Logarithmic Amplifying Circuit Based on the Bias-Offset Technique

1. Field of the Invention

The present Invention relates to a logarithmic amplifying circuit, and relates in particular to a logarithmic amplifying circuit suitable to be formed on a bipolar integrated circuit or a CMOS integrated circuit.

2. Description of the Related Art

Logarithmic amplifying circuits have been used in a variety of fields for outputting a logarithmic answer in response to an input signal. For realizing a logarithmic amplifying circuit, there are methods such as a first method which uses a logarithmic characteristic region included in a current-voltage characteristic curve for a PN junction, and a second method which provides a plurality of nonlinear amplifiers to combine output signals from each amplifier in order to obtain the approximate logarithmic characteristic of the output signals. According to the first method, it is possible to get logarithmic output signals with accuracy in the range from picoampere levels to several hundred microamperes. However, inasmuch as the frequency bandwidth for the output signals is narrow and depends on signals, it cannot be applied to, for example, an IF (intermediate frequency) stage of a radio receiver. In contrast, according to the second method, it is possible to expand the frequency band, the method being also applicable to the IF amplifier or the like. Further, as a third method for realizing a logarithmic amplifying circuit, there is for example the method disclosed in U.S. Pat. No. 4,794,342 and Japanese Patent Laid-open Gazettes No. 292010/87 (JP, A, 62-292010) and No. 165805/92 (JP, A, 4-165805) all by the present inventor. According to the method, a plurality of cascade-connected differential amplifiers and a plurality of rectifiers are provided for the full-wave square-law detection of a differential input of each stage and a differential output of the last stage, and then output signals of these rectifiers are summed up to obtain a synthetic logarithmic output.

FIG. 1 is a block diagram showing a basic structure of a logarithmic amplifying circuit according to the third method. In FIG. 1, there are m number of differential amplifiers 91.sub.1 to 91.sub.m cascade-connected in such a manner that a differential output signal from a former stage is inputted to a subsequent stage as a differential input signal. In FIG. 1, (m+1) number of rectifiers 92.sub.1 to 92.sub.m+1 are also provided and differential input signals to be inputted into differential amplifiers 91.sub.1 to 91.sub.m are also inputted to rectifiers 92.sub.1 to 92.sub.m respectively. Further, a differential output signal of the differential amplifier 91.sub.m of the last stage is inputted into the (m+1)-th rectifier 92.sub.m+1. Rectifiers 92.sub.1 to 92.sub.m+1 are structured such that each rectifier outputs, as a current signal, a full-wave rectified signal which is proportional to the square of the input voltage when the input voltage is within a range of the upper and lower limits of predetermined values. Each rectifier also outputs an electric current of a first predetermined value when the input voltage is over the upper limit and outputs an electric current of a second predetermined value when the input voltage is less than the lower limit. Further, an adder 93 is provided to add all output currents I.sub.RS1 to I.sub.RS(m+1) from respective rectifiers 92.sub.1 to 92.sub.m+1 for calculating the total sum. The output from the adder 93 becomes the output signal of the logarithmic amplifying circuit.

FIG. 2 shows, with reference to the logarithmic amplifying circuit of FIG. 1, the relation between the output currents I.sub.RS1 to I.sub.RS(m+1) from respective rectifiers 92.sub.1 to 92.sub.m+1 and an addition current I.sub.RSSI of the adder 93. The horizontal axis of FIG. 2 is expressed in a dB scale, i.e., in a logarithmic scale. In the diagram, a symbol G.sub.O represents an amplification factor (gain) of each of differential amplifiers 91.sub.1 to 91.sub.m+1. As shown in the diagram, the synthetic amplification factor is given as (m+1)G.sub.O, proving that logarithmic amplification has been achieved. Inasmuch as the logarithmic amplifying circuit implements logarithmic approximation by means of a number of rectifiers, the propriety of the logarithmic characteristic or the propriety of the approximation to the logarithmic characteristic is determined depending on the circuits or the rectifying characteristics of each full-wave rectifier. It is to be noted that the synthetic amplification factor can be adjusted by changing the amplification factor of each of differential amplifiers 91.sub.1 to 91.sub.m+1.

As a rectifier to be used in the logarithmic amplifying circuit of this kind having a full-wave square rectification characteristic, as shown in the above Laid-open documents, there is a rectifier familiar that is well known to the public in which input signals supplied to two pairs of unbalanced differential pairs composed of bipolar transistors or MOS transistors are cross-coupled and output signals therefrom are parallel-connected. The unbalanced differential pair can be prepared, if necessary, by using bipolar transistors each having an emitter of a different area or by using MOS transistors each having a different ratio of channel width W to channel length L, that is, a different value of W/L. The relation between the ratios of emitter areas or (W/L)s and the full-wave rectification characteristics of transistors composing the unbalanced differential pair is disclosed in detail in the literature by the present inventor, "K. Kimura, IEEE Transactions on Circuits and Systems-I, Vol. 39, No. 9, pp. 771-777, September 1992". By referring to the literature, it becomes possible to constitute a proper half- and full-wave rectifiers.

