Transistor differential circuit with exponential transfer characteristic

To provide for exactly exponential relationship between the collector current and the voltage applied between the bases of the differential circuit, two branches are provided, each containing a series connected circuit including transistors of respectively opposite conductivity type, and resistances positioned in each branch of such value that the sum of the voltage drops of connection and contact resistances arising in the respective branches are compensated. The values of the resistances are so selected that the voltage drop across the respective resistance matches the sum of the voltage drops due to the connection and contact resistances of the opposite branch.

Reference to related patent publications: 
U.S. Pat. No. 3,684,974--Solomon and Davis 
U.S. Pat. No. 3,714,462--Blackmer 
The present invention relates to a transistor differential circuit having 
exponential transfer relation, and more particularly to such a circuit in 
which the relationship between the collector current-ratio of the 
transistors and the base voltage-difference applied to the transistors has 
an exactly exponential relationship. 
BACKGROUND 
Differential stages are used in many applications of which a few 
representative ones are given. 
For use in analog signal processing, see for example: 
(1) Wong & Ott: Function Circuits, McGraw-Hill Book Co., 1976 
(2) Blackmer, D. E.: Multiplier Circuits, U.S. Pat. No. 3,714,462 
(3) Solomon & Davis: Automatic Gain Control Amplifier, U.S. Pat. No. 
3,684,974 
In analog computer technology, for example for logarithmic circuits, 
anti-log circuits, and multipliers--see: 
(4) Wong & Ott: Function Circuits, McGraw-Hill Book Co., 1976. 
In audio technology, e.g. for voltage controlled amplifiers--see: 
(5) Blackmer, D. E.: Multiplier Circuits, U.S. Pat. No. 3,714,462. 
(6) In high frequency technology with automatic gain control (AGC) 
circuits--see the aforementioned Solomon & Davis "Automatic Gain Control 
Amplifier", U.S. Pat. No. 3,684,974. 
In voltage-controlled, voltage-current transfer circuits, in control 
technology, for integrators with voltage-controlled time constant; in 
filter technology, for filters with voltage-controlled limiting frequency, 
and in instrumentation, for function generators and sinusoidal oscillators 
with controlled frequency--see the above reference. 
Ordinary differential circuits frequently do not have exact exponential 
relationship between control voltage to the base of the transistor circuit 
and the collector current; this, apparently, is due to the base connection 
resistances, and emitter contact and connection resistances. These 
connection resistances are also referred to as bulk resistances. The 
voltage drops over these resistances can be compensated--see the "Wong & 
Ott" reference above. This reference discloses that the error due to the 
voltage drop on the emitter and the base bulk resistances can be 
compensated by applying an equal and opposite voltage to the base of the 
transistor which generates the logarithmic function. 
Compensating circuits as previously known have the disadvantage that the 
base connections of the transistors are used for compensation and thus 
cannot be connected to other circuits according to freely selectable 
design requirements. Compensation can be carried out with resistors only 
if a voltage proportional to the current through these resistors is 
available. The voltage, additionally, must be of the proper polarity. 
Junctions with a proportional voltage are loaded by the current flowing 
through the resistors. 
The additional transistors are connected in parallel and thus cause twice 
the current which then must be connected through a current mirror circuit. 
Compensation at high frequency becomes inaccurate and is difficult to uses 
in integrated networks. The compensation error is additionally a function 
of temperature and is different in NPN and PNP differential circuits. 
THE INVENTION 
It is an object to provide a differential circuit utilizing transistors in 
which the relationship between base voltage difference and collector 
current ratio is accurately exponential and which, preferably, can be used 
over a wide range of frequencies and is essentially temperature 
independent. 
Briefly, two branches are provided, each including a first transistor of a 
first conductivity type, for example an NPN, and, serially connected with 
the collector-emitter path thereof, a second transistor of the opposite 
conductivity type, that is, in the selected example a PNP transistor. 
Networks are provided which are connected to one of the transistors of 
each branch and which are dimensioned to control operation of the 
respective transistors to compensate for voltage drops of the connection 
or bulk resistances arising in the respective branches. Typically, the 
connections include resistors of low value which are, respectively, 
connected to the collectors and bases of the PNP transistors and 
additional resistors which interconnect the collector of one PNP 
transistor with the base of the other PNP transistor; in another form of 
the invention, low-resistance resistors are serially connected with the 
collectors of the PNP transistors and, further, the collectors of the PNP 
transistors in the respective branches are connected to the bases of the 
PNP transistors of the other branches, thereby effecting a cross 
connection. 
