NEGATIVE DIFFERENTIAL RESISTANCE IN TWO-TRANSISTOR
ONE-PORTS WITH NO INTERNAL SOURCES *
Ljiljana Trajkovit and Alan N. Willson, Jr.
Electrical Engineering Department
School of Engineering and Applied Science
University of California, Los Angeles, California 90024
Abstract
In this paper we examine the occurrence of Negative Difletential Resistance (NDR) in transistor one-ports which contain
resistors and only two transistors. For practical transistor circuits certain types of transistors and circuit topology are necessary for the occurrence of NDR behavior. We prove two recent
conjectures of Chua et al. [ l ,21 related to these issues, namely
that in such circuits similar types of transistors are required
for an N-type NDR, and complementary types for an S-type
NDR behavior. We also give necessary topological conditions
for achieving biasing of the transistor pn junctions appropriate
for the occurrence of NDR. These then lead to a specification of
the minimum number of resistors that must be present in such
circuits.
I.
I
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Figure 1: Various types of NDR behavior.
Introduction
When a transistor circuit is structured as a one-port, with
the only independent sources placed at the port terminals,
certain limitations are placed on the one-port’s ability t o
realize various types of NDR. In this paper we examine
these limitations in the case of two-transistor one-ports.
In general, two-transistor one-ports can exhibit various types of NDR. The shape of the one-port’s v vs. i
curve depends on the circuit’s topology, parameter values,
and type of transistors. We distinguish between a voltagecontrolled (or N-type) curve, a current-controlled (or S
type) curve, and a multiple-valued (or C-type) curve, as
shown in Figure 1.
We consider practical NDR transistor one-ports-ones
which can provide NDR behavior for practical values of
circuit parameters, and practical values of the applied independent voltage or current source. Such circuits must
possess a feedback structure. Moreover, they must be able
to provide appropriate biasing of the transistor pn junctions [3].
Taking a slightly different perspective, we consider the
possible location of a single independent source in a twotransistor circuit (thus defining the sourceless one-port’s
port), and we prove that certain locations preclude the occurrence of NDR. These restrictions on port location are
related to the orientation of existing feedback structures
in a particular circuit. We examine the location of the independent source with respect to the dominant feedback
structure 41 in a particular circuit. We first prove that
a practica two-transistor circuit having a single independent source connected directly to a transistor base cannot
exhibit NDR behavior. We then prove Chua’s conjectures.
Our proof is based on the no-gain property of transistors
[5] and the biasing conditions necessary for the occurrence
of NDR in practical two-transistor circuits (31.
c -
11.
Transistor Types Necessary for the
Occurrence of NDR
A series or parallel connection of a resistor with an NDR
one-port would only diminish the prominence of the oneport’s NDR behavior [l].It follows therefore that we may
assume, without loss in generality, that the connection of
the circuit’s independent source is made only to transistor
terminals.
The following lemma is related to the permissible locations of the port and the occurrence of N-type and S t y p e
NDR. It follows immediately from the topological requirements for the occurrence of N-type and S t y p e NDR in
transistor circuits (i.e., for any N-type NDR, when the
port is open-circuited the resulting circuit must possess
a feedback structure, and for an S t y p e NDR a feedback
structure must be present when the port is short circuited).
Lemma 1. If, for a two-transistor circuit, the feedback
structure ( i j ) is dominant [4] the corresponding sourceless
one-port cannot exhibit N-type (Stype) NDR behavior if
the port is positioned in any of the branches indicated by
Y (by I ) in Figure 2.
The following theorem is based on the biasing theorem
[3] and the no-gain property of transistors [5]. If the only
independent source in a circuit is connected t o the base
of a transistor, as illustrated by the example of Figure 3,
then this particular transistor will operate either in the
diode or the 08state, depending on the source polarities.
In the case of practical transistor circuits this operational
state implies that the one-port that faces the source cannot
exhibit NDR behavior.
I
Theorem 1. A practical two-transistor circuit having a
single independent source connected directly to a transistor base cannot exhibit NDR behavior.
We now examine possible positions of the independent source with respect to the active feedback structure
‘This research was supported by NSF Grants ECS 82-06207 and
DCI 86-03639.
747
ISCAS’88
CH2458-8/8810000-0747$1 .OO 0 1988 IEEE
i*
j*
Figure 2: Positions of an ihdependent source not permitting NDR behavior due to circuit's topology.
