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Structural Versus Behavioral Device Models

Newly-discovered physical principles are often the driving cause for the invention of new kinds of devices such as the transistor. It is one of the highlights in the history of electronics, and illustrates how device models develop in engineering and why they are often a key to technological advancement.

The semiconductor phenomenon was known early in the twentieth century, but the successful development of alternative technology, thermionic valves or electron tubes, detracted from its emergence until the late 1940s. The transistor, like any new breakthrough in technology, was not well understood in the 1950s. The very first transistors were bipolar, not field-effect, devices. They were manufactured using a simple, highly obsolete process that gave them the name of “point-contact” transistors. As the benefits of transistors were quickly realized, great effort was put into their development, and the bipolar junction transistor (BJT) soon replaced it. The quirky point-contact transistor could show some of the negative-resistance effects of tunnel diodes under the right conditions, adding to its mysteriousness and obscuring the phenomena essential to transistor behavior. From an engineering viewpoint, what was needed was to understand the essential principles underlying transistors. In this regard, the development of electronics is no different than any other area of engineering and leads to device modeling.

Inadequate Models Can Limit Design Insight

Engineers start with a repertoire of elemental devices and processes, and seek new combinations of them that produce new functional capabilities. In a new engineering discipline, the combinations are simple and the number of elemental devices comprising them are relatively few. The demanding power and space requirements of electron tubes limited their number to, at most, hundreds in a given device. Giant, unreliable electron-tube computers of the 1950s demonstrated the limits of their applicability. Yet in 1920, the leading use of tubes, in radios, required no more than a dozen of them. By the 1950s, radio design had been optimized to where multiple brands used the minimalist “All-American Five” combination of tubes. Color televisions and laboratory-quality oscilloscopes by then increased the demand for tubes per device to a few dozen. Electron tubes were better understood in the 1950s than in the early radio days, and this allowed engineers to make more effective use of them. Without a clear understanding of one's building blocks, one is left in doubt as to how best to “stack” them. A device model is the basis for this kind of engineering understanding.

The full range of possibilities for a new kind of device is unknown early in its development. Nobody could foresee in the early 1900s that the electron tube would ultimately pose reliability limits for its use, performing functions – namely, computing – that escaped even the speculations of sci-fi writers of the early tube days. Foresight into the fuller exploitation of technology is limited by a murky conception of its devices. Knowledge of the building blocks we have to work with is the key to the refinement of technology, and in engineering this takes the form of a model : a representation of a physical device. Sometimes, models themselves are physical as representations, but of a different scale, as in the popular meaning to the word. Architects often build such models. Essentially all electronics device modeling, however, is mathematical, graphical, or computational. The goal is to capture the full range of behaviors of a device in the model.

The development of device models parallels the growing discoveries of how to apply devices. As models improve, they provide deeper insight into the possible uses for devices. Better models also can lead to a better understanding of the context in which devices are used. As transistor models improved, the understanding of the circuits they were used in became simplified, as the simpler models led to simpler circuit concepts. This is elegantly illustrated in the history of BJT model development. Before this illustration can best be appreciated, we first briefly consider another aspect of model development.

Physical Versus Behavioral Models

There are different kinds of models. Ultimately, a model shows us what a device is in itself and is a structural model . This kind of model is context-independent. It is valid no matter how the device is used or configured, or the circuit in which it is embedded. In the early history of a new kind of device, not much is known about it physically. It appears as a “black box” wherein the contents are largely unknown. Semiconductor physics was not sufficiently advanced when the BJT first made its appearance. Instead of immediately developing a good physical model, engineers were forced to instead revert to a behavioral model for the BJT. A behavioral model is based on what a device does , not what it is .

Suppose you have a black box with three terminals. This was the BJT to engineers in the 1950s. From earlier work, the concept of two-port networks was available. With three leads, a transistor, like an electron-tube triode, could be modeled as a network with two ports, where one of the BJT terminals is shared in common by the ports. A port is simply a pair of terminals with defined polarities for voltage and current. With three terminals, BJTs have three possible two-port configurations, named after the BJT terminal common to both ports. The two ports correspond to input and output circuit networks. Electrical quantities associated with the common BJT terminal appear in both networks. The basic scheme is shown below.

Modeling then proceeds by characterizing BJTs by their two-port parameters. The parameters are coefficients within network equations based on a given BJT configuration. Electronics books in the '50s and '60s often presented transistor theory using hybrid- or h-parameters . Instead of the physical parameter, β , there was the equivalent parameter, hfe of the common-emitter (CE) configuration. The difference is that whereas β appears in physical models as a consequence of device structure, hfe appears as a characterization of port behavior under certain external conditions. Whereas β is derived from insight into the inner working of the BJT, hfe merely captures BJT terminal behavior for a given external circuit configuration.

The h -parameters are configuration-dependent, and hence circuit-dependent, and could be applied equally to electron tubes or transistors. This is not a desirable feature, for it fails to account for the differences in physical structure of the devices. To be optimized, models should be specific to the device and not based on a general network characterization technique. Device models should also be modular – self-contained and not dependent upon external conditions imposed by their environment – by the circuit they are a part of.

Of the multiple sets of possible two-port parameters (h , y , z, etc.), the h parameters were chosen for BJT modeling because the four h parameters best characterized BJT behavior. The four port quantities are each a function of the other quantities, leading to two equations with two terms each. For an example of two-port parameter characterization, the impedance- or z -parameter two-port incremental (small-signal) model equations are

The four parameters (all impedances) result from choosing vo and vi to be functions of io and ii . Then each z parameter can be found from port-quantity measurements. For one, transimpedance, with output port open, is

The condition that incremental current io = 0 – that is, that io be held constant, or Δ io = 0 – allows zm to be derived from the first equation. The other parameters are derived similarly. In general, for dependent variable y (x 1 , x 2 ), either x 1 , or x 2 , is nulled (set to zero) in order to find y with respect to the other independent variable.

