output capacitance, Coss** parameter
values. These two parameterization methods are related as follows:

*C*=_{GD}*Crss**C*=_{GS}*Ciss*–*Crss**C*=_{DS}*Coss*–*Crss*

For the four terminals parameterization, the **Input capacitance, Ciss**,
**Reverse transfer capacitance, Crss**, and **Output capacitance,
Coss** are obtained using these equations:

*C*=_{GD}*Crss**C*+_{GS}*C*=_{GB}*Ciss*–*Crss**C*=_{DB}*Coss*–*Crss*

A simplified Meyer's capacitance model is used to describe the gate-source capacitance,
*C _{GS}*, the gate-bulk capacitance,

Gate-bulk and gate-source capacitance change instantaneously.

The two fixed capacitance options (```
Specify fixed input, reverse transfer and
output capacitance
```

or ```
Specify fixed gate-source, gate-drain and
drain-source capacitance
```

) let you model gate junction capacitance as a fixed
gate-source capacitance *C _{GS}* and either a fixed or a
nonlinear gate-drain capacitance

`Gate-drain charge function is nonlinear`

option for the
For instructions on how to map a time response to device capacitance values, see the N-Channel IGBT block reference page. However, this mapping is only approximate because the Miller voltage typically varies more from the threshold voltage than in the case for the IGBT.

**Note**

Because this block implementation includes a charge model, you must model the impedance of the circuit driving the gate to obtain representative turn-on and turn-off dynamics. Therefore, if you are simplifying the gate drive circuit by representing it as a controlled voltage source, you must include a suitable series resistor between the voltage source and the gate.

The surface-potential-based variant of the block provides a greater level of model fidelity than the simple square-law (threshold-voltage-based) model. The surface-potential-based block variant includes the following effects:

Fully nonlinear capacitance model (including the nonlinear Miller capacitance)

Charge conservation inside the model, so you can use the model for charge sensitive simulations

Velocity saturation and channel-length modulation

The intrinsic body diode

Reverse recovery in the body diode model

Temperature scaling of physical parameters

For the thermal variant, dynamic self-heating (that is, you can simulate the effect of self-heating on the electrical characteristics of the device)

This model is a minimal version of the world-standard PSP model (see https://briefs.techconnect.org/papers/introduction-to-psp-mosfet-model/), including only certain effects from the PSP model in order to strike a balance between model fidelity and complexity. For details of the physical background to the phenomena included in this model, see [2].

The basis of the model is Poisson equation:

$$\frac{{\partial}^{2}\psi}{\partial {x}^{2}}+\frac{{\partial}^{2}\psi}{\partial {y}^{2}}=\frac{q{N}_{A}}{{\epsilon}_{Si}}\left[1-\mathrm{exp}\left(\frac{-\psi}{{\varphi}_{T}}\right)+\mathrm{exp}\left(\frac{\psi -2{\varphi}_{B}-{V}_{CB}}{{\varphi}_{T}}\right)\right]$$

$${\varphi}_{T}=\frac{{k}_{B}T}{q}$$

where:

*ψ*is the electrostatic potential.*q*is the magnitude of the electronic charge.*N*is the density of acceptors in the substrate._{A}*ɛ*is the dielectric permittivity of the semiconductor material (for example, silicon)._{Si}*ϕ*is the difference between the intrinsic Fermi level and the Fermi level in the bulk silicon._{B}*V*is the quasi-Fermi potential of the surface layer referenced to the bulk._{CB}*ϕ*is the thermal voltage._{T}*k*is Boltzmann’s constant._{B}*T*is temperature.

