# Section 3: How The New Model Data Were Extracted

## β vs. I_{C} and I_{C} vs. V_{BE} Curves of Model

The best way to see how SPICE models the variation of β with collector current is to look at Gummel plots of the device. A Gummel plot is a plot of ln(I_{C}) and ln(I_{B}) vs. V_{BE} at a collector-base voltage V_{CB} of zero. Figure 8 below shows such a plot for an idealized transistor whose β is constant over all values of I_{C}. This plot was derived from a simulation of a transistor whose SPICE model was created just for this purpose. The SPICE parameters ISE, NE and IKF were omitted for the simulation of Figure 8. Doing so makes the simulated β independent of I_{C}.

In this figure, "log" is the LTspice simulator's notation for natural log, so this plot represents ln(I_{C}) and ln(I_{B}) vs. V_{BE}. At any fixed value of V_{BE}, the difference ln(I_{C}) - ln(I_{B}) is the same as ln(I_{C}/I_{B}), which is just ln(β). That is, ln(β) is the vertical displacement between the two curves at a fixed V_{BE}. Thus β is constant for all I_{C} because ln(β) does not change.

A more interesting plot introduces the SPICE parameters ISE, NE and IKF, which model the reduction of β at very low and very high currents. This is shown below in Figure 9. As in Figure 8, the collector-base voltage V_{CB} is zero.

As in Figure 8, ln(β) is the vertical distance between the two curves at any fixed value of V_{BE}. It can be seen that β is roughly constant for I_{C} values in the middle of the range, and gets smaller at very low and very high I_{C}. For small collector currents, ln(I_{C}) vs. V_{BE} fits a line very well, but deviates from the line at higher collector currents. Conversely, at higher base currents, ln(I_{B}) vs. V_{BE} fits a line very well, but deviates from that line at lower base currents. Therefore at very low values of I_{C}, β gets smaller primarily due to deviations of ln(I_{B}) vs. V_{BE} from a straight line, while at high values of I_{C}, β gets smaller primarily due to deviations of ln(I_{C}) vs. V_{BE} from a straight line.

It's worth noting how the Early effect causes the Gummel plots to change. To illustrate this, Figure 10 shows plots of ln(I_{C}) and ln(I_{B}) vs. V_{BE} with V_{CB} as a parameter. For this test case, the Early Voltage V_{A} is 20 Volts, and V_{CB} is stepped from 0 to 50 Volts in 5 Volt steps.

There are a number of very interesting observations about the Early effect that can be made from Figure 10. To simplify things, assume that the junction temperature of the device does not change with the varying currents and voltages. This might correspond to the data having been taken very fast, as would be the case if the measurement were performed in pulsed mode using standard equipment for that purpose.

The first observation is that variations in V_{CB} do not affect the relationship between I_{B} and V_{BE} at all. Only the relationship between I_{C} and V_{BE} is affected. Therefore, if I_{B} is fixed, V_{BE} does not change either, even with variations in V_{CB}. This leads to some observations about the common-emitter characteristic curves of a transistor displayed on a curve tracer. The curves in question are plots of I_{C} vs. V_{CE} with I_{B} as a parameter, such as Figure 1. In contrast to Figure 1, we'll assume the device is in the active region. The Early effect is often described in textbooks in terms of the "tilt" of each characteristic curve at a fixed value of I_{B}. But since a fixed value of I_{B} corresponds to a fixed value of V_{BE}, we can relate the characteristic curve behavior back to the Gummel plot of Figure 10. Specifically, varying V_{CE} at a fixed I_{B} on the curve tracer causes a variation in I_{C}. This corresponds exactly to holding V_{BE} fixed in the Gummel plot of Figure 10 while varying V_{CB}. Since the vertical displacement between the ln(I_{C}) and ln(I_{B}) curves at a fixed V_{BE} and V_{CB} is exactly ln(β), we can interpret the variation of V_{CB} at a fixed V_{BE} (fixed I_{B}) as modulation of β by V_{CB}. When measured on a curve tracer using common-emitter mode, this corresponds to a change in I_{C} proportional to the change in β (since I_{B} is fixed). This also explains why the common-base characteristic curves are so much flatter than the common-emitter curves on a curve tracer. For the common-base characteristic curves, each curve corresponds to a constant I_{E}. When V_{CB} varies, β varies according to Figure 10. But since I_{C} = I_{E} * β / (β + 1), the change in I_{C} is much less than it would be in common-emitter mode.

