Exponential Function Circuitry

This invention relates to circuits that generate electrical currents proportional to an exponential function of one or more input currents.

If I.sub.d is the current flowing through a diode, then the voltage across the diode is equal to V.sub.t .multidot.1n[I.sub.d +I.sub.s)/I.sub.s], where I.sub.s is the saturation current of the diode. V.sub.t =K.sub.B T/q, where K.sub.B is Boltzmann's constant, T is the temperature, and q is the charge of an electron. Since I.sub.s is typically in the range of 10.sup.-18 to 10.sup.-16 amperes, and I.sub.d >>I.sub.s, the voltage across the diode closely approximates V.sub.t .multidot.1n(I.sub.d /I.sub.s). Likewise, the voltage across the base-emitter junction of a transistor closely approximates V.sub.t .multidot.1n(I.sub.d /I.sub.s) where I.sub.c is the current flowing into the collector of the transistor.

FIG. 1 shows a circuit 100 that produces an output current I.sub.o equal to the square root of the product of currents I.sub.1 and I.sub.2. The saturation current I.sub.s is the same for all of the transistors in the circuit. Current source 102 produces current I.sub.1 and current source 104 produces current I.sub.2. Current source 102 is connected between a voltage source 106 and the collector of transistor 108. The emitter of transistor 108 is connected to ground. The voltage at the base of transistor 108 is therefore V.sub.t .multidot.1n(I.sub.1 /I.sub.s). The base of transistor 108 is connected to the emitter of transistor 110. Current source 104 is connected between the emitter of transistor 110 and ground. The collector of transistor 110 is connected to the voltage source 106. The voltage at the base of transistor 110 is therefore V.sub.t .multidot.1n(I.sub.1 /I.sub.s)+V.sub.t .multidot.1n(I.sub.2 /I.sub.s). The base of transistor 110 is connected to current source 102 and the base of transistor 112. The emitter of transistor 112 is connected to the collector and base of transistor 114, which functions as a diode. The emitter of transistor 114 is connected to ground. The voltage at the base of transistor 112 is therefore 2V.sub.t .multidot.1n(I.sub.o /I.sub.s). Thus,

Other circuits produce an output voltage equal to the square root of an input voltage. For example an operational amplifier can be connected with a diode in its feedback loop, so that the operational amplifier produces an output proportional to the logarithm of an input voltage. The logarithm output is connected to a voltage divider that produces an output voltage equal to one-half of the input voltage to the voltage divider. The output of the voltage divider is connected to the inverting input of a second operational amplifier through a diode, so that the second amplifier produces an output proportional to the antilogarithm of the output of the voltage divider. Thus,

In another circuit, an input voltage V.sub.in is connected through a resistor to the inverting input of an operational amplifier. The output, V.sub.out, of the operational amplifier is connected to a multiplier circuit whose output is equal to -(V.sub.out).sup.2. The output of the multiplier circuit is connected through a resistor to the inverting input of the operational amplifier. V.sub.out equals V.sub.in 1/2.

In one aspect the invention features a circuit that generates an electrical current representative of an exponential function of an input current. The circuit includes an input diode chain and an output diode chain. Each of the diodes in the input diode chain has an input current passing therethrough, creating a voltage drop across the input diode chain. A voltage driving circuit drives a voltage drop across the output diode chain that has a predetermined relationship to the voltage drop across the input diode chain. The voltage drop across the output diode chain results in a current through the output diode chain that is proportional to an exponential function of the input current or currents.

In another aspect, the invention features a circuit for generating electrical currents representative of an exponential function of an electrical input current, in which each diode in an input diode chain is connected in series with an input current source. The input current source or sources are connected below the cathode of the diode. An output diode chain has a voltage drop across itself proportional to the voltage drop across the input diode chain.

In one embodiment of the invention, the input diode chain includes first and second input subchains. A first current source pulls a first input current through the first and second input subchains. A second current source pulls a second input current through the second input subchain only. The first and second subchains of the input diode chain each have a number of diodes equal to one-half the number of diodes in the output diode chain. The current through the output diode chain is equal to the square root of the product of the first input current and the sum of the first and second input currents.

