# PNP Bipolar Transistor

PNP bipolar transistor using enhanced Ebers-Moll equations

Libraries:
Simscape / Electrical / Semiconductors & Converters

## Description

The PNP Bipolar Transistor block uses a variant of the Ebers-Moll equations to represent an PNP bipolar transistor. The Ebers-Moll equations are based on two exponential diodes plus two current-controlled current sources. The PNP Bipolar Transistor block provides the following enhancements to that model:

• Early voltage effect

• Optional base, collector, and emitter resistances.

• Optional fixed base-emitter and base-collector capacitances.

The collector and base currents are [1]:

$\begin{array}{l}{I}_{C}=-IS\left[\left({e}^{-q{V}_{BE}/\left(k{T}_{m1}\right)}-{e}^{-q{V}_{BC}/\left(k{T}_{m1}\right)}\right)\left(1+\frac{{V}_{BC}}{{V}_{A}}\right)-\frac{1}{{\beta }_{R}}\left({e}^{-q{V}_{BC}/\left(k{T}_{m1}\right)}-1\right)\right]\\ {I}_{B}=-IS\left[\frac{1}{{\beta }_{F}}\left({e}^{-q{V}_{BE}/\left(k{T}_{m1}\right)}-1\right)+\frac{1}{{\beta }_{R}}\left({e}^{-q{V}_{BC}/\left(k{T}_{m1}\right)}-1\right)\right]\end{array}$

Where:

• IB and IC are base and collector currents, defined as positive into the device.

• IS is the saturation current.

• VBE is the base-emitter voltage and VBC is the base-collector voltage.

• βF is the ideal maximum current gain BF

• βR is the ideal maximum current gain BR

• VA is the forward Early voltage VAF

• q is the elementary charge on an electron (1.602176e-19 Coulombs).

• k is the Boltzmann constant (1.3806503e-23 J/K).

• Tm1 is the transistor temperature, as defined by the Measurement temperature parameter value.

You can specify the transistor behavior using datasheet parameters that the block uses to calculate the parameters for these equations, or you can specify the equation parameters directly.

If qVBC / (kTm1) > 40 or qVBE / (kTm1) > 40, the corresponding exponential terms in the equations are replaced with (–qVBC / (kTm1) – 39)e40 and (–qVBE / (kTm1) – 39)e40, respectively. This helps prevent numerical issues associated with the steep gradient of the exponential function ex at large values of x. Similarly, if qVBC / (kTm1) < –39 or qVBE / (kTm1) < –39 then the corresponding exponential terms in the equations are replaced with (–qVBC / (kTm1) + 40)e–39 and (–qVBE / (kTm1) + 40)e–39, respectively.

Optionally, you can specify fixed capacitances across the base-emitter and base-collector junctions. You also have the option to specify base, collector, and emitter connection resistances.

### Modeling Capacitance and Charge

You model capacitance and charge using the Base-collector junction capacitance and Base-emitter junction capacitance parameters. You can also represent reverse recovery charge and dynamics using the Total forward transit time and Total reverse transit time parameters. This equation defines the base-collector charge,

${Q}_{\text{BC}}={t}_{\text{R}}{I}_{\text{CE}}+{C}_{\text{BC}}{V}_{\text{BC}},$

where:

• tR is the Total reverse transit time parameter value.

• ICE is the collector-emitter current.

• CBC is the Base-collector junction capacitance parameter value.

• VBC is the base-collector voltage.

This equation defines the base-collector charge and capacitor current,

${I}_{{Q}_{\text{BC}}}=\frac{d{Q}_{\text{BC}}}{dt}.$

This equation defines base-emitter charge,

${Q}_{\text{BE}}={t}_{\text{F}}{I}_{\text{C}}+{C}_{\text{BE}}{V}_{\text{BE}},$

where:

• tF is the Total forward transit time parameter value.

• IC is the collector current.

• CBE is the Base-emitter junction capacitance parameter value.

• VBE is the base-emitter voltage.

This equation defines the base-emitter charge and capacitor current,

${I}_{{Q}_{\text{BE}}}=\frac{d{Q}_{\text{BE}}}{dt}.$

### Modeling Temperature Dependence

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. You can optionally 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 measurement temperature value, Tm1, is replaced with the simulation temperature, Ts. The saturation current, IS, and the forward and reverse gains (βF and βR) become a function of temperature according to the following equations:

$I{S}_{Ts}=I{S}_{Tm1}\cdot {\left({T}_{s}/{T}_{m1}\right)}^{XTI}\cdot \mathrm{exp}\left(-\frac{EG}{k{T}_{s}}\left(1-{T}_{s}/{T}_{m1}\right)\right)$

${\beta }_{Fs}={\beta }_{Fm1}{\left(\frac{{T}_{s}}{{T}_{m1}}\right)}^{XTB}$

${\beta }_{Rs}={\beta }_{Rm1}{\left(\frac{{T}_{s}}{{T}_{m1}}\right)}^{XTB}$

where:

• Tm1 is the temperature at which the transistor parameters are specified, as defined by the Measurement temperature parameter value.