The equations that determine the behavior shown in Figure 10 are as follows:

(1) | $${I}_{B}={I}_{S}\left[\frac{1}{{\beta}_{F}}\left({e}^{\frac{q{V}_{BE}}{{n}_{F}kT}}-1\right)-\frac{1}{{\beta}_{R}}\right]+{I}_{SE}\left({e}^{\frac{q{V}_{BE}}{{n}_{EL}kT}}-1\right)$$ |

(2) | $${I}_{C}={I}_{S}\left[\frac{1}{{q}_{b}}\left({e}^{\frac{q{V}_{\mathit{BE}}}{{n}_{F}kT}}\right)+\frac{1}{{\beta}_{R}}\right]+{I}_{\mathit{SC}}$$ |

where

(3) | $${q}_{b}=\frac{{q}_{1}}{2}\left(1+\sqrt{1+4{q}_{2}}\right)$$ |

and

(4) | $${q}_{1}={\left(1-\frac{{V}_{\mathit{BC}}}{{V}_{A}}-\frac{{V}_{\mathit{BE}}}{{V}_{B}}\right)}^{-1}$$ |

and

(5) | $${q}_{2}=\frac{{I}_{S}}{{I}_{\mathit{KF}}}\left({e}^{\frac{q{V}_{\mathit{BE}}}{{n}_{F}kT}}-1\right)-\frac{{I}_{S}}{{I}_{\mathit{KR}}}$$ |

The correspondence between the variables in equations (1)-(5) and the SPICE parameter names is given in Table 1 below.

Variable Name | SPICE Parameter Name |
---|---|

I_{S} |
IS |

β_{F} |
BF |

n_{F} |
NF |

β_{R} |
BR |

I_{SE} |
ISE |

n_{EL} |
NE |

I_{SC} |
ISC |

V_{A} |
VAF |

V_{B} |
VAR |

I_{KF} |
IKF |

I_{KR} |
IKR |

Table 1. SPICE Names for DC Parameters

In equation (2), q_{b} is given by (3), (4) and (5). The bend of the curve of ln(I_{C}) vs. V_{BE} in the high current region (high-level injection) is modeled by IKF. Specifically, IKF represents the "knee" of the curve where the asymptotic straight line behaviors in the low-current region and the high-level injection region intersect. Only the knee can be varied. The slope of the straight line in the high-level injection region is fixed in the model. The parameter q_{b} models the combined contributions of the Early effect via q_{1} and high-level injection via q_{2}. Note that the reverse Early voltage plays a part in the determination of I_{C} vs. V_{BE} in the forward region. For the base current curves, NE controls the slope of ln(I_{B}) vs. V_{BE} in the low-current region, while the combination of ISE and NE controls the knee of the curve. The asymptotic maximum β is determined by BF. It should be noted that in cases where there is significant overlap between the non-ideal behaviors of ln(I_{C}) vs. V_{BE} and ln(I_{B}) vs. V_{BE}, BF can be much higher than the actual measured or simulated maximum β.

### Terminal resistors RE, RC and RB

SPICE includes bulk resistances RE, RC and RB connecting the intrinsic emitter, collector and base respectively with their associated external terminal. It should be noted that the voltages V_{CB} and V_{BE} in the equations above refer to the *intrinsic* rather than the *external* nodes of the corresponding name. When converting measured voltages to their intrinsic counterparts, it's important to subtract off the voltage drops of these bulk resistances before substituting values into equations such as (1) and (2). More will be mentioned about this in a following subsection. The values of RE, RC and RB were computed as follows.