In another embodiment, the first current source pulls the first input current through the first input subchain only. The second current source pulls the second input current through the second input subchain only. The current through the output diode chain is equal to the square root of the product of the first and second input currents.

In preferred embodiments, the voltage driving circuit is a differential amplifier having first and second npn transistors. The differential amplifier is configured to force the voltage at the base of the second transistor equal to the voltage at the base of the first transistor. The base of the first transistor is connected to the cathode of the bottommost diode in the input diode chain. The base of the second transistor is connected to the cathode of the bottommost diode in the output diode chain. The anode of the topmost diode in the input diode chain is connected to the anode of the topmost diode in the output diode chain.

Circuits according to the invention can exhibit a high degree of precision, the precision being enhanced by increasing the number of diodes in the input and output diode chains. Since the input current sources are connected below the cathodes of the diodes through which the input current sources pull the input currents, the input current sources can be npn transistors, rather than more expensive current sources that utilize high-speed pnp transistors or high-speed amplifiers. Because the differential amplifier also consists of npn transistors, circuits according to the invention can exhibit a high-speed response to changes in the input currents. The transistors into which the output currents flow require very little head room. The head room can be as low as 0.2 volts.

Other advantages and features will become apparent from the following detailed description and from the claims when read in connection with the accompanying drawings.

We first briefly describe the drawings.

FIG. 1 is a circuit diagram of a prior art circuit that produces an output current equal to the square root of the product of two input currents.

FIG. 2 is a circuit diagram of a circuit according to the invention that produces output currents proportional to the square root of the product of a first input current and the sum of the first input current and a second input current.

FIG. 3 is a circuit diagram of a circuit according to the invention that produces output currents proportional to the square root of the product of two input currents.

FIG. 4 is a circuit diagram of a circuit according to the invention that produces output currents proportional to an exponential function of a product or a ratio of input currents.

FIG. 2 is a circuit diagram of a multiple-output square root circuit according to the invention. The circuit includes an input diode chain 14 and an output diode chain 18. The diodes may be the base-emitter junctions of npn transistors, where the base of each transistor is connected to the transistor's collector. Diode chain 14 consists of two input sub-chains 20 and 22, each having N diodes, where N is any number greater than or equal to 1. Output diode chain 18 has 2N diodes. The voltage at the top of input diode chain 14 equals the voltage at the top of output diode chain 18. A voltage driving circuit in the form of a differential amplifier 24 forces the voltage at the bottom of diode chain 18 equal to the voltage at the bottom of diode chain 14, as explained in greater detail below.

A first input current I.sub.in1 passes through the entire length of input diode chain 14, while a second input current I.sub.in2 passes only through input subchain 20. Thus, the current through input subchain 20 is equal to I.sub.in1 plus I.sub.in2, and the current through input subchain 22 is equal to I.sub.in1. The small base current to transistor 26 is negligible compared to the input currents I.sub.in1 and I.sub.in2, and can thus be ignored. The current sources that produce currents I.sub.in1 and I.sub.in2 can be npn transistors having a resistor connected between the emitter and ground and having a fixed voltage applied to the base.

The voltage across each diode is equal to V.sub.t .multidot.1n[(I.sub.d +I.sub.s)/I.sub.s ]. V.sub.t =k.sub.B T/q, where k.sub.B is Boltzmann's constant, T is the temperature, and q is the charge of an electron. I.sub.d is the current through the diode, and I.sub.s is the saturation current of the diode. I.sub.s for each diode is proportional to the diode area.