However, in two pairs of unbalanced differential pairs of the conventional logarithmic amplifying circuit, since collectors of a larger size or drains of a larger size are coupled with each other, the resulting capacity becomes large and the drive current tends to be increased for expanding the frequency characteristic to the high frequency area.

In the U.S. Pat. No. 4,990,803 by B. Gilbert, there is disclosed a logarithmic amplifying circuit in which a full-wave rectifier is structured with two pairs of bipolar differential pairs parallelly connected, and this full-wave rectifier is disposed in each stage of cascade-connected differential amplifiers. In the full-wave rectifier of the circuit by B. Gilbert, a DC offset voltage with a different sign is superimposed on the base of each transistor which constitutes the differential pair, and a logarithmic dynamic range per full-wave rectifier is about 10 dB. Therefore, in order to provide a logarithmic amplifying circuit, it is necessary to cascade-connect differential amplifiers each having a gain of about 10 dB. Consequently, a number of differential amplifiers are required to obtain a logarithmic dynamic range of a desired level resulting in the increase of the dissipation current.

Thus, the conventional logarithmic amplifying circuit described above has a structure which is disadvantageous for realizing a low dissipation current.

An object of the present invention is to constitute a full-wave rectifier used in a logarithmic amplifying circuit with transistors having collectors or drains of a small area and provide a logarithmic amplifying circuit which can easily be formed on an integrated circuit and is driven by a small current.

An object of the present invention is achieved by a logarithmic amplifying circuit which comprises a differential amplifier, a plurality of rectifiers provided corresponding to input and output terminals of the logarithmic amplifying circuit, and an adder for adding the output from each of the rectifiers; wherein each rectifier further includes a differential pair consisting of a plurality of transistors which are emitter-coupled or source-coupled, and offset means for superimposing a DC offset voltage on a differential input voltage to be supplied to the differential pair.

An object of the present invention can also be achieved according to a logarithmic amplifying circuit which comprises a plurality of cascade-connected differential amplifiers, a plurality of rectifiers for receiving a differential input of a differential amplifier of each stage and a differential output of the differential amplifier of the last stage, respectively, and outputting differential currents corresponding to the received input signals and an adder for adding the differential current of each rectifier and outputting signals according to the sum thus obtained; wherein the rectifier includes a differential pair consisting of a single tail current source and two MOS transistors whose sources are connected to the tail current source, and offset means for superimposing a DC offset voltage on a differential input voltage to be supplied to the differential pair, gates of the MOS transistors are provided for receiving input signals of the rectifier, and drains of the MOS transistors is provided for transmitting output signals of the rectifier.

An object of the present invention can also be achieved according to a logarithmic amplifying circuit which comprises a plurality of cascade-connected differential amplifiers, a plurality of rectifiers for receiving a differential input of a differential amplifier of each stage and a differential output of the differential amplifier of the last stage, respectively, and outputting differential currents corresponding to the received input signals and an adder for adding the differential current of each rectifier and outputting signals according to the sum thus obtained; wherein the rectifier comprises a differential pair including a single tail current source and two bipolar transistors whose emitters are connected to the tail current source, and offset means for superimposing a DC offset voltage on a differential input voltage to be supplied to the differential pair, gates of the bipolar transistors are provided for receiving the differential input of the rectifier and emitters of the bipolar transistors are provided for transmitting the differential output of the rectifier.

An object of the present invention can also be achieved according to a logarithmic amplifying circuit which comprises a plurality of cascade-connected differential amplifiers, a plurality of full-wave rectifiers for receiving a differential input of a differential amplifier of each stage and a differential output of the differential amplifier of the last stage, respectively, and outputting differential currents corresponding to the received input signals and an adder for adding the differential current of each full-wave rectifier and outputting signals according to the sum thus obtained; wherein the full-wave rectifier further comprises a double differential pair consisting of two parallel-connected differential pairs of the same characteristic and offset means provided for each of the differential pairs for superimposing a DC offset voltage on the differential input voltage of the corresponding differential pair; wherein each of the differential pairs is consisting of a tail current source and two transistors whose sources or emitters are commonly connected to the tail current source, and the polarity of the DC offset voltage for one differential pair is reverse to the polarity of the DC offset voltage for the other differential pair.

The above and other objects, features and advantages of the present invention will become apparent from the following description referring to the accompanying drawings which illustrate examples of preferred embodiments of the present invention.

FIG. 1 is a block diagram showing the structure of a logarithmic amplifying circuit which employs cascade-connected differential amplifiers, and rectifiers.

FIG. 2 is a diagram showing a characteristic of rectified currents in the logarithmic amplifying circuit of FIG. 1.

FIG. 3 is a block diagram