### RC value

Reference [2] makes the distinction between the RC value measured in saturated mode and linear mode. The saturated-mode RC is generally much less than its linear-mode counterpart. The recommended method of calculating RC is to use linear mode, by taking the reciprocal of the slope of I_{C} vs. V_{CE} with I_{B} adjusted to force β to unity [2]. Unfortunately, quasi-saturation causes problems with this approach. The value of RC in the ON Semiconductor model for the MJL3281A is around 0.2 Ohms, which causes problems as follows. The simulated plot of β vs. I_{C} at V_{CE} = 5 Volts shows a downward slope not present in the measured data. Even after removing IKF from the model to nominally prevent the simulated data from showing a reduction of β at high currents, the incorrect downward slope remains. This turns out to be an undesired interaction between the too-high value of RC and the Early effect. As collector current increases, the voltage drop across RC increases, decreasing the intrinsic V_{CB} as a result. This decrease in intrinsic V_{CB} causes the simulated value of β to decrease due to Early effect modeling. To find an appropriate RC value, simulated plots of the transistor characteristic curves were made, and the value of RC was adjusted in an attempt to match the slopes of the simulated and measured characteristic curves near saturation. This resulted in an RC value for both the MJL3281A and MJL1302A of 0.06 Ohms. After substituting this smaller value of RC back into the MJL3281A model and running the simulated β vs. I_{C} curves again, the undesired decrease of β with increasing I_{C} was eliminated.

### RE value

The datasheet doesn't provide the information necessary to calculate RE, so the modified model uses the default value of zero.

### RB value

SPICE uses several parameters to model the resistance between the base region and the base terminal. These are RB, RBM and IRB. Their purpose is to model a base resistance that varies with base current. RB is the zero-bias base resistance. RBM is the minimum base resistance at high currents. IRB is the current where the base resistance falls halfway to its minimum value. Datasheet measurements do not give enough information to determine how this base resistance varies with current, so RBM and IRB are omitted from the modified model. The result is a constant base resistance of value RB. Since the MJL3281A and MJL1302A can operate at high collector currents and β gets smaller as current increases, the base current becomes large enough that the voltage drop across RB cannot be neglected. Therefore determining RB is an integral part of fitting SPICE parameters to the measured curves of V_{BE} vs. I_{C} from the datasheet. This will be discussed in more detail in the next subsection. One important thing should be noted about RB however. It causes an undesired interaction between the extraction procedures needed to determine the SPICE parameters that relate I_{C} and I_{B} to the *external* V_{BE}. Fitting the measured I_{C} vs. V_{BE} to the model requires determining RB in addition to IS, NF, VA, and IKF. Once this information is obtained, BF, ISE and NE determine how I_{B} varies with the intrinsic V_{BE} in accordance with equation (1). This matches up the measured and simulated β vs. collector current. However, modifying the values of BF, ISE and NE affects how the base current varies with the intrinsic V_{BE}. This change, combined with the presence of RB, causes the curve of I_{C} vs. the *external* V_{BE} to change. For this reason, it's often necessary to go back to the measured I_{C} vs. V_{BE} curves and find new values of IS, NF, VA and IKF if the parameters that determine the base current are changed. This in turn may require yet another recalculation of BF, ISE and NE to match the simulated and measured β vs. I_{C} again. Thus the presence of RB turns what would ordinarily be a direct procedure into an iterative one. A step-by-step algorithm for getting these parameters will be outlined in the next subsection.

## I_{C} vs. V_{BE} and β vs. I_{C}

Obtaining the model parameters that control I_{C} vs. V_{BE} and β vs. I_{C} involves two procedures.

- Manual adjustment of model parameters to get reasonably good agreement between simulated and measured data
- An optimization procedure that uses the Excel solver to adjust the model parameters to get a much more accurate solution

Both procedures use the simulation setup shown below in Figure 11.

This simulation setup sweeps the emitter current from 100mA to 21A. The VCVS named E1 provides a voltage reference for the DC voltage VCE to allow the collector-emitter voltage to remain constant as V_{BE} varies. The circuit is used for plots of V_{BE} vs. I_{C} and β vs. I_{C}, both of which duplicate the datasheet condition of V_{CE} = 5 Volts.