[6] in any particular circuit. The following theorem holds
for practical one-ports containing only resistors and two
transistors.
Theorem 2. If a single independent voltage (current)
source is connected, by making a parallel (series) insertion,
in a circuit containing only resistors and two transistors,
an active feedback structure (i,j) can result only if it connects to a t least one of the nodes i, j (only if it is made in
the path i - j ) .
The proof of Theorem 2 first considers N-type NDR
circuits. It follows from Lemma 1 that the only possible
position of an independent voltage source which contradicts the conditions of Theorem 2 is one where the voltage
source is connected to node i* and node j * , as shown in
Figure 4. Biasing conditions [3] require that, in this case,
transistor 2'1 is npn. In practical transistor circuits resistances R, and Rb should be very small, which in turns
implies that T I will be ofi having its base connected to the
negative potential of the source. From the biasing theorem
[3] such a state cannot lead to NDR behavior.
For an S t y p e NDR it follows from Lemma 1 that the
only position of the independent current source contradicting Theorem 2 is one of the two shown in Figure 5. It
then follows, for example, that transistor 2'1 has to be an
npn type, which, again, implies that it is in the of state.
From the biasing requirement [3] such a state cannot lead
to NDR behavior.
We now prove, for practical transistor circuits, two
recent conjectures by Chua
et al. [l, 21 related to the
type of transistors necessary for the occurrence of NDR in
two-transistor circuits:
Figure 3: Example of an independent voltage source (a),
or current source (b), connected directly to a transistor
base.
Our proof is based on the no-gain property and the
permissible locations of the independent source in the case
of practical transistor circuits, as illustrated in Figure 6
and Figure 7.
In the case of an N-type NDR the only possible connections of the independent source are as in Figure 6. Feedback structure i - j is dominant only if R f < Re [4]. In
a practical transistor circuit the difference between resistances R f and Re should be very large. Such values of the
resistances in a circuit result in the potential of node j being close to the potential of node i. Therefore, the current
flows and the voltage distributions indicate that for the
occurrence of NDR the transistors should be of a similar
type.
In the case of an S t y p e NDR the only possible connections of the independent current source, after possibly
relabeling 2'1 and Tz,are as shown in Figure 7. In the
case of Figure 7(a), due to the no-gain property of transistors, the necessary biasing conditions for the occurrence of
NDR are satisfied only if the two transistors are of complementary type. In the case of Figure 7(b), by the same
reasoning as in the case of N-type NDR, the potential of
node i is close to the potential of node j , and hence, for the
occurrence of NDR the two transistors must be of complementary type.
It is easy to show that the conjectures do not hold if
internal independent sources are present in a two-transistor
one-port [7].In addition to NDR circuits with BJT devices one can
synthesize various NDR one-ports employing FET devices
[S,91. While it is essential to use the appropriate types
of BJT devices in order to obtain a desired type of NDR
behavior, in the case of circuits employing FET devices it
is possible to construct one-ports exhibiting either N-type
Conjecture [I, 21 In order to obtain a twotransistor type-S negative resistance device, the
two transistors must be complementary, i.e., one
is p-n-p, the other is n-p-n.
Conjecture [I, 2) In order to obtain a twotransistor type-N negative resistance device, the
two transistors must be of the same type, i.e.,
both p-n-p or both n-p-n.
We now proceed to prove these conjectures, which we
have restated in the following theorem:
Theorem 3. A one-port containing only resistors and two
transistors can exhibit N-type (Stype) NDR behavior only
if the transistors are of similar (complementary) type.
'A partial proof for these conjectures was given by Chua, et al.
in [2] for a limited class of circuits. Our comments on the proof are
given in 161.
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Figure 4: Position of the independent voltage source not
permitting N-type NDR behavior due to inappropriate biasing.
Figure 6: The appropriate connections of the voltage
source for an N-type NDR.
(4
Figure 5: Positions of the independent current source not
permitting S t y p e NDR behavior due t o inappropriate biasing.
Re
or S t y p e NDR behavior with two similar or two complementary FETdevices. In one-ports with FET devices their
threshold voltage is a parameter which can be used t o vary
the shape of the one-port’s NDR characteristics, but again,
the sign of the threshold voltage does not determine the
type of NDR behavior the one-port will exhibit.
