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# NPN Bipolar Transistor

Model NPN bipolar transistor using enhanced Ebers-Moll equations

## Library

Semiconductor Devices

## Description

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

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 forward current gain BF

• βR is the ideal maximum reverse 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 parasitic fixed capacitances across the base-emitter and base-collector junctions. You also have the option to specify base, collector, and emitter connection resistances.

### 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:

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

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 port tab to the block dialog box.

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 tab parameters, see Simulating Thermal Effects in Semiconductors.

## Basic Assumptions and Limitations

The NPN Bipolar Transistor model has the following 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.

## Dialog Box and Parameters

### Main Tab

Parameterization

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 [1]. 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.

Forward current transfer ratio h_fe

Small-signal current gain. This parameter is only visible when you select Specify from a datasheet for the Parameterization parameter. The default value is 100.

Derivative of the collector current with respect to the collector-emitter voltage for a fixed base current. This parameter is only visible when you select Specify from a datasheet for the Parameterization parameter. The default value is 5e-05 1/Ω.

Collector current at which h-parameters are defined

The h-parameters vary with operating point, and are defined for this value of the collector current. This parameter is only visible when you select Specify from a datasheet for the Parameterization parameter. The default value is 1 mA.

Collector-emitter voltage at which h-parameters are defined

The h-parameters vary with operating point, and are defined for this value of the collector-emitter voltage. This parameter is only visible when you select Specify from a datasheet for the Parameterization parameter. The default value is 5 V.

Voltage Vbe

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. This parameter is only visible when you select Specify from a datasheet for the Parameterization parameter. The default value is 0.55 V.

Current Ib for voltage Vbe

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. This parameter is only visible when you select Specify from a datasheet for the Parameterization parameter. The default value is 0.5 mA.

Forward current transfer ratio BF

Ideal maximum forward current gain. This parameter is only visible when you select Specify using equation parameters directly for the Parameterization parameter. The default value is 100.

Saturation current IS

Transistor saturation current. This parameter is only visible when you select Specify using equation parameters directly for the Parameterization parameter. The default value is 1e-14 A.

Forward Early voltage VAF

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. This parameter is only visible when you select Specify using equation parameters directly for the Parameterization parameter. The default value is 200 V.

Reverse current transfer ratio BR

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.

Measurement temperature

Temperature Tm1 at which Vbe and Ib, or IS, are measured. The default value is 25 C.

### Ohmic Resistance Tab

Collector resistance RC

Resistance at the collector. The default value is 0.01 Ω.

Emitter resistance RE

Resistance at the emitter. The default value is 1e-4 Ω.

Zero bias base resistance RB

Resistance at the base at zero bias. The default value is 1 Ω.

### Capacitance Tab

Base-collector junction capacitance

Parasitic capacitance across the base-collector junction. The default value is 5 pF.

Base-emitter junction capacitance

Parasitic capacitance across the base-emitter junction. The default value is 5 pF.

Total forward transit time

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]. The default value is 0 μs.

Total reverse transit time

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]. The default value is 0μs.

### Temperature Dependence Tab

Parameterization

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.

Forward current transfer ratio, h_fe, at second measurement temperature

Small-signal current gain at the second measurement temperature. This parameter is only visible when you select Specify from a datasheet for the Parameterization parameter on the Main tab. 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. The default value is 125.

Voltage Vbe at second measurement temperature

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. This parameter is only visible when you select Specify from a datasheet for the Parameterization parameter on the Main tab. The default value is 0.45 V.

Current Ib for voltage Vbe at second measurement temperature

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. This parameter is only visible when you select Specify from a datasheet for the Parameterization parameter on the Main tab. The default value is 0.5 mA.

Second measurement temperature

Second temperature Tm2 at which h_fe,Vbe, and Ib are measured. This parameter is only visible when you select Specify from a datasheet for the Parameterization parameter on the Main tab. The default value is 125 C.

Current gain temperature coefficient, XTB

Current gain temperature coefficient value. This parameter is only visible when you select Specify using equation parameters directly for the Parameterization parameter on the Main tab. The default value is 0.

Energy gap, EG

Energy gap value. This parameter is only visible when you select Specify using equation parameters directly for the Parameterization parameter on the Main tab. The default value is 1.11 eV.

Saturation current temperature exponent, XTI

Saturation current temperature coefficient value. This parameter is only visible when you select Specify using equation parameters directly for the Parameterization parameter on the Main tab. The default value is 3.

Device simulation temperature

Temperature Ts at which the device is simulated. The default value is 25 C.

## Ports

The block has the following ports:

B

Electrical conserving port associated with the transistor base terminal.

C

Electrical conserving port associated with the transistor collector terminal.

E

Electrical conserving port associated with the transistor emitter terminal.

## Examples

See the NPN Bipolar Transistor Characteristics example.

## 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.