Before going into detail about the procedures, it should be noted that the datasheet graphs of V_{BE} vs. I_{C} do not allow for reading the V_{BE} values with high resolution. For this reason, the value of the model parameter NF will be assumed equal to unity, as computing it accurately depends on getting high-resolution V_{BE} data. Also, no Excel optimization procedure will be used to match up the simulated V_{BE} vs. I_{C} to the measured data. The Excel optimization will only be used to match up the simulated and measured β vs. I_{C}. The initial manual adjustment of model parameters is performed as follows.

- Keep the manufacturer's model values for the device's forward parameter VAF and reverse parameters BR, NR, VAR, IKR, ISC and NC.
- Assume NF = 1
- Permanently remove RBM and IRB from the model.
- Set RB initially to zero and temporarily remove IKF, NE, and ISE.
- Pick an initial value for BF equal to the maximum measured β at room temperature.
- Start with the manufacturer's value of IS.
- Using the circuit of Figure 11, plot V
_{BE}vs. I_{C}. - Adjust IS so that V
_{BE}at I_{C}= 100 mA matches the datasheet value. - Adjust RB so that V
_{BE}at I_{C}= 1 A matches the datasheet value. - Using the circuit of Figure 11, plot β vs. I
_{C}. - Adjust IKF so the reduction of β at high currents matches the datasheet as closely as possible.
- Using the circuit of Figure 11, plot V
_{BE}vs. I_{C}. - Adjust IKF and RB so that the V
_{BE}values at I_{C}= 1A and I_{C}= 10 A match the datasheet values as closely as possible. - Change the setup of Figure 11 so that the V
_{CE}source takes on values of 5 V and 20 V. - Adjust VA so that the variation in V
_{BE}at I_{C}= 20 A as V_{CE}goes from 5 V to 20 V matches the datasheet. - Change the setup of Figure 11 back so that V
_{CE}is fixed at 5 V again. - Plot β vs. I
_{C}. - Adjust ISE, NE, IKF and BF so that β vs. I
_{C}matches the datasheet as closely as possible. - Plot V
_{BE}vs. I_{C}. - Adjust IS, IKF and RB so that the V
_{BE}values at I_{C}= 100 mA, I_{C}= 1A and I_{C}= 10 A match the datasheet value as closely as possible. - Repeat steps 17-20 until both V
_{BE}vs. I_{C}and β vs. I_{C}match the datasheet as closely as possible.

While performing the procedure above, one finds that varying RB, IS and IKF to obtain the desired V_{BE} vs. I_{C} converges on the desired result very quickly. However, when varying ISE, NE and BF to get the desired β vs. I_{C}, many iterations are required to get reasonable results. This is especially true of the combination of ISE and NE. Once a reasonable result has been obtained, it's difficult to tell whether the result is close to optimum or not. One way around this dilemma is to use an optimizer to get a combination of ISE, NE and BF that gives some kind of "best fit" to β vs. I_{C}. An initial attempt was made using the Mathcad software's Minerr() function. However, this function was found to be inflexible and unreliable. Investigation led to a description on the since-deleted Analog Services web site explaining how to use the Excel solver for this application. The Excel solver serves as both a solver and an optimizer. The analysis at the Analog Services site did not deal with the issue of the non-negligible voltage drop across RB as described above, but that situation can be remedied easily. The performance and usability of the Excel solver is good enough to allow for a much better fit to the measured β vs. I_{C} curves than can be obtained by hand. The procedure for fitting the data is summarized below. The optimization procedure below assumes that the manual procedure above has already been performed, such that only minor refinements to parameter values are needed.