Since I.sub.s is typically in the range of 1O.sup.-18 to 10.sup.-16 amperes, and I.sub.d >>I.sub.s, the voltage across each diode closely approximates V.sub.t .multidot.1n(I.sub.d /I.sub.s). The voltage across diode subchain 20 is therefore NV.sub.t .multidot.1n[(I.sub.in1 +I.sub.in2)/I.sub.s20 ], and the voltage across input subchain 22 is NV.sub.t .multidot.1n(I.sub.in1 /I.sub.s22), where I.sub.s20 and I.sub.s22 are the saturation currents of each of the diodes in diode subchain 20 and each of the diodes in diode subchain 22, respectively. Since the differential amplifier 24 forces the voltage at the bottom of output diode chain 18 equal to the voltage at the bottom of input diode chain 14, a current I.sub.o flows through output diode chain 18 to the collector of transistor 29. The voltage across output diode chain 18 is therefore equal to 2NV.sub.t .multidot.1n(I.sub.o /I.sub.s18), where I.sub.s18 is the saturation current of each of the diodes in output diode chain 18. The small base current to transistor 28 can be ignored.

Let V.sub.os be the offset voltage between the base of transistor 26 and the base of transistor 28 in differential amplifier 24. Since the voltage across input diode chain 14 plus the offset voltage V.sub.os of the differential amplifier is equal to the voltage across output diode chain 18,

Thus, 0.51n{[(I.sub.in1 +I.sub.in2).multidot.I.sub.in1 ]/(I.sub.s20 .multidot.I.sub.s22)}+0.5.multidot.V.sub.os /(NV.sub.t)=1n(I.sub.o /I.sub.s18). If V.sub.os =0, then

Thus, I.sub.o =[I.sub.s18 /(I.sub.s20 .multidot.I.sub.s22)1/2].multidot.[(I.sub.in1 +I.sub.in2).multidot.I.sub.in1 ]1/2. If the saturation current is the same for all of the diodes in input subchains 20 and 22 and output diode chain 18, then I.sub.o =[(I.sub.in1 +I.sub.in2).multidot.I.sub.in1 ]1/2.

The current I.sub.o flows into the collector of transistor 29. The actual output currents of the square root circuit, I.sub.o1, and I.sub.o2, flow into the collectors of transistors 30 and 32, which have their bases connected to the base of transistor 29. Resistors 34, 36, and 38 connect the emitters of transistors 29, 30, and 32, respectively, to ground. If the resistors 34, 36, and 38 all have the same resistance, and if the emitter areas of all three transistors 29, 30, and 32 are the same, then output currents I.sub.o1, and I.sub.o2, which enter the collectors of transistors 30 and 32, respectively, will both be equal to the current I.sub.o that enters the collector of transistor 29. By decreasing the resistance of resistor 36 or 38 relative to the resistance of resistor 34, or by using a transistor 30 or 32 having an emitter area greater than the emitter area of transistor 29, output current I.sub.o1 or I.sub.o1, respectively, can be made greater than but proportional to I.sub.o. Likewise, by increasing the resistance of resistor 36 or 38 relative to the resistance of resistor 34, or by using a transistor 30 or 32 having an emitter area smaller than the emitter area of transistor 29, output current I.sub.o1 or I.sub.o2, respectively, can be made less than but proportional to I.sub.o. For example, if the resistance of resistor 36 is 1/k times the resistance of resistor 34, and the emitter area of transistor 30 is k times the emitter area of transistor 29, where k is a constant, the output current I.sub.o1 will be k times I.sub.o. Note that if the voltage across resistor 36 or resistor 38 is low enough, the voltage at the collector of transistor 30 or transistor 32 can be as low as 0.2 volts without transistors 30 or 32 becoming saturated. Thus, transistors 30 and 32 provide output current sources that can drive low output voltages.

In addition to input diode chain 14 and output chain 18, the square root circuit includes diode chains 12 and 16. Diode chain 12 is used to provide sufficient head room for the proper operation of the input current sources, as described below. "Head room" as used in this specification and in the claims refers to the voltages above the input current sources as shown in the Figures, e.g., the voltage at the base of transistor 26 and the voltage at the point between input diode subchains 20 and 22 in FIG. 2. Diode chain 16 is used to ensure that transistors 26 and 28 of differential amplifier 24 are not saturated, and to reduce error in the offset voltage V.sub.os of differential amplifier 24, as described below.