The network thus leaves free the bases of the main transistors--in the 
example of the NPN transistors--which can be connected to any other 
circuit element since the base connections are not needed for 
compensation. No additional auxiliary voltages or auxiliary currents 
external to the differential stage are used. 
Voltages proportional to the currents flowing through the connection and 
contact resistances are generated by the collectors of the series of 
connected transistors and additional resistors. If the entire circuit is 
placed on the integrated chip, the connection resistances are formed by 
connection track resistances on the chip. The series connected PNP 
transistors can be looked at as diodes for purposes of the user of the 
compensated differential stage, and generally do not cause additional 
power or heating losses in the overall circuit which are in excess of 
neglectable power losses. The compensation error as a function of 
temperature in the N-differential stage and the P-differential stage is 
the same if each branch in the differential stages has the same number of 
NPN and PNP transistors, which is the case in the circuit of the present 
invention. The compensation is accurate also at high frequency, since no 
phase shift due to current mirror circuits occurs. The differential stages 
can readily be integrated with processes with dielectric isolation.

The N-differential stage of FIG. 1 has two NPN transistors 1, 2, and two 
PNP transistors 3, 4, and six compensating resistors 31, 32, 33 and 41, 
42, 43. Resistors 31, 32 are the collector and base resistors, 
respectively, for a PNP transistor 3. Resistors 41, 42 are the collector 
and base resistors for a second PNP transistor 4. The resistor 33 is 
connected between the collector of one PNP transistor 3 in one branch of 
the circuit and the base of the other PNP transistor 4 in the other 
branch. Resistor 43 is connected between the collector of the second PNP 
transistor 4 and the base of the first PNP transistor 3. The four 
connecting resistors 31, 32, 41, 42 are connected together and at one 
terminal as a single junction 9, which forms the emitter of the 
N-differential stage of FIG. 1. The two connecting resistors 31, 41 have, 
for example, a value of 1 ohm each. The two resistors 32, 42 have, for 
example, a value of 100 ohms each. The importance circuit configuration in 
the example of FIG. 1 is, however, that the voltage drop across resistor 
42, for example, is equal to the sum of the voltage drops of the 
connection and contact resistances of the two transistors 1, 3; similarly, 
the voltage drop across resistor 32 should be the same as the sum of the 
voltage drops of the connection and the contact resistances of the 
transistors 2 and 4. The two other resistors 33, 43 must have a resistance 
which permits meeting the foregoing requirement. In the example of FIG. 1, 
the resistors 33, 43 have a value of 50 ohms each. The two resistors need 
not have the same resistance value. The N-differential stage of FIG. 1 has 
base connections 12, 13 which are connected to the bases of the NPN 
transistors 1, 2. The collector connections 10, 11 of the N-differential 
stage are directly connected to the respective collectors of transistors 
1, 2. 
FIG. 2 shows an N-differential stage which has two NPN transistors 5, 6 and 
two PNP transistors 7, 8, and two compensating resistors 71, 81. FIG. 2 
has a circuit which is simpler than that of FIG. 1. The resistance values 
of the two resistors 71, 81 in each branch must be so dimensioned that the 
voltage drop over the compensation resistor of one branch is equal to the 
sum of the voltage drops of the connecting and the contact resistances of 
the transistors of the respective branch. The base of transistor 8 is 
connected to the junction of the collector of transistor 7 and one 
terminal of resistor 71 by a connecting line 72; the base of transistor 7 
is connected to the junction of the collector of transistor 8 and one 
terminal of the resistor 81 by a cross connecting line 82. The other 
terminals of the two resistors 71, 81 are connected together and to form 
the emitter terminal 9 of the overall N-differential stage. The collector 
terminals 10, 11 of the stage are directly connected to the collectors of 
the respective transistors 5, 6. The base terminals 12, 13 of the 
differential stage are connected directly to the bases of the transistors 
5, 6. 
The resistance value of the respective resistors 71, 81 is in the order of 
about 0.6 ohms. These resistors can be formed by discrete resistors, as 
shown in FIG. 2, or may be formed by suitable contact resistances, for 
example within the contact connection on the same semiconductor chip which 
includes the emitter terminal 9. 