- Using the simulation setup of Figure 11, plot V
_{BE}and I_{B}vs. I_{C}. - The user enters into the spreadsheet the first-guess values of ISE, NE and BF obtained in the procedure above. These will be varied by the Excel solver later.
- The user records the datasheet values of β into the spreadsheet at each of the collector current values 100mA, 200mA, 300mA, 400mA, 500mA, 600mA, 700mA, 800mA, 900mA, 1A, 2A, 3A, 4A, 5A, 6A, 7A, 8A, 9A and 10A.
- The user records the simulated V
_{BE}and I_{B}into the spreadsheet for each of the collector current values above. This fixed I_{B}value is only used in the next step below. - The spreadsheet computes the
*intrinsic*V_{BE}from the external V_{BE}by subtracting RB * I_{B}(obtained above) at each of the above collector currents. - The spreadsheet computes the adjustable simulated I
_{B}(variable with ISE, NE and BF) from equation (1) at each V_{BE}value corresponding to the above collector currents. - The spreadsheet computes the adjustable simulated β at each of the above collector currents by dividing the collector current by the adjustable simulated I
_{B}. - The spreadsheet computes the difference between the adjustable simulated β and the datasheet β at each of the above currents to obtain the β error.
- The spreadsheet multiplies the β error by a weighting factor to obtain the weighted β error at each of the above currents. The weighting factor is 1 at low and mid currents and steadily decreases for very high currents. This forces the β error to be smallest in the normal operating region at the expense of slightly higher β error at very high currents.
- The spreadsheet computes the RMS value of the weighted β error over the collector currents specified above and saves it in a cell.
- A plot is used to simultaneously display the computed and measured β vs. I
_{C}for comparison purposes. - The Excel solver is invoked. The solver varies ISE, NE and BF to adjust the simulated I
_{B}vs. V_{BE}. The solver's goal is set to minimize the RMS value of the weighted RMS β error. - The user tries different initial guesses for ISE, NE and BF to obtain the solution with the smallest weighted β error. This process is aided by observing the graph that compares simulated and measured β

A partial view of the spreadsheet is shown below in Figure 12. The data for collector currents above 0.5 A were omitted to keep the width of the graphic small. The spreadsheets can be downloaded here for the MJL3281A and here for the MJL1302A.

## Junction Capacitances vs. Reverse Voltage

### Formulas for Junction Capacitance

The equations for the collector-base and base-emitter junction capacitance as used by SPICE each have the same form. For the collector-base capacitance, the equation is (reference [2], equation 2-51):

(6) | $${C}_{\mathit{CB}}=\frac{{C}_{\mathit{JC}}\left(0\right)}{{\left(1-\frac{{V}_{\mathit{BC}}}{{\varphi}_{C}}\right)}^{{m}_{C}}}+{\tau}_{R}{g}_{\mathit{mR}}$$ |

For the base-emitter capacitance, the equation is ([2], equation 2-50):

(7) | $${C}_{\mathit{BE}}=\frac{{C}_{\mathit{JE}}\left(0\right)}{{\left(1-\frac{{V}_{\mathit{BE}}}{{\varphi}_{E}}\right)}^{{m}_{E}}}+{\tau}_{F}{g}_{\mathit{mF}}$$ |

In (6) and (7) above, the g_{m} terms are proportional to the DC current through the corresponding forward-biased junction. The voltages V_{BC} and V_{BE} are negative for the reverse-biased condition. The measurements from the datasheet are for the reverse-biased condition only, so the g_{m} terms are both zero. The parameters τ_{R} and τ_{F} are the reverse and forward transit times respectively. The parameter τ_{R} is not normally available from the datasheet. The relationship between the parameter τ_{F} and the asymptotic maximum f_{T} will be shown in the next subsection. The correspondence between the formula parameters in (6) and (7) and the SPICE parameters is given below in Table 2.

