Technical Feature
Fixing the Curtice FET Model
Convergence difficulties are common in nonlinear analyses of circuits using a Curtice model. The problem is not the model itself, but the limitations of the parameters describing the gate I/V characteristic. In this article, we show why that failure occurs, and how the model can be modified to prevent it.
Steve Maas
Applied Wave Research
El Segundo, CA
The venerable Curtice-Ettenberg FET model (traditionally called the Curtice model)^{1} has existed for almost two decades. During that time, it has been used extensively for all kinds of microwave circuit designs. It is as fundamental to GaAs MESFET circuit design as the Shichman and Hodges model has been for circuits using silicon FETs. The model, in one form or another, has been implemented in virtually all circuit simulators. Many have extended the model to include nonlinear capacitances and other effects that were not in the original publication.
Still, there are perennial problems in using the model. Designers frequently encounter convergence difficulties and errors from incorrect transconductance, especially near pinch-off. Most of these problems can be eliminated surprisingly easily; in this article, we describe some simple modifications that remove most of the problems.
Fig. 1 Equivalent circuit of a FET used in the Curtice model.
The Curtice Model
Most simulators' "Curtice" model is actually an extension of the model described in the original publication. Even so, the basic equations for the channel current - the part of the model most interesting to us - remains unchanged. As with most of the classical FET models, the Curtice model consists of an equivalent circuit for the device that includes linear elements, nonlinear capacitances, diodes and a nonlinear current source representing the channel current. The linear elements model such things as contact resistances and some of the parasitic capacitances, while the diodes model the gate-to-channel junction. Figure 1 shows the equivalent circuit.
The channel current (which we can loosely call the drain current) is given by
I_{d} (V_{gs} ,V_{ds} ) = (A_{0} + A_{1} V_{1} +A_{2} V^{2} _{1} + A_{3} V^{3} _{1} ) tanh ( gV_{ds} ) (1 + lV_{ds} ) (1)
where
V_{1} = V_{gs} (t- t)(1 + b(V_{ds0} -V_{ds} )) (2)
V_{gs} = internal gate voltage
V_{ds} = drain voltage (not including voltage drops across the gate, drain and source resistances)
V_{ds0} = particular voltage (usually the one where the A_{n} polynomial coefficients were determined)
b = constant
t = gate-to-drain time delay
Note that V_{1} = V_{gs} at DC when V_{ds} = V_{ds0} . The remaining terms, g and l, are also constants.
If we ignore Equation 2 for a moment (say, by assuming that b = 0), Equation 1 can be put in the form
I_{d} = f_{g} (V_{gs} ) f_{d} (V_{ds} ) (3)
where
f_{g} (V_{gs} ) = A_{0} + A_{1} V_{1} + A_{2} V^{2} _{1} + A_{3} V^{3} _{1} (4)
and
f_{d} (V_{ds} ) = tanh ( gV_{ds} )(1 + lV_{ds} ) (5)
This is a convenient expression, as it allows us to separate the effects of the gate and drain I/V characteristics and to determine them easily; f_{g} (V_{gs} ) can be found from a plot of I_{d} at fixed V_{ds} and f_{d} (V_{ds} ) at fixed V_{gs} . We can also see, quite clearly, the effect of the two functions. f_{g} (V_{gs} ), which we call the gate I/V characteristic, is simply a cubic polynomial, which should be easy to fit to any smoothly varying set of measured data. f_{d} (V_{gs} ), the drain I/V characteristic, is a hyperbolic tangent function. This function is used simply because any FET's drain I/V characteristic looks like a hyperbolic tangent curve when V_{ds} > 0. The remaining term, 1 + lV_{ds} , accounts for DC drain-to-source resistance.
Now, what about Equation 2? This expression simply offsets the gate voltage, at high drain voltages, to account for the tendency of power FETs to conduct at gate voltages below pinch-off. In Curtice and Ettenberg1, this is described as a shift in pinch-off voltage, but it could also be called a kind of sub-threshold conduction. The correction is necessary only in power devices; models for small-signal FETs should have b = 0.
Other elements of the equivalent circuit are straightforward, and we can dispense with them quickly. The drain, source and gate resistances are well known parasitics. C_{ds} , the drain-to-source capacitance, is largely a capacitance between metallizations on the chip, as is C_{gd} , the gate-to-drain capacitance. C_{gd} has a significant nonlinearity only near V_{ds} = 0. C_{gs} , the largest capacitance, is moderately nonlinear. It is often modeled as a conventional Schottky-junction capacitance, although that approach is terribly accurate, especially near pinch-off. R_{ds,f} and C_{i} model the dramatic decrease in drain-to-source resistance at RF and microwave frequencies, and the diodes model conduction of the gate-to-channel junction.