The h parameters are defined by equations for io (ii , vo ) and vi (ii , vo ):

In the common-emitter (CE) circuit configuration, BJT quantities io = ic and ii = ib . Then

which in the structural models is β . Furthermore for the CE, hoe = 1/ro , hie = r π , and hre ≅ –1/μ , where

The negative sign accounts for base-to-collector voltage inversion, to make μ a positive number. μ and 1/hre are not exactly equal (nor are their signs) because the condition, taken from the hre parameter, is not the same as for calculation of μ . For μ , ic = 0 (open-circuit output port), to prevent voltage drop across ro and allow the dependent collector voltage-source voltage to be applied across the collector (output) terminal. The h model and physical model parameters do not correspond exactly and hre is of limited usefulness.

Not only are the h -parameter models dependent upon BJT configuration (hfb = α , not β ), they only approximate, if that, the actual structural parameters. Because of this, they tend to obscure rather than reveal basic BJT-circuit insights. A simple structural model for a BJT is the Τ model with ro , as shown below.

The dependent current source of the Τ model allows for the application of the β transform. The β transform is half of the reduction theorem , the network theorem for controlled current sources whereby resistance on the emitter side of the circuit appears to be β + 1 times larger at the base:

More generally, the β transform can be diagrammed as shown below, where N1 is the base network, and N2 is the emitter network. The dependent current source can be removed and the resistances of one network referred to the other if the resistances are scaled appropriately by β + 1.

By applying the β transform, we can express the relationship between physical model parameters rπ and re as

In other words, rπ is simply re referred to the base, as shown in the hybrid-π model above. (Note the word “hybrid” in hybrid-π, reflecting historic development of the physical model from the two-port model.)

The Thevenin equivalent hybrid-π model makes use of the dual parameter to β , that of μ The other half of the reduction theorem is the voltage dual, the μ transform, depicted below. This applies to the transistor model (both BJT and JFET, and also to the triode model) in that network N1 is the base (or grid) circuit and N2 the collector (or plate) circuit.

The dependent voltage source can be removed by referring collector resistances to the emitter (middle drawing) or emitter resistances to the collector (bottom). In other words, emitter resistance, RE , referred to the collector appears as collector resistance,

with ib = 0, or the base open. Also ro can be referred by the μ transform to the emitter as incremental emitter resistance, re . (For electron tubes, the equivalent equation is rp = (μ + 1)x rk , where rp is plate resistance, and rpk = cathode incremental resistance). For FETs, transresistance, rm = 1/gm = rs λ , and ro = μxrm , where

λ is the voltage dual of α = β /( β + 1) of the β transform. (Consequently, for electron tubes, rp = μxrm .) λ is the transmittance from the gate voltage to the source voltage. At the source, the μ transform applies. (A fuller explanation can be found in my book, Designing Amplifier Circuits , the first book in the book-set Analog Circuit Design , at scitechpubs.)

For JFETs and triodes, ro is less than for BJTs and MOSFETs, and has more effect on circuit behavior. The FET incremental physical model can be derived simply from the Thevenin-equivalent hybrid- π BJT model by letting r π → ∞.

Closure

What is elegant about the use of parameter ro over 1/hoe is that it results in a simple, physical circuit model for the BJT (or FET) to which the μ transform, a general network principle, can be applied. (Electron-tube models by the '50s were mature and were physical models to which the μ transform was also commonly applied.) Instead of trying to think in terms of BJT behavior based on circuit conditions imposed on two-port equations, physical or structural models give us insight into their necessary behavior, represented as circuits. Structural circuit models are modular and can be placed into any larger circuit, in any configuration, and be valid. Computer circuit simulator programs make use of structural models at least for this reason.

Another advantage of structural models over behavioral models is that as additional device behaviors, or secondary effects, are accounted for in a more refined model, the circuit model can incorporate them more simply and optimally. For a two-port equivalent model, the number of equations and independent variables would have to grow, and could become unwieldy without the simplifications physical models provide. Neither do two-port models inculcate a causal understanding of circuits because they merely capture behavior. In contrast, physical models, expressed as circuits, capture structure (as interconnected circuit elements), which allows us to infer causes for behavior from our understanding of the circuit elements themselves. Physical models enhance our intuition about circuits.

Obsolete concepts often die hard, and over a half century later some newly published books still use two-port modeling in explaining transistors. The h -parameter model faded away because it was replaced by a clearly better transistor model, just as the electron tube was replaced by (in almost all respects) a clearly better active device. Despite this, it is worthwhile to have two-port network theory in your conceptual toolbox. Why? As a coping mechanism for the next novel device or circuit. Perhaps it will be a molecular mechanical or quantum device for which the physical principles are, for a while, not well understood. (Quantum mechanics is still not really understood.) Engineers are not likely to twiddle their thumbs waiting for a respectable structural model before they commercially exploit the novelty. In the interim, a behavioral model might be all that is possible, but it is better than no device characterization at all. Yet it is a stop-gap measure, a place to start in the search for a physical model. A two-port model lets us most easily choose some arbitrary device quantities and relate them, then characterize their behavior parametrically. This can guide our search for the right physical parameters. Our engineering instincts, however, are not satisfied until we have that simplified, conceptually appealing, structural model for our new building block.

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