Variable Name | SPICE Parameter Name |
---|---|

C_{JC}(0) |
CJC |

φ_{C} |
VJC |

m_{C} |
MJC |

τ_{R} |
TR |

C_{JE}(0) |
CJE |

φ_{E} |
VJE |

m_{E} |
MJE |

τ_{F} |
TF |

Table 2. SPICE Names For Junction Capacitance Parameters

Three parameters are needed to characterize each of the junction capacitances in the reverse-biased region. To determine the SPICE parameters for the collector-base capacitance C_{CB}, the three quantities C_{JC}(0), φ_{C} and m_{C} must be found. These can be determined by taking three data points from the datasheet curves of capacitance vs. reverse voltage corresponding to Figures 3 and 6 of this document. The three data points should be chosen as the two endpoints of the curve, together with a point in the middle chosen for ease of reading the data from the graph. Given these three data points, three nonlinear equations in three unknowns result. These can be solved numerically. In the case of the models presented here, the parameter values were determined using the Mathcad Minerr() function. To prevent the solver from reaching an incorrect solution, the capacitance data needed to be scaled to units of pF. These data were obtained before bumping into the serious limitations of Mathcad's Minerr() that were found later when trying to fit model parameters to the measured β vs. I_{C}. The Excel solver could be used just as well for the capacitance parameter fitting also. The manual scaling required by Mathcad could be done automatically with Excel by choosing the "use automatic scaling" Excel solver options.

For the base-emitter capacitance C_{BE}, the same procedure can be followed. However, the g_{mF} _{*} τ_{F} term also affects the simulated f_{T} vs. I_{C}. The specifics are explained in the next subsection.

## Adjusting SPICE Parameters to Match Simulated and Measured f_{T} vs. I_{C}

### Low- and Mid-Current Regions of f_{T}

The hybrid-π model used internally by SPICE for small-signal analysis is shown below in Figure 13 from reference [2].

We will compute f_{T} for this circuit using some simplifying approximations. First, neglect r_{BB'} so that C_{JX} and C_{μ} appear in parallel as a capacitor C_{CB}. The capacitor C_{π} is the same as C_{BE} of equation (7). Also, neglect r_{C} and r_{E}. These assumptions make node B the same as B', C the same as C' and E the same as E'. Let port 1 be assigned to B' and port 2 be assigned to C'. By grounding the emitter, the h-parameter h_{21} becomes h_{fe}. By the definition of the h-parameters:

(8) | $${h}_{21}={\frac{{I}_{2}}{{I}_{1}}|}_{{V}_{2}=0}$$ |

so we short both the collector and emitter to ground in order to find h_{fe}. This has the effect of shorting out g_{0} and C_{JS}. Analysis of the circuit with the above assumptions results in the small-signal transfer function relating I_{C}(s) to I_{B}(s) with terminals E and C shorted to ground. That expression is:

(9) | $$\frac{{I}_{C}\left(s\right)}{{I}_{B}\left(s\right)}=\frac{{g}_{\mathit{mF}}-{g}_{\mu}-s{C}_{\mathit{CB}}}{{g}_{\pi}+{g}_{\mu}+s\left({C}_{\mathit{BE}}+{C}_{\mathit{CB}}\right)}$$ |

where g_{μ} = 1 / r_{μ}, g_{π} = 1 / r_{π} and C_{CB} and C_{BE} are given by equations (6) and (7) respectively. Equation (9) has a right-half-plane zero at a very high frequency which will be neglected. The value of f_{T} is found by multiplying the DC value of (9) by its pole frequency in Hz. The result is:

(10) | $${f}_{T}=\frac{{g}_{\mathit{mF}}-{g}_{\mu}}{2\pi \left({C}_{\mathit{BE}}+{C}_{\mathit{CB}}\right)}$$ |

Since g_{mF} >> g_{μ}, (10) can be approximated as:

(11) | $${f}_{T}=\frac{{g}_{\mathit{mF}}}{2\pi \left({C}_{\mathit{BE}}+{C}_{\mathit{CB}}\right)}$$ |

Now equation (7) can be split up as the sum of two parts as follows:

(12) | $${C}_{\mathit{BE}}={\tau}_{F}{g}_{\mathit{mF}}+{C}_{\mathit{JE}}\left({V}_{\mathit{BE}}\right)$$ |

The function C_{JE}(V_{BE}) is modified by SPICE so that its complete form in the forward-biased region is more complex than the first term of equation (7). The purpose of this modification is to avoid the singularity that would occur for a forward-biased value of V_{BE} equal to φ_{E}. The modification avoids the singularity while at the same time making C_{JE}(V_{BE}) and its derivative continuous [2]. The form of C_{JE}(V_{BE}) is modified as follows ([2], equation 2-58):

