Transistor Switching Analysis
∗
Carver A. Mead
Abstract
—With the widespread application of junction transistors in switching applications, the need
for a general method of analysis useful in the region of collector voltage saturation has become apparent.
Linear equivalent circuits using lumped elements have long been used for small signal calculations of
normally-biased transistors, but a comparable method for saturated transistors has been lacking. Recently,
Linvill [
3
] proposed the method of lumped models which allow the analysis of complex switching problems
with the ease of linear circuit calculations. The method is shown to be equivalent to a well-known linear
equivalent circuit under normal bias conditions. Examples of the application of the method and the use of
approximations are drawn from practical circuit problems. Emphasis is placed upon the understanding of
the physical phenomena involved, a necessary prerequisite to intelligent circuit design.
1 Introduction
With the first analysis of a junction transistor triode, it was recognized that such a device was capable of
symmetrical
operation; that is, either the “emitter” junction or “collector” junction could act as a source
of minority carriers in the base region. Thus, modes of operation are available in a transistor which have
never existed in the vacuum tube. For example, a saturated transistor (both emitter and collector forward
biased) will carry signals well in both directions, while if both junctions are reverse biased, essentially no
signal is allowed to pass in either direction. The inherently low voltage drop across a saturated transistor
makes possible the control of very high powers with low dissipations. For these reasons, transistors find
switching service a most important and useful application.
The first detailed analysis of the large signal properties of transistors was done by Ebers and Moll [
1
].
Later, Linvill [
3
] proposed a technique by which the same results may be obtained, but which has the
advantage that a linear model is used and physical insight into the behavior of the device is more readily
gained. The purpose of the present article is to extend this method to the general treatment of diodes
and transistors in practical circuit applications, to present results in special cases of importance, and to
illustrate applications of the analysis in sufficient detail to be generally useful to the design engineer.
1.1 Requirements of the Analytical Method
Let us examine the requirements on a method of analysis to be used in problems of this nature. Clearly,
what is needed is a model, similar to an equivalent circuit such as used for small signal work, yet appropriate
for all conditions encountered in transistor operation; i.e., either junction may be forward or reverse biased,
currents may be either small or large,
a-c, d-c,
or both.
∗
Re-typeset from original material by Donna Fox, October 2016. This paper was originally published in
Semiconductor
Products,
Vol. 3 (9), pp. 43-47, 1960 (Part 1); Vol. 3 (10), pp. 38-42, 1960 (Part 2); and, Vol. 3 (11), pp. 28-32, 1960 (Part
3).
1
Together with a model, the method must include a procedure for evaluating elements in the model and
for making approximations where an exact analysis would be too cumbersome. Furthermore, in order to
be of general utility, the method of analysis should possess the following qualities:
a) The principal variables should be related in a linear manner so linear circuit theory may be used.
b)
Non-linearities in the system should be easily and accurately approximated by a simple piecewise
linear idealization.
c)
Elements in the model should be readily obtainable in terms of simple, easily-measured device
parameters.
d)
Variables in the model should possess physical significance, and results of the analysis should enhance
one’s physical insight into the problem.
e) The model should reduce to familiar form for special cases, e.g., normal bias, small signal.
The variables necessary to solve normal semiconductor problems are:
a) Junction voltage
b) Junction current
c) Minority carrier density
provided we deal only with devices in which the diffusion current predominates. Transistors of this type
are typically used in switching service where the method is most generally useful.
1.2 Review of Basic Processes
Before launching into the details of analysis, let us briefly review the basic processes which occur within a
semiconductor. If we limit our discussion to one-dimensional diffusion flow, a complete description of the
motion of minority carriers within the material consists of:
1. The continuity equation for minority carriers
∂N
∂t
=
N
0
−
N
τ
+
D
∂
2
N
∂x
2
(1)
where
N
is the density (number per unit volume) of minority carriers as a function of
x
and
t
N
0
is the density of minority carriers at thermal equilibrium
τ
is the “lifetime” of minority carriers
D
is the diffusion constant of minority carriers
x
is the distance through the semiconductor
2. The condition that any macroscopic volume element of the material be electrically neutral.
2
1.3 Introduction to Lumped Models
The continuity equation is a partial differential equation, involving both time and space derivatives and
its solutions are in general both difficult and messy, resulting in carrier densities at all points as a function
of time.