FIG. 3 is identical to FIG. 1 but includes one possible set of resistance 
values for specific types of transistors. The transistors are BC337 and 
BC327 types with well-matched base-emitter-voltages. 
FIG. 4 is identical to FIG. 2 but includes the resistance values for the 
same transistor types as above. The resistor 71 has a value of about 0.8 
ohms and represents the sum of the emitter-bulk-resistances of the PNP- 
and NPN- transistors and the sum of the base-bulk-resistances divided by 
the current gain of the transistors. These four terms of the sum have 
about the same value of about 0.2 ohms with this (large) 0.8A- transistor 
types. With collector currents up to 10 mA these values are nearly 
constant. Note that the voltage drop across a base-bulk-resistance is 
caused by the base current, which equals the collector current divided by 
the current gain. The base-bulk-resistance has a value of about 30 ohms 
(independent of current up to 10 mA). The current gain has a value of 
about 150. So the quotient has a value of about 0.2 ohms. 
FIG. 5 shows the application of the invention in the Multiplier Circuit of 
U.S. Pat. No. 3,714,462. For easier identification, the same topology and 
the same element numbering has been used. 
The original circuit uses two differential circuits: a P-type differential 
circuit is formed with transistors Q1 and Q4; and a N-type differential 
circuit is formed with transistors Q2 and Q5. 
FIG. 5 shows an improved Multiplier or Voltage Controlled Amplifier with 
two differential circuits as described in connection with FIGS. 1-4 
thereof. The application of the concept of the present invention improves 
the distortion performance over the prior art by a factor of about fifty, 
or 32 dB. 
Various changes and modifications may be made in the circuit, depending on 
the technology for which the circuit are being used and in a network into 
which they are to be connected. 
The circuit of FIG. 1 appears more complex, but it is easier to manufacture 
with discrete components, since for example the higher resistance values 
of the resistors 33, 43 can be adjusted according to the resistance values 
of the transistors. 
The circuit of FIG. 2 can be constructive on a single chip and the 
resistors 71, 81 are automatically well matched to the resistances of the 
transistors. This embodiment thus may be preferred when quantities justify 
the cost of single-chip manufacture. 
The referenced book by Wong and Ott describes, in connection with functions 
circuits, an arrangement to compensate for errors due to voltage drops on 
the emitter and base bulk resistances. If the terminal collector-base 
voltage of a transistor connected, for example, in circuit with an 
operational amplifier, is essentially zero, the only collector-base 
voltage variation will be due to the collector resistance r.sub.cs. The 
voltage drop on the collector resistance produces a change in the terminal 
base-emitter voltage of 
EQU .mu..sub.ic R.sub.cs. 
The total base-emitter voltage is the sum of the intrinsic base-emitter 
voltage, the voltage drop on the bulk resistance R.sub.B and the feedback 
voltage .mu..sub.ic R.sub.cs : 
##EQU1## 
For modern silicon transistors the collector saturation current is only 
about 0.1 pA and the collector saturation resistance is of the order of 5 
to 100 .OMEGA., which results in the term (1+ql.sub.cs r.sub.cs /kT) being 
different from 1 by less than 10.sup.-9. The effective bulk resistance 
r.sub.B ranges between 0.25 and 10 .OMEGA., depending on the size of the 
transistor, and is generally large compared to .mu.r.sub.cs, since the 
feedback factor is typically 3.times.10.sup.-4 and as a result 
.mu.r.sub.cs is in the range of only 0.0015 to 0.03.mu.. The conclusion is 
that the only significant error in the simple expression for the 
base-emitter voltage given in the above equation is the effect of bulk 
resistance in the emitter and base, provided the voltage drop on the 
collector saturation resistance as previously discussed is less than kT/q, 
which is about 26 mV at +25.degree. C. 
By applying a voltage of the proper magnitude to the base of the transistor 
with the logarithmic transfer function, it is possible to compensate for 
the error due to the voltage drop on the emitter and base bulk 
resistances. Such a resistor, connected between the base and ground, or 
reference, should have a resistance value which is kept small, since it 
increases the effective value of r.sub.B by the amount of the resistance 
value divided by .beta., in which .beta. is the common emitter-current 
gain of the transistor. 
For an exhaustive discussion of this subject matter, reference is made to 
the aforementioned textbook by Wong and Ott, "Function Circuits".