• Ts is the simulation temperature.

• ISTm1 is the saturation current at the measurement temperature.

• ISTs is the saturation current at the simulation temperature. This is the saturation current value used in the bipolar transistor equations when temperature dependence is modeled.

• βFm1 and βRm1 are the forward and reverse gains at the measurement temperature.

• βFs and βRs are the forward and reverse gains at the simulation temperature. These are the values used in the bipolar transistor equations when temperature dependence is modeled.

• EG is the energy gap for the semiconductor type measured in Joules. The value for silicon is usually taken to be 1.11 eV, where 1 eV is 1.602e-19 Joules.

• XTI is the saturation current temperature exponent.

• XTB is the forward and reverse gain temperature coefficient.

• k is the Boltzmann constant (1.3806503e–23 J/K).

Appropriate values for XTI and EG depend on the type of transistor and the semiconductor material used. In practice, the values of XTI, EG, and XTB need tuning to model the exact behavior of a particular transistor. Some manufacturers quote these tuned values in a SPICE Netlist, and you can read off the appropriate values. Otherwise you can determine values for XTI, EG, and XTB by using a datasheet-defined data at a higher temperature Tm2. The block provides a datasheet parameterization option for this.

You can also tune the values of XTI, EG, and XTB yourself, to match lab data for your particular device. You can use Simulink® Design Optimization™ software to help tune the values.

### Thermal Port

You can expose the thermal port to model the effects of generated heat and device temperature. To expose the thermal port, set the Modeling option parameter to either:

• No thermal port — The block does not contain a thermal port and does not simulate heat generation in the device.

• Show thermal port — The block 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.

For more information on using thermal ports and on the Thermal Port parameters, see Simulating Thermal Effects in Semiconductors.

### Variables

To set the priority and initial target values for the block variables before simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Note

To satisfy all your initial targets, do not set the priority to High for more initial targets than the total number of differential variables in the block equations. If you set the Base-collector junction capacitance or Total reverse transit time parameter to a nonzero value and set the Base-emitter junction capacitance or Total forward transit time parameter to a nonzero value, the number of differential variables in the block equations is two. Choose one of these options:

• To set the terminal current initial condition, in the Initial Targets section, set the priority of only the Base current and Collector current variables to High.

• To set the terminal voltage initial condition, in the Initial Targets section, set the priority of only the Collector-emitter voltage and Base-emitter voltage variables to High.

• To set the total charge initial condition, in the Initial Targets section, set the priority of only the Base-collector charge and Base-emitter charge variables to High. (since R2024a)

Use nominal values to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources. One of these sources is the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

### Plot Basic I-V Characteristics

You can plot the basic I-V characteristics of the PNP Bipolar Transistor block without building a complete model. Use the plots to explore the impact of your parameter choices on device characteristics. If you parameterize the block from a datasheet, you can compare your plots to the datasheet to check that you parameterized the block correctly. If you have a complete working model but do not know which manufactured part to use, you can compare your plots to datasheets to help you decide.

To enable this option, set the Modeling option parameter of the PNP Bipolar Transistor block to No thermal port. To plot the basic characteristics, right-click the block and select Electrical > Basic characteristics from the context menu. For more information about the Basic characteristics option, see Plot Basic I-V Characteristics of Semiconductor Blocks.

## Assumptions and Limitations

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

• You may need to use nonzero ohmic resistance and junction capacitance values to prevent numerical simulation issues, but the simulation may run faster with these values set to zero.

## Ports

### Conserving

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Electrical conserving port associated with the transistor base terminal.

Electrical conserving port associated with the transistor collector terminal.

Electrical conserving port associated with the transistor emitter terminal.

Thermal conserving port.

#### Dependencies

To enable this port, set Modeling option to Show thermal port.

## Parameters

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Whether to enable the thermal port of the block and model the effects of generated heat and device temperature.

### Main

Select one of the following methods for block parameterization:

• Specify from a datasheet — Provide parameters that the block converts to equations that describe the transistor. The block calculates the forward Early voltage VAF as Ic/h_oe, where Ic is the Collector current at which h-parameters are defined parameter value, and h_oe is the Output admittance h_oe parameter value [2]. The block sets BF to the small-signal Forward current transfer ratio h_fe value. The block calculates the saturation current IS from the specified Voltage Vbe value and the corresponding Current Ib for voltage Vbe value when Ic is zero. This is the default method.

• Specify using equation parameters directly — Provide equation parameters IS, BF, and VAF.

Small-signal current gain.

#### Dependencies

This parameter is visible only when you select Specify from a datasheet for the Parameterization parameter.

Derivative of the collector current with respect to the collector-emitter voltage for a fixed base current.