Diode chain 16 has M diodes, and diode chain 12 has M+2N+2 diodes. The number M can be any number greater than or equal to zero. The value of M determines the voltage at the base of transistor 26 and the voltage at the point between input diode subchains 20 and 22, and hence the value of M determines the amount of head room available for the input current sources.

Current flows from supply voltage 48, through resistor 44, and through the diodes in diode chain 12 to ground. The voltage at the top of diode chain 12 is equal to (M+2N+2).multidot.V.sub.be, where V.sub.be is the voltage across each diode. As explained above, V.sub.be varies with the amount of current that passes through each diode, but since V.sub.be varies logarithmically with the current, V.sub.be can be assumed to be approximately the same for each diode in the circuit for purposes of the analysis to follow. The voltage at the emitter of transistor 42 is equal to (M+2N+1)V.sub.be, because the Voltage drop across the base emitter junction of transistor 42 is V.sub.be. The voltage at the base of transistor 26 is (M+1)V.sub.be, because the voltage across each of the 2N diodes in input diode chain 14 is V.sub.be. Thus, diode chain 12 sets up a common reference voltage at the top of diode chains 14 and 18, and provides for a voltage at the bottom of input diode chain 14 that leaves sufficient head room for the proper operation of the input current source associated with I.sub.in1.

Current source 50 causes current to flow from supply voltage 48 through transistor 46 and diode chain 16. The voltage at the base of transistor 46 is equal to (M+2)V.sub.be plus the voltage across resistor 34, since the voltage across each diode in diode chain 16 and across the base-emitter junctions of transistors 28 and 46 is V.sub.be. Since the base of transistor 46 is connected to the bases of transistors 54 and 56, the voltage at the emitter of transistor 54 and the voltage at the emitter of transistor 56 will equal (M+1)V.sub.be plus the voltage across resistor 34. Thus, the voltage at the collectors of transistors 26 and 28 will never be less than the voltages at the bases of transistors 26 and 28. (Recall that the differential amplifier 24 forces the voltage at the base of transistor 28 approximately equal to the voltage at the base of transistor 26.) Transistors 26 and 28 therefore will never be saturated. Moreover, since the voltages at the collectors of transistors 26 and 28 are the same, error in the offset voltage V.sub.os of differential amplifier 24 is minimized.

Differential amplifier 24 consists of transistors 26, 28, 54, and 56, current sources 52 and 58, and compensation capacitor 60. Current source 52 delivers current from supply voltage 48 through transistor 54 to the collector of transistor 26. Current source 58 produces a current equal to twice the current produced by current source 52, so that a current flows into the collector of transistor 28 that is equal to the current flowing into the collector of transistor 26. Since the current flowing through transistor 26 equals the current flowing through transistor 28, the base-emitter voltage drop of transistor 26 equals the base-emitter voltage drop of transistor 28. Thus, differential amplifier 24 drives the voltage at the base of transistor 28 approximately equal to the voltage at the base of transistor 26. Because the differential amplifier 24 is a closed-loop system subject to possible oscillation effects, a compensation capacitor 60 is used to stabilize the differential amplifier 24.

The accuracy of the square root circuit can be enhanced by increasing the number N of diodes in the input diode subchains 20 and 22. Recall that NV.sub.t .multidot.1n[(I.sub.in1 +I.sub.in2)/I.sub.s ]+NV.sub.t .multidot.1n(I.sub.in1 /I.sub.s)+V.sub.os =2NV.sub.t .multidot.1n(I.sub.o /I.sub.s), where V.sub.os is the offset voltage of differential amplifier 24. If V.sub.os is not exactly equal to zero, then the term V.sub.os introduces error into the result I.sub.o =[(I.sub.in1 +I.sub.in2).multidot.I.sub.in1 ]1/2. As N increases, however, the error caused by the term V.sub.os is minimized. The maximum number of diodes in diode chains 14 and 18 is limited only by the supply voltage 48. Thus, if N is large enough, the circuit can achieve a high degree of precision. Moreover, since the differential amplifier 24 consists entirely of npn transistors, the square root circuit exhibits a high-speed response to changes in the input currents I.sub.in1 and I.sub.in2.