The Model's Characteristics
Users of the Curtice model, and especially those who create parameter sets for it, should be aware of some of its characteristics. First, there are only two parameters for adjusting the drain I/V characteristic, l and g, and these have very different effects. As shown in Figure 2 , l adjusts the slope of the curve, while g adjusts the location of the knee. Clearly, one cannot adjust, say, g to match the knees of a set of drain I/V curves measured at several values of V_{gs} ; therefore, it is best to match the most important curve, and not worry about the rest. In a power amplifier, the most important curve is the one representing the highest value of V_{gs} , because the knee of that curve affects calculations of output power. In small-signal circuits, the drain current and voltage may never come close to the knee, so it doesn't matter much where it is. Modelers should be mindful of what the device is intended to do, and create the model accordingly.
Fig. 2 Increasing g moves the knee of the FET's drain I/V characteristic to the left; increasing l tilts the current-saturation (flat) part of the curve.
A second difficulty is that the cubic polynomial used for the gate I/V characteristic does not guarantee that I_{d} pinches off at V_{gs} = V_{p} , where V_{p} is the pinch-off voltage. (We assume for the rest of this article that b = 0 so V_{1} = V_{gs} . This does not present any difficulties, since the b term simply shifts V_{1} relative to V_{gs} . As long as V_{1} is treated properly, it does not matter what value of V_{gs} corresponds to any particular value of V_{1} .) Furthermore, it is literally impossible for a cubic polynomial to remain at zero when V_{gs} < V_{p} . Thus, in virtually all implementations of this model, the user can enter a value of V_{p} , and the simulator then returns zero when V_{gs} < V_{p} .
Now the problems begin. Even if we force f_{g} (V_{gs} ) to be zero at V_{gs} = V_{p} , we have not guaranteed that the transconductance G_{m} is also zero at V_{gs} = V_{p} . This is a serious problem, as a simulation would indicate that the device could amplify at pinch-off, something that real FETs just don't do. We need to guarantee that both G_{m} and I_{d} reach zero at V_{gs} = V_{p} .
This problem was recognized in Curtice and Ettenberg^{1} , but no solution was presented. In fact, it is fairly easy to solve. We first recognize that we have, as a minimum, three basic requirements
I_{d} = 0 at V_{gs} = V_{p}
G_{m} = 0 at V_{gs} = V_{p}
I_{d} = I_{dss} at V_{gs} = 0
Imagine that we can factor Equation 4 so it has the form
f_{g} (V_{gs} ) = I_{0} (V_{gs} -V_{z1} )(V_{gs} -V_{z2} )(V_{gs} -V_{z3} ) (6)
We must find the zeros and the term I_{0} ; once they are known, we can expand Equation 6 to obtain the A_{n} coefficients. The first of these requirements can be met by making one zero of the polynomial, say, V_{z3} , equal to V_{p} , the pinch-off voltage. As it happens, making a second zero equal to V_{p} satisfies the second requirement. Satisfying the third requires
where we have assumed that V_{z2} = V_{z3} = V_{p} and
Î_{dss} = I_{dss} /f_{d} (V_{d0} ) (8)
This leaves us only one parameter, V_{z1} , to adjust the shape of the gate I/V characteristic to match that of the measured data. Although we might conclude that it is difficult to obtain a good "fit" to measured data with a Curtice model, it turns out that by placing the remaining zero at a large positive voltage (that is, a few volts), it is usually easy to obtain good results. Conversely, it makes it quite easy to fit a Curtice model using a simple spreadsheet or MathCAD page^{2} ; no fancy numerical methods are needed.
Unfortunately, we're not out of the woods yet. A set of model parameters determined in this way might not work very well. To understand this, we must make a small digression into the subject of nonlinear circuit analysis.
It is a distressing fact of life in the world of nonlinearity that it is rarely - almost never, in fact - possible to solve nonlinear sets of equations directly. Invariably, some type of iterative solution is necessary. To obtain any such solution, we must first estimate it, and then find some way to improve the solution step-by-step, until we decide that it is good enough. For this reason, solutions to sets of nonlinear equations are never perfect; they always come with some amount of residual error. The trick is to make that error small enough to be negligible.
The key to solving such equations is to find a process that improves a given, inaccurate solution. It does not have to solve it completely; as long as it can reliably improve an imperfect solution, we can start with a rough estimate of the solution, and apply the method to it repeatedly until the problem is beaten into submission. One of the best techniques is Newton's Method, a process that involves using the derivatives of the equations to determine the manner in which to modify the variables for which the equations must be solved.