#### Dependencies

This parameter is visible only when you select Specify from a datasheet for the Parameterization parameter.

The h-parameters vary with operating point, and are defined for this value of the collector current.

#### Dependencies

This parameter is visible only when you select Specify from a datasheet for the Parameterization parameter.

The h-parameters vary with operating point, and are defined for this value of the collector-emitter voltage.

#### Dependencies

This parameter is visible only when you select Specify from a datasheet for the Parameterization parameter.

Base-emitter voltage when the base current is Ib. The [ Vbe Ib ] data pair must be quoted for when the transistor is in the normal active region, that is, not in the saturated region.

#### Dependencies

This parameter is visible only when you select Specify from a datasheet for the Parameterization parameter.

Base current when the base-emitter voltage is Vbe. The [ Vbe Ib ] data pair must be quoted for when the transistor is in the normal active region, that is, not in the saturated region.

#### Dependencies

This parameter is visible only when you select Specify from a datasheet for the Parameterization parameter.

Ideal maximum forward current gain.

#### Dependencies

This parameter is visible only when you select Specify using equation parameters directly for the Parameterization parameter.

Transistor saturation current.

#### Dependencies

This parameter is visible only when you select Specify using equation parameters directly for the Parameterization parameter.

In the standard Ebers-Moll equations, the gradient of the Ic versus Vce curve is zero in the normal active region. The additional forward Early voltage term increases this gradient. The intercept on the Vce-axis is equal to –VAF when the linear region is extrapolated.

#### Dependencies

This parameter is visible only when you select Specify using equation parameters directly for the Parameterization parameter.

Ideal maximum reverse current gain. This value is often not quoted in manufacturer datasheets because it is not significant when the transistor is biased to operate in the normal active region. When the value is not known and the transistor is not to be operated on the inverse region, use the default value of 1.

Temperature Tm1 at which Vbe and Ib, or IS, are measured.

### Ohmic Resistance

Resistance at the collector.

Resistance at the emitter.

Resistance at the base at zero bias.

### Capacitance

Parasitic capacitance across the base-collector junction.

Parasitic capacitance across the base-emitter junction.

Represents the mean time for the minority carriers to cross the base region from the emitter to the collector, and is often denoted by the parameter TF [1].

Represents the mean time for the minority carriers to cross the base region from the collector to the emitter, and is often denoted by the parameter TR [1].

### Temperature Dependence

Select one of the following methods for temperature dependence parameterization:

• None — Simulate at parameter measurement temperature — Temperature dependence is not modeled, or the model is simulated at the measurement temperature Tm1 (as specified by the Measurement temperature parameter on the Main tab). This is the default method.

• Model temperature dependence — Provide a value for simulation temperature, to model temperature-dependent effects. You also have to provide a set of additional parameters depending on the block parameterization method. If you parameterize the block from a datasheet, you have to provide values for a second [ Vbe Ib ] data pair and h_fe at second measurement temperature. If you parameterize by directly specifying equation parameters, you have to provide the values for XTI, EG, and XTB.

Small-signal current gain at second measurement temperature. It must be quoted at the same collector-emitter voltage and collector current as for the Forward current transfer ratio h_fe parameter on the Main tab.

#### Dependencies

This parameter is visible only when you select Specify from a datasheet for the Parameterization parameter on the Main tab.

Base-emitter voltage when the base current is Ib and the temperature is set to the second measurement temperature. The [Vbe Ib] data pair must be quoted for when the transistor is in the normal active region, that is, not in the saturated region.

#### Dependencies

This parameter is visible only when you select Specify from a datasheet for the Parameterization parameter on the Main tab.

Base current when the base-emitter voltage is Vbe and the temperature is set to the second measurement temperature. The [ Vbe Ib ] data pair must be quoted for when the transistor is in the normal active region, that is, not in the saturated region.

#### Dependencies

This parameter is visible only when you select Specify from a datasheet for the Parameterization parameter on the Main tab.

Second temperature Tm2 at which h_fe,Vbe, and Ib are measured.

#### Dependencies

This parameter is visible only when you select Specify from a datasheet for the Parameterization parameter on the Main tab.

Current gain temperature coefficient value.

#### Dependencies

This parameter is visible only when you select Specify using equation parameters directly for the Parameterization parameter on the Main tab.

Energy gap value.

#### Dependencies

This parameter is visible only when you select Specify using equation parameters directly for the Parameterization parameter on the Main tab.

Saturation current temperature coefficient value.

#### Dependencies

This parameter is visible only when you select Specify using equation parameters directly for the Parameterization parameter on the Main tab.

Temperature Ts at which the device is simulated.

## References

[1] G. Massobrio and P. Antognetti. Semiconductor Device Modeling with SPICE. 2nd Edition, McGraw-Hill, 1993.

[2] H. Ahmed and P.J. Spreadbury. Analogue and digital electronics for engineers. 2nd Edition, Cambridge University Press, 1984.

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