(13) | $${C}_{\mathit{JE}}\left({V}_{\mathit{BE}}\right)=\{\begin{array}{ll}{C}_{\mathit{JE}}\left(0\right){\left(1-\frac{{V}_{\mathit{BE}}}{{\varphi}_{E}}\right)}^{-{m}_{E}}\text{,}& {V}_{\mathit{BE}}<\mathit{FC}\times {\varphi}_{E}\\ \frac{{C}_{\mathit{JE}}\left(0\right)}{{F}_{2}}\left({F}_{3}+\frac{{m}_{E}{V}_{\mathit{BE}}}{{\varphi}_{E}}\right)\text{,}& {V}_{\mathit{BE}}\ge \mathit{FC}\times {\varphi}_{E}\end{array}$$ |

The quantity FC is the SPICE parameter of the same name. It is a number between 0 and 1 that determines the value of V_{BE} at which the transition between the two formulas occurs in (13). Its default value is 0.5, but computations show that it has little effect on the f_{T} data until it becomes close to 1. If FC is set equal to 1, the singularity in C_{JE}(V_{BE}) will occur at V_{BE} = φ_{E}. The original value for FC of 0.1 from the manufacturer-provided SPICE models was retained, although some experimentation was done with other values. The quantities F_{2} and F_{3} are given by ([2], equation 2-42):

(14) | $${F}_{2}={\left(1-\mathit{FC}\right)}^{1+{m}_{E}}$$ |

(15) | $${F}_{3}=1-\mathit{FC}\left(1+{m}_{E}\right)$$ |

Substituting (12) into (11) and dividing the numerator and denominator of the result by g_{mF} gives:

(16) | $${f}_{T}=\frac{1}{2\pi \left[{\tau}_{F}+\frac{1}{{g}_{\mathit{mF}}}\left({C}_{\mathit{JE}}\left({V}_{\mathit{BE}}\right)+{C}_{\mathit{CB}}\right)\right]}$$ |

where C_{JE}(V_{BE}) is given by (13), (14) and (15) and C_{CB} is given by (6) with g_{mR} set to zero (due to the reverse-biased collector-base junction when in the active region).

Equation (16) provides insight into the variation of f_{T} with collector current. We know that g_{mF} is directly proportional to collector current, being approximately I_{C} / 26mV at room temperature. Therefore, at very low collector current, the denominator of (16) becomes large, making f_{T} small. As collector current increases, the denominator of (16) decreases, making f_{T} increase. Finally, as g_{mF} approaches infinity, f_{T} approaches 1/(2π*τ_{F}). An offsetting effect occurs from the increase of V_{BE} as I_{C} increases. This causes an increase in C_{JE}(V_{BE}). But V_{BE} is a logarithmic function of I_{C}, so the linear increase of g_{mF} with I_{C} dominates. Therefore, at low currents f_{T} starts out low and increases as I_{C} increases, finally reaching a constant value of 1/(2π*τ_{F}) at high currents. This does not predict the subsequent reduction of f_{T} at very high currents however. SPICE modifies equation (16) to account for this effect as well as the change of f_{T} with varying V_{CB}. This will be described in the next subsection. This section will only deal with low- and mid-current variations of f_{T}.

We now see how equation (16) together with (13)-(15) predict the variation of f_{T} with I_{C}. Using Table 2 we find the corresponding SPICE parameters are VJE, CJE, MJE and TF. While the effects of C_{CB} show up in (16), the numerical value of C_{JE}(V_{BE}) in the forward-bias condition is much larger than C_{CB} and swamps out its effect.