In nonlinear circuit analysis, we use the derivatives of the FET's I/V characteristic to estimate new values of the drain and gate voltages. It should be no surprise that those derivatives must be well behaved; if they have "kinks" in them or are discontinuous, the estimate of the new solution may be very different on the two sides of the kink. If we are unlucky, and the solution process traverses the kink as it modifies the voltages, the estimated solution changes suddenly, and the simulator can lose convergence. This is true of both harmonic-balance analysis and transient (SPICE) analysis.
By using a polynomial to describe the gate I/V characteristic, and setting it to zero below V_{p} , we have created a huge kink in the transconductance - the derivative of I_{d} with respect to V_{gs} - at V_{gs} = V_{p} . Satisfying the three conditions above are not enough; we must satisfy a fourth, namely dG_{m} /dV_{gs} = 0 at V_{gs} = V_{p} . This extra condition would use up our remaining zero, leaving nothing to adjust for fitting the measured I/V characteristic to the model. This just won't do. (In fact, we may have overstated the case a bit. The Curtice model usually works surprisingly well in spite of this flaw, and as long as the model parameters are well chosen, convergence failure at reasonable power levels is unlikely. Still, by fixing this problem, the model can be made significantly more robust, so convergence occurs reliably even at high power levels.)
Fixing the Model
From this discussion, it appears that we need a couple of specific things:
- The earlier requirements at V_{gs} = V_{p} and V_{gs} = 0 must be satisfied;
- The additional requirement that dG_{m} /dV_{gs} = 0 at V_{gs} = V_{p} must be met;
- More parameters must be freed, to provide more degrees of freedom for fitting the model to measured I/V data.
These requirements have subtle implications. Specifically, the second implies that the FET must have a "soft" pinch-off characteristic; that is, I_{d} decays gradually around V_{gs} = V_{p} , and the abrupt pinch-off of the classical model cannot exist. That's actually good, because FETs pinch-off gradually, not abruptly. A model that meets these new requirements therefore is likely to describe the FET better near pinch-off.
One way to meet some of these requirements is to limit V_{gs} to V_{gs} > V_{p} in some numerically acceptable way, before using it in (1) and (2). Clearly, we cannot simply define a new V_{gs} variable, say V_{gs,x} and say
because that would reintroduce the kink in G_{m} . A better solution is to use a function of the form
This function limits V_{gs,x} to a maximum value of V_{p} , while creating a gradual transition near V_{p} . The transition is controlled by d; a large d provides a more gradual transition, while d = 0 creates an abrupt one, equivalent to Equation 9.
With this change, I_{d} trails off gradually at V_{p} , even as V_{gs} drops below V_{p} and G_{m} drops to zero. It does not, however, guarantee I_{d} to be zero below pinch-off. In fact, this change alone can make matters worse - if the model parameters have not been extracted to ensure that I_{d} = 0 at V_{gs} = V_{p} , the FET may show substantial, even negative, current at pinch-off. We need to make certain that the latter condition is also satisfied.
There are two ways to implement this second part of the solution. The first is simply to modify the gate I/V characteristic in some way so that the I_{d} = 0 condition at V_{gs} = V_{p} is met. This approach probably requires the creation of a new expression for the gate I/V characteristic, making users' existing Curtice model libraries obsolete. Recognizing that this solution will not endear the authors with circuit designers, we may consider a second approach - modify the existing parameter set in a minimal manner, so that it satisfies these obviously necessary requirements but does not change the I/V characteristic more than necessary. We should do this in a way that does not modify the parameters if they already meet the pinch-off conditions.
Fig. 3 An example of a poorly conceived model.
We begin by assuming that the gate I/V characteristic simply fails to provide zero current when V_{gs} = V_{p} , but it is otherwise well conceived. This condition is shown in Figure 3 , where the drain current does not reach zero at pinch-off, and the transconductance has a strong discontinuity at that point. A simple circuit using this FET model would not converge to a solution in harmonic-balance analysis. (A_{0} = 0.022, A_{1} = 0.046, A_{2} = 0.037, A_{3} = 0.030 and V_{p} = -0.6 V.) Of course, many other errors are possible, many of which are frightening. For example, it is possible for f_{g} (V_{gs} ) to have a minimum between V_{p} and V_{gs} = 0, so it is actually decreasing at pinch-off as V_{gs} increases. We will not attempt to fix all such errors; at some point, the user has to take responsibility for creating a model that is at least a reasonable representation of the device.
After some experimentation, we have determined that a simple and effective way to correct a problem like the one shown is simply to distribute the error linearly across the range V_{gs} = (V_{p} , V_{max} ), where V_{max} is some maximum gate voltage. In many cases, V_{max} = 0 is the best value to use, as it preserves the I_{dss} of the model. However, this value may not be practical in devices having low pinch-off voltages. In such devices, a V_{max} value of a few tenths of a volt may be preferable.