There is shown in FIG. 3 an alternative configuration of input diode chain 14. The bottom of input diode subchain 20 is connected to the base of transistor 62, rather than being connected directly to the top of input diode subchain 22. The top of diode subchain 22 is connected to the emitter of transistor 62. The collector of transistor 62 is connected to the emitter of transistor 42. Ignoring the small base currents to transistors 26 and 62, the current through input subchain 20 is equal to I.sub.in1, and the current through input subchain 22 is equal to I.sub.in2. Note that there are N-1 diodes, rather than N diodes, in input diode subchain 22, because the current I.sub.in2 passes through the base-emitter junction of transistor 62, which functions as one diode voltage drop. With this configuration of diode chain 14, the current I.sub.o through diode chain 18 will equal (I.sub.in1 .multidot.I.sub.in2)1/2.

There is shown in FIG. 4 an alternative configuration of output diode chain 18 that results in output currents proportional to exponential functions of products or ratios, where the exponential function need not be a square root function. Output diode chain 18 includes subchain 64 and subchain 66. The top of diode subchain 64 connects with the emitter of transistor 42. The bottom of diode subchain 64 connects with the base of transistor 68. The collector of transistor 68 connects with the emitter of transistor 42, and the base-emitter junction of transistor 68 forms the first diode drop in diode subchain 66. The bottom of subchain 66 connects with the base of transistor 28 of differential amplifier 24.

An input current I.sub.in3 passes through diode subchain 64. The voltage across each diode in diode subchain 64 is V.sub.t .multidot.1n(I.sub.in3 /I.sub.s64), where I.sub.s64 is the saturation current of each of the diodes in subchain 64. Likewise, the voltage across each diode in diode subchain 66 is V.sub.t .multidot.1n(I.sub.o /I.sub.s66), where I.sub.s66 is the saturation current of each of the diodes in subchain 66. If diode subchain 20 has A diodes, diode subchain 22 has B diodes, diode subchain 64 has C diodes, and diodes subchain 66 has D diodes, then A.multidot.V.sub.t .multidot.1n(I.sub.in2 /I.sub.s20) +B.multidot.V.sub.t .multidot.1n(I.sub.in1 /I.sub.s22)=C.multidot.V.sub.t .multidot.1n(I.sub.in3 /I.sub.s64)+D.multidot.V.sub.t .multidot.1n(I.sub.o /I.sub.s66). Thus, (I.sub.in2).sup.A (I.sub.in1).sup.B /(I.sub.s20).sup.A (I.sub.s22).sup.B =(I.sub.in3).sup.C (I.sub.o).sup.D /(I.sub.s64).sup.C (I.sub. s66).sup.D. Hence, I.sub.o =[(I.sub.s64).sup.C (I.sub.s66).sup.D /(I.sub.s20).sup.A (I.sub.s22).sup.B ].multidot.[(I.sub.in2).sup.A (I.sub.in1).sup.B /(I.sub.in3).sup.C ].sup.1/D. Since the saturation currents are constants, I.sub.o =k[(I.sub.in2).sup.A (I.sub.in1).sup.B /(I.sub.in3).sup.C ].sup.1/D, where k is a constant. The circuit of FIG. 4 produces a current I.sub.o that is proportional to an exponential function of a product or ratio of input currents. The nature of the exponential function (square root, cube root, etc.) depends on the values of A, B, C, and D. Note that FIG. 3 is a special case of FIG. 4 with I.sub.in3 =0, C=0, 2A=2B=D, and I.sub.o =k(I.sub.in1 .multidot.I.sub.in2)1/2.

Other embodiments are within the following claims.