Poisson equation is used to derive the surface-potential equation:

$${\left({V}_{GB}-{V}_{FB}-{\psi}_{s}\right)}^{2}={\gamma}^{2}\left[{\psi}_{s}+{\varphi}_{T}\left(\mathrm{exp}\left(\frac{-{\psi}_{s}}{{\varphi}_{T}}\right)-1\right)+{\varphi}_{T}\mathrm{exp}\left(-\frac{2{\varphi}_{B}+{V}_{CB}}{{\varphi}_{T}}\right)\left(\mathrm{exp}\left(\frac{{\psi}_{s}}{{\varphi}_{T}}\right)-1\right)\right]$$

where:

*V*is the applied gate-body voltage._{GB}*V*is the flatband voltage._{FB}*ψ*is the surface potential._{s}*γ*is the body factor,

$$\gamma =\frac{\sqrt{2q{\epsilon}_{Si}{N}_{A}}}{{C}_{ox}}$$

*C*is the capacitance per unit area._{ox}

The block uses an explicit approximation to the surface-potential equation, to avoid the need for numerical solution to this implicit equation.

Once the surface potential is known, the drain current
*I _{D}* is given by

$${I}_{D}=\frac{W{\mu}_{0}}{L{G}_{\Delta L}{G}_{mob}\sqrt{1+{\left({\theta}_{sat}\Delta \psi \right)}^{2}}}\left[-{\overline{Q}}_{inv}\Delta \psi +{\varphi}_{T}\left({Q}_{invL}-{Q}_{inv0}\right)\right]$$

where:

*W*is the device width.*L*is the channel length.*μ*is the low-field mobility._{0}*θ*is the velocity saturation._{sat}*Δψ*is the difference in the surface potential between the drain and the source.*Q*and_{inv0}*Q*are the inversion charge densities at the source and drain, respectively._{invL}$${\overline{Q}}_{inv}$$ is the average inversion charge density across the channel.

*G*is the mobility reduction factor. For more information, see the_{mob}**Surface roughness scattering factor**parameter description in the Main (Surface-Potential-Based Variant) section.*G*is the channel-length modulation._{ΔL}

$${G}_{\Delta L}=1-\frac{\Delta L}{L}=1-\alpha \mathrm{ln}\left[\frac{{V}_{DB}-{V}_{DB,eff}+\sqrt{{\left({V}_{DB}-{V}_{DB,eff}\right)}^{2}+{V}_{p}^{2}}}{{V}_{p}}\right]$$

where:

*α*is the channel-length modulation factor.*V*is the drain-body voltage._{DB}*V*is the drain-body voltage clipped to a maximum value corresponding to velocity saturation or pinch-off (whichever occurs first)._{DB,eff}*V*is the channel-length modulation voltage._{p}

The block computes the inversion charge densities directly from the surface potential.

The block also computes the nonlinear capacitances from the surface potential. Source and drain charge contributions are assigned via a bias-dependent Ward-Dutton charge-partitioning scheme, as described in [3]. These charges are computed explicitly, so this model is charge-conserving. The capacitive currents are computed by taking the time derivatives of the relevant charges. In practice, the charges within the simulation are normalized to the oxide capacitance and computed in units of volts.

The MOSFET gain, β, is given by

$$\beta =\frac{W{\mu}_{0}{C}_{ox}}{L}$$

The threshold voltage for a short-circuited source-bulk connection is approximately given by

$${V}_{T}={V}_{FB}+2{\varphi}_{B}+2{\varphi}_{T}+\gamma \sqrt{2{\varphi}_{B}+2{\varphi}_{T}}$$

where:

2

*ϕ*is the surface potential at strong inversion._{B}

The overall three and four terminal models consist of an intrinsic MOSFET defined by the surface-potential formulation, a body diode, series resistances, and fixed overlap capacitances, as shown in the schematics.