Figure 1: Transmission line and equivalent lumped element representation.
One is reminded of the transmission line problem where again a partial differential equation must be
solved, and the results are voltage (or current) at any point on the line as a function of time. Pursuing
the analog still further, the analysis of a transmission line is greatly simplified if we are content to find
approximate
voltages and currents at
certain specified
points along the line, instead of
exact
voltages and
currents at
all
points along the line. Since normally, only the ends of the line are of interest, such a
procedure seems highly desirable. To this end, we approximate the line by a ladder network as shown in
Fig. 1.
The line possesses a series inductance and resistance per unit length, and a shunt capacitance per unit
length. These distributed parameters are represented in the ladder network by the
lumped elements
L,R
and
C
. Thus, we have created a
lumped model
as an approximation to the real transmission line. The
voltage at any node, or the current through any element, may be obtained by standard circuit analysis
techniques. Hence, we have transformed a problem in partial differential equations into a problem in
simple circuit theory.
Each section of the lumped model corresponds to a given length of transmission line. As the number of
sections is increased, the length of line to which each section corresponds is decreased, and the accuracy
of the approximation is improved. In the limit, as the number of sections becomes infinite, each section
represents an infinitesimal length of time, and we are again faced with the solution of a partial differential
equation. Perhaps the most significant feature of the lumped model is that the approximations have
been made before any equations were written, and each element in the model has definite physical
significance.
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1.4 Semiconductor Lumped Model
In order to simplify our expressions for minority carrier flow, it is convenient to write the continuity
equation in terms of a new variable, the
excess density
,
N
−
N
0
.
∂
(
N
−
N
0
)
∂t
=
−
(
N
−
N
0
)
τ
+
D
∂
2
(
N
−
N
0
)
∂x
2
(2)
For a
p
-type semiconductor:
N
=
n,N
0
=
n
p
Therefore,
∂η
∂t
=
η
τ
+
D
∂
2
η
∂x
2
η
=
n
−
n
p
(3)
For an
n
-type semiconductor:
N
=
p,N
0
=
n
n
Therefore,
∂ρ
∂t
=
−
ρ
τ
+
D
∂
2
ρ
∂x
2
ρ
=
p
−
p
n
(4)
Now let us examine the physical significance of each term in the continuity equation. Consider a long bar
of
n
-type semiconductor with unit cross-sectional area containing volume elements 1 and 2.
Suppose within the bar there exists a distribution of excess holes
ρ
(
x
) as shown in Fig. 2, which is a
function of
x
, but independent of
y
and
z
. The volume elements will possess average excess densities
ρ
1
and
ρ
2 respectively.
Figure 2: Semiconductor bar with excess minority carrier distribution.
Storage.
Due to the charge neutrality requirement, an excess density of minority carriers implies an
equal excess density of majority carriers. Thus, a
change in minority excess density with time produces a
majority carrier current into the volume element.
Such a change in density with time is represented by
the first term in the continuity equation, and the resulting current is the
rate of change of stored charge
in the volume element.
Recombination.
According to the simple linear recombination law, the recombination rate is proportional
to the excess density. Again, the charge neutrality condition requires that if the minority carrier density
remains fixed, for each recombination a new majority carrier must enter the volume element. However, in
order for the minority carrier density to remain fixed, a new minority carrier must also enter the volume
element. Thus, since the carriers possess opposite charges, the net effect of recombination is to bring
minority carrier current into the volume element and force an equal majority current out of the volume
element. This effect is represented by the second term in the continuity equation.
4