111.
Topology Necessary for the
Occurrence of NDR
(b)
Figure 7: The appropriate connections of the current
source for an S-type NDR.
We now examine the topology necessary for the occurrence
of NDR in two-transistor one-ports with no internal independent sources. It is known [lo] that practical twotransistor circuits with no resistors cannot exhibit NDR.
Therefore, the presence of resistors in such a circuit is essential.
We first examine N-type NDR one-ports. It has been
observed by Chua et al. 11, 2, 81 by means of an “exhaustive searchn that a one-port which exhibits N-type NDR
must have a t least three resistors. Such a one-port and
its SPICE generated response is shown in Figure 8. By
examining the transistor biasing we can observe that the
resistors RI and Rz provide necessary biasing for transistor T I by precluding the direct connection of the voltage
source t o the transistor base. Resistor R3 provides the appropriate biasing of transistor Tz (in its absence Tz cannot
be on). Hence, a two-transistor one-port having the only
independent source connected at the terminals of the oneport can exhibit N-type NDR behavior only if the circuit
obtained by open-circuiting the port has the topology of
Figure 9, and it possesses, at least, the three resistors R I ,
R z , and RB.
In the case of S t y p e NDR one-ports, an example of
a one-port having one resistor is shown in Figure 10. Its
U vs. i port curve is obtained by SPICE simulation. We
have considered the possible locations of a single resistor
and we obtained the topology necessary for the occurrence
of NDR. A series insertion of a resistor in a branch will not
suffice, since the magnitude of the largest current which
can then flow through the circuit will be on the order of a
transistor saturation current. Therefore, a tw-transistor
one-port having the only independent source connected a t
the terminals of the one-port can exhibit S t y p e NDR behavior only if the circuit obtained by short-circuiting the
port has a t least one resistor and the topology of Figure 11.
In the case when internal independent sources are permitted in a one-port, the presence of resistors is no longer
a requirement for NDR behavior. It is not difficult t o construct examples of one-ports exhibiting NDR behavior by
not having resistors, or by having fewer than specified. [7].
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Figure 8: N-type NDR circuit with three resistors.
Figure 9:
Necessary topological structure for an
open-circuited N-type NDR sourceless one-port.
IV.
Concluding Remarks
We have examined the occurrence of NDR in transistor
circuits structured as one-ports which contain only resistors and two transistors. The location of the port in such
circuits plays an important factor in the one-port’s ability t o exhibit NDR behavior. We first established port
locations appropriate for the occurrence of NDR. Using
the no-gain property of transistors we then deduced the
appropriate biasing of transistor pn junctions for various
port locations, and proved that in practical transistor circuits similar types of transistors are required for an N-type
NDR, and complementary types for an S t y p e NDR behavior. These considerations then lead t o a specification of the
minimum number of resistors that must be present in such
circuits.
Figure 11:
Necessary topological structure for a
short-circuited S t y p e NDR sourceless one-port.
Trans. Circuits Syst., vol. CAS-22, pp. 6 7 8 4 8 7 , Aug.
1975.
[6] Lj. TrajkoviC, “Negative Differential Resistance in
Transistor Circuits,” Electr. Eng. Dept., UCLA, Los
Angeles, CA, 1986.
171 Lj. TrajkoviC and A. N. Willson, Jr., “Behavior of
Nonlinear Transistor One-Ports: Things Are Not Always As Simple As Might Be Expected,” Proc. 30th
Midwest Symposium on Circuits and Systems, Syracuse, NY, Aug. 1987, t o be published.
[8] L. 0. Chua, J. B. Yu, and Y. Y. Yu, “Bipolar-JFETMOSFET Negative Resistance Devices,” IEEE Trans.
Circuits Syst., vol. CAS-32, pp. 46-61, Jan. 1985.
[9] C. J. Wu and C. Y. Wu, “The New General Realization Theory of FET-like Integrated VoltageControlled Negative Differential Resistance Devices,”
IEEE Trans. Circuits Syst., vol. CAS-28, pp. 382-390,
May 1981.
[lo] Lj. Trajkovif. and A. N. Willson, Jr., “A TwoTransistor Circuit Having No Resistors Cannot Model
an SCR,” Proc. IEEE Int. Symp. on Circuits and Systems, Philadelphia, PA, 1987, pp. 888-891.
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750