The block models the body diode as an ideal, exponential diode with both junction and diffusion capacitances:

$${I}_{dio}={I}_{s}\left[\mathrm{exp}\left(-\frac{{V}_{DB}}{n{\varphi}_{T}}\right)-1\right]$$

$${C}_{j}=\frac{{C}_{j0}}{\sqrt{1+\frac{{V}_{DB}}{{V}_{bi}}}}$$

$${C}_{diff}=\frac{\tau {I}_{s}}{n{\varphi}_{T}}\mathrm{exp}\left(-\frac{{V}_{DB}}{n{\varphi}_{T}}\right)$$

where:

*I*is the current through the diode._{dio}*I*is the reverse saturation current._{s}*V*is the drain-body voltage._{DB}*n*is the ideality factor.*ϕ*is the thermal voltage._{T}*C*is the junction capacitance of the diode._{j}*C*is the zero-bias junction capacitance._{j0}*V*is the built-in voltage._{bi}*C*is the diffusion capacitance of the diode._{diff}*τ*is the transit time.

The capacitances are defined through an explicit calculation of charges, which are then differentiated to give the capacitive expressions above. The block computes the capacitive diode currents as time derivatives of the relevant charges, similar to the computation in the surface-potential-based MOSFET model.

The default behavior is that dependence on temperature is not modeled, and the device is
simulated at the temperature for which you provide block parameters. To model the dependence on
temperature during simulation, select `Model temperature dependence`

for the **Parameterization** parameter on the **Temperature
Dependence** tab.

**Threshold-Based Model**

For threshold-based variant, you can include modeling the dependence of the transistor static behavior on temperature during simulation. Temperature dependence of the junction capacitances is not modeled, this being a much smaller effect.

When including temperature dependence, the transistor defining equations remain the same.
The gain, *K*, and the threshold voltage,
*V _{th}*, become a function of temperature according to
the following equations:

$${K}_{Ts}={K}_{Tm1}{\left(\frac{{T}_{s}}{{T}_{m1}}\right)}^{BEX}$$

*V _{ths}* =

where:

*T*is the temperature at which the transistor parameters are specified, as defined by the_{m1}**Measurement temperature**parameter value.*T*is the simulation temperature._{s}*K*is the transistor gain at the measurement temperature._{Tm1}*K*is the transistor gain at the simulation temperature. This is the transistor gain value used in the MOSFET equations when temperature dependence is modeled._{Ts}*V*is the threshold voltage at the measurement temperature._{th1}*V*is the threshold voltage at the simulation temperature. This is the threshold voltage value used in the MOSFET equations when temperature dependence is modeled._{ths}*BEX*is the mobility temperature exponent. A typical value of*BEX*is -1.5.*α*is the gate threshold voltage temperature coefficient,*d**V*/_{th}*d**T*.

For the four terminals parameterization, *V _{th}* is
obtained using these equations:

V Range_{BS} | V Equation_{th} |
---|---|

$${V}_{BS}\le 0$$ | $$\frac{d{V}_{th}}{dT}=\frac{d{V}_{T0}}{dT}-\frac{\gamma}{2\sqrt{2{\varphi}_{B}}}\frac{d2{\varphi}_{B}}{dT}+\frac{\gamma}{2\sqrt{2{\varphi}_{B}-{V}_{BS}}}\frac{d2{\varphi}_{B}}{dT}$$ |

$$0<{V}_{BS}\le 4{\varphi}_{B}$$ | $$\frac{d{V}_{th}}{dT}=\frac{d{V}_{T0}}{dT}-\frac{\gamma {V}_{BS}}{4}{\left(2{\varphi}_{B}\right)}^{-\frac{3}{2}}\frac{d2{\varphi}_{B}}{dT}$$ |

$${V}_{BS}>4{\varphi}_{B}$$ | $$\frac{d{V}_{th}}{dT}=\frac{d{V}_{T0}}{dT}-\frac{\gamma}{2\sqrt{2{\varphi}_{B}}}\frac{d2{\varphi}_{B}}{dT}$$ |

Where:

$${\varphi}_{B}=\frac{kT}{q}\mathrm{ln}\left(\frac{{N}_{B}}{{n}_{i}}\right)$$ is the surface potential and $$\frac{d2{\varphi}_{B}}{dT}=\frac{1}{T}\left[2{\varphi}_{B}-\left(\frac{{E}_{g}(0)}{q}+\frac{3kT}{q}\right)\right]$$.