The gate I/V characteristic is modified as
where V_{1} (which we have been assuming to be equal to V_{gs} ) is as used in Equation 1 and DI_{d} is the error in I_{d} at V_{gs} = V_{p} . This change can be implemented simply by modifying the polynomial coefficients
The modification also works when DI_{d} < 0. Additionally, we have found that it provides relief in many situations that are even uglier than the one shown. Note that the coefficients are not modified if they satisfy pinch-off conditions as provided.
These two modifications guarantee that at least two of the polynomial coefficients in Equation 4 can be determined independently, while retaining proper pinch-off characteristics and I_{dss} . The other two coefficients cannot be treated independently, as they are constrained by Equation 12, but the user still has somewhat more freedom than before in selecting those coefficients. If the f_{g} (V_{gs} ) polynomial is well determined, although perhaps without satisfying the pinch-off conditions, these operations force f_{g} (V_{gs} ) to satisfy them with minimal modification.
Fig. 4 Uncorrected gate I/V characteristic.
Example
The above modification has been included in the Curtice model used in Microwave Office, version 5.0.^{2} These modifications could be included in any other circuit simulator that provides a capability for users to write their own models.
In the Microwave Office implementation, V_{max} is normally zero, but is modified so that V_{max} - V_{p} > 0.5 V in all cases. V_{max} is not a user-modifiable parameter; that would make it vulnerable to the same kind of misuse that gives rise to the errors it is designed to correct. Similarly, d is set to 2.5 • 10^{-3} V^{2} _{P} . This gives a smooth curve at pinch-off, without making the gate I/V characteristic too soft.
Figure 4 shows the gate I/V characteristic of a conventional Curtice model that does not pinch-off properly at V_{gs} = V_{p} (A_{0} = 0.08, A_{1} = 0.06756, A_{2} = 0.00612, A_{3} = -0.00383 and V_{p} = -1.4 V.) The current clearly does not reach zero at V_{gs} = V_{p} , and a little mental differentiation shows that the transconductance also is non-zero at pinch-off. In fact, both are discontinuous. Figure 5 shows the same set of parameters with the above modifications. It is clear that both the current and transconductance are now well behaved near pinch-off, and the characteristic has been modified minimally. The drain current error of 10 mA at the pinch-off voltage of -1.4 V has been corrected, and the change in drain current is only about 3 mA above -1V. The value of I_{dss} , 80 mA, is preserved in the corrected characteristic.
Conclusion
The venerable Curtice model can have convergence difficulties when parameter extraction is imperfect. In the classical model, the user is forced to choose between a parameter set that matches the entire range of the measured I/V characteristics, or one that operates properly at pinch-off. We can correct the model, however, by limiting the value of V_{gs} in a numerically acceptable manner, and modifying the A_{0} and A_{1} polynomial coefficients in a way that minimizes changes in the I/V characteristic. Increased accuracy and robustness result.
References
1. W.R. Curtice and M. Ettenberg, "A Nonlinear GaAs FET Model for Use in the Design of Output Circuits for Power Amplifiers," IEEE Transactions on Microwave Theory and Techniques , Vol. MTT-33, 1985, p. 1383.
2. See http://www.nonlintec.com/curtice.med to download an example.
3. Applied Wave Research, 1960 E. Grand Ave., Suite 430, El Segundo, CA 90245.
I am very flattered by Dr. Maas' comments on my FET model of 1985. Furthermore, I am very pleased to see it appear in Microwave Office. If anything, it is a simple model for a very complex device. I believe that is why it has continued to be used, and why some more accurate models that are extremely complex do not get used as much. In the years since 1985, I have added a sub-threshold characteristic, charge-conservative nonlinear capacitance and self-heating effects. All of these effects are needed in today's GaAs circuit designs. Dr. Maas has improved the pinch-off behavior of the 1985 model. The fundamental problem was always that the (cubic) polynomial is not well behaved outside of the calibration range. I presently add code to handle near pinch-off and sub-threshold behavior and maximum channel current conditions. My experience with simulators using my 1985 model is that the convergence problems are usually due to not handling pinch-off properly. That occurs because pinch-off is a function of drain-source voltage and, thus, in the model (and device), is not a constant value. It is interesting that the present form of the model has proved useful in modeling today's GaN HEMT devices (see Green1, for example). Who knows, maybe the model will last another 20 years? 1. B.M. Green, H. Kim, K.K. Chu, H.S. Lin, V. Tilak, J.R. Shealy, J.A. Smart and L. Eastman, "Validation of an Analytical Large-signal Model for AlGaN/GaN HEMTs," IEEE MTT-S International Microwave Symposium Digest, 2000 . Walter Curtice |