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Model NPN bipolar transistor using enhanced Ebers-Moll equations
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:
$$\begin{array}{l}{I}_{C}=IS\left[\left({e}^{q{V}_{BE}/(k{T}_{m1})}-{e}^{q{V}_{BC}/(k{T}_{m1})}\right)\left(1-\frac{{V}_{BC}}{{V}_{A}}\right)-\frac{1}{{\beta}_{R}}\left({e}^{q{V}_{BC}/(k{T}_{m1})}-1\right)\right]\\ {I}_{B}=IS\left[\frac{1}{{\beta}_{F}}\left({e}^{q{V}_{BE}/(k{T}_{m1})}-1\right)+\frac{1}{{\beta}_{R}}\left({e}^{q{V}_{BC}/(k{T}_{m1})}-1\right)\right]\end{array}$$
Where:
I_{B} and I_{C} are base and collector currents, defined as positive into the device.
IS is the saturation current.
V_{BE} is the base-emitter voltage and V_{BC} is the base-collector voltage.
β_{F} is the ideal maximum forward current gain BF
β_{R} is the ideal maximum reverse current gain BR
V_{A} 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).
T_{m1} 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 qV_{BC} / (kT_{m1}) > 40 or qV_{BE} / (kT_{m1}) > 40, the corresponding exponential terms in the equations are replaced with (qV_{BC} / (kT_{m1}) – 39)e^{40} and (qV_{BE} / (kT_{m1}) – 39)e^{40}, respectively. This helps prevent numerical issues associated with the steep gradient of the exponential function e^{x} at large values of x. Similarly, if qV_{BC} / (kT_{m1}) < –39 or qV_{BE} / (kT_{m1}) < –39 then the corresponding exponential terms in the equations are replaced with (qV_{BC} / (kT_{m1}) + 40)e^{–39} and (qV_{BE} / (kT_{m1}) + 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.
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, T_{m1}, is replaced with the simulation temperature, T_{s}. The saturation current, IS, and the forward and reverse gains (β_{F} and β_{R}) become a function of temperature according to the following equations:
$$I{S}_{Ts}=I{S}_{Tm1}\cdot {({T}_{s}/{T}_{m1})}^{XTI}\cdot \mathrm{exp}\left(-\frac{EG}{k{T}_{s}}(1-{T}_{s}/{T}_{m1})\right)$$
$${\beta}_{Fs}={\beta}_{Fm1}{\left(\frac{{T}_{s}}{{T}_{m1}}\right)}^{XTB}$$
$${\beta}_{Rs}={\beta}_{Rm1}{\left(\frac{{T}_{s}}{{T}_{m1}}\right)}^{XTB}$$
where:
T_{m1} is the temperature at which the transistor parameters are specified, as defined by the Measurement temperature parameter value.
T_{s} is the simulation temperature.
IS_{Tm1} is the saturation current at the measurement temperature.
IS_{Ts} 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.
β_{Fm1} and β_{Rm1} are the forward and reverse gains at the measurement temperature.
β_{Fs} and β_{Rs} 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 T_{m2}. 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 T_{m1} at which Vbe and Ib, or IS, are measured. The default value is 25 C.
Resistance at the collector. The default value is 0.01 Ω.
Resistance at the emitter. The default value is 1e-4 Ω.
Resistance at the base at zero bias. The default value is 1 Ω.
Parasitic capacitance across the base-collector junction. The default value is 5 pF.
Parasitic capacitance across the base-emitter junction. The default value is 5 pF.
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.
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.
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 T_{m1} (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 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.
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.
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 temperature T_{m2} 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 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 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 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.
Temperature T_{s} at which the device is simulated. The default value is 25 C.
[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.