Recall that we previously found a way to determine VJE, CJE and MJE. This involved setting g_{mR} to zero in (6) and finding the solution of three nonlinear equations in three unknowns. These three equations are found from three datasheet values of base-emitter capacitance in the reverse-biased region. Next, the TF value can be found by first setting the asymptotic maximum value of the computed f_{T} (= 1/(2π*τ_{F})) to a value slightly higher than the actual maximum datasheet value. We then have all the parameters needed to characterize f_{T} vs. I_{C} in the low- and mid-current range. However, when observing Figures 5 and 6, we saw that it's possible to get an excellent match for the measured reverse-biased base-emitter capacitance vs. voltage and yet have non-negligible errors in the low-current f_{T} values determined by the same parameters. One could attempt to adjust TF to compensate, but the effects of TF on the simulated f_{T} are negligible at low currents. However, there is a way out. First, observe that the value of the SPICE parameter CJE is the zero-bias value of the base-emitter capacitance. The datasheet graphs from which CJE, VJE and MJE are computed show data only for reverse V_{BE} values from 2 to 10 Volts. Therefore CJE is an *extrapolated* parameter. Because of this fact, it's possible to make small adjustments in CJE (= C_{JE}(0)). Then, assuming C_{JE}(0) is known, one can use (6) to solve *two* nonlinear equations in *two* unknowns using the capacitance values at the graph endpoints of 2 and 10 Volts. Using this approach, it's possible for equation (6) to still predict the base-emitter capacitances very accurately at reverse voltages from 2 to 10 Volts. At the same time, the simulated f_{T} at low currents can be made to match the measured values very closely. This procedure is summarized below.

- Use the Excel solver to solve equation (7) for C
_{JE}(0), m_{E}and φ_{E}with g_{mR}set to zero. This is done by equating (7) to the datasheet capacitances at three different values of reverse V_{BE}. The endpoints of 2 and 10 Volts, together with some convenient intermediate voltage should be used. This determines the SPICE parameters CJE, VJE and MJE respectively. - Retain the values of ITF, XTF and VTF from the manufacturer-provided models. These parameters will be explained in the next subsection.
- Estimate the asymptotic maximum value of f
_{T}(= f_{T(max)}) using a number slightly higher than the actual maximum value of f_{T}from the datasheet. - Compute τ
_{F}from τ_{F}= 1/(2π*f_{T(max)}). - Plug the computed values of CJE, MJE, VJE and TF into the model and simulate f
_{T}at 100mA and 300mA. - Adjust TF to get the simulated maximum f
_{T}as close to the datasheet value as possible. - Estimate a new value of CJE based on the data for f
_{T}at I_{C}= 100mA. - Use the Excel solver to solve equation (7) for m
_{E}and φ_{E}using V_{BE}values of 2 and 10 Volts. Assume CJE is known and use its value determined in the previous step. - Repeat steps (5)-(8) until the simulated f
_{T}at values of I_{C}from 100mA to about 3A matches the datasheet as closely as possible.

### High-Current Region of f_{T}

The decrease of f_{T} at very high current levels is modeled in SPICE by scaling the forward transit time τ_{F} in (16) by a factor called ATF. ATF is given by ([2], equation 2-120):

(17) | $$\mathit{ATF}=1+{X}_{\tau F}{e}^{\left(\frac{{V}_{\mathit{BC}}}{1.44{V}_{\tau F}}\right)}{\left(\frac{{I}_{\mathrm{CC}}}{{I}_{\mathrm{CC}}+{I}_{\tau F}}\right)}^{2}$$ |

The relationship between the formula parameters X_{τF}, V_{τF} and I_{τF} and the corresponding SPICE parameters is given below in Table 3.

Variable Name | SPICE Parameter Name |
---|---|

X_{τF} |
XTF |

V_{τF} |
VTF |

I_{τF} |
ITF |

Table 3. SPICE Names for High-Current f_{T} Adjustment

It's important to realize that I_{CC} in (17) above is not the actual collector current, but rather the ideal forward diffusion current given by (18) below.

(18) | $${I}_{\mathrm{CC}}={I}_{S}\left({e}^{\frac{{\mathit{qV}}_{\mathit{BE}}}{{n}_{F}\mathit{kT}}}-1\right)$$ |

The method used for computing ITF, VTF and XTF was the following:

- Pick a very large value for ITF, say ITF = 1000.
- Using the simulation setup of Figure 11, simulate f
_{T}at various I_{C}values with V_{CE}values of 5 and 10 Volts. - Adjust XTF and VTF to match the simulated and datasheet curves of f
_{T}vs. I_{C}as closely as possible using V_{CE}values of 5 and 10 Volts. - Repeat step 3 until the best match of simulated and measured data is found.