*E*is the extrapolated zero degree band-gap, which is equal to_{g}(0)`1.16`

`eV`

for silicon.*V*is the bulk-source voltage._{BS}

For most MOSFETS, you can use the default value of `-1.5`

for
*BEX*. Some datasheets quote the value for *α*, but most
typically they provide the temperature dependence for drain-source on resistance,
*R _{DS}(on)*. Depending on the block parameterization
method, you have two ways of specifying

If you parameterize the block from a datasheet, you have to provide

*R*at a second measurement temperature. The block then calculates the value for_{DS}(on)*α*based on this data.If you parameterize by specifying equation parameters, you have to provide the value for

*α*directly.

If you have more data comprising drain current as a function of gate-source voltage for
more than one temperature, then you can also use Simulink^{®}
Design Optimization™ software to help tune the values for *α* and
*BEX*.

**Surface-Potential-Based Model**

The surface-potential-based model includes temperature effects on the capacitance characteristics, as well as modeling the dependence of the transistor static behavior on temperature during simulation.

The **Measurement temperature** parameter on the **Main**
tab specifies temperature *T*_{m1} at which the other
device parameters have been extracted. The **Temperature Dependence** tab
provides the simulation temperature, *T*_{s}, and the
temperature-scaling coefficients for the other device parameters. For more information, see
Temperature Dependence (Surface-Potential-Based Variant).

The block has an optional thermal port, hidden by default. To expose the thermal port, right-click the block in your model, and select the appropriate block variant:

For a model based on threshold voltage and with exposed thermal port, select

**Simscape**>**Block choices**>**Threshold-based with thermal**.For a thermal variant of the model based on surface potential, select

**Simscape**>**Block choices**>**Surface-potential-based with thermal**.

This action displays the thermal port **H** on the block icon, and exposes
the **Thermal Port** parameters.

Use the thermal port to simulate the effects of generated heat and device temperature. For
more information on using thermal ports and on the **Thermal Port**
parameters, see Simulating Thermal Effects in Semiconductors.

When modeling temperature dependence for the threshold-based block variant, consider the following:

The block does not account for temperature-dependent effects on the junction capacitances.

When you specify

*R*at a second measurement temperature, it must be quoted for the same working point (that is, the same drain current and gate-source voltage) as for the other_{DS}(on)*R*value. Inconsistent values for_{DS}(on)*R*at the higher temperature will result in unphysical values for_{DS}(on)*α*and unrepresentative simulation results. Typically*R*increases by a factor of about 1.5 for a hundred degree increase in temperature._{DS}(on)You may need to tune the values of

*BEX*and threshold voltage,*V*_{th}, to replicate the*I*–_{DS}*V*relationship (if available) for a given device. Increasing_{GS}*V*moves the_{th}*I*-_{DS}*–V*plots to the right. The value of_{GS}*BEX*affects whether the*I*–_{DS}*V*curves for different temperatures cross each other, or not, for the ranges of_{GS}*V*and_{DS}*V*considered. Therefore, an inappropriate value can result in the different temperature curves appearing to be reordered. Quoting_{GS}*R*values for higher currents, preferably close to the current at which it will operate in your circuit, will reduce sensitivity to the precise value of_{DS}(on)*BEX*.

[1] Shichman, H. and D. A. Hodges. “Modeling and simulation of
insulated-gate field-effect transistor switching circuits.” *IEEE J. Solid State
Circuits*. SC-3, 1968.

[2] Van Langevelde, R., A. J. Scholten, and D. B .M. Klaassen.
"Physical Background of MOS Model 11. Level 1101."* Nat.Lab. Unclassified Report
2003/00239*. April 2003.

[3] Oh, S-Y., D. E. Ward, and R. W. Dutton. “Transient analysis
of MOS transistors.” *IEEE J. Solid State Circuits*. SC-15, pp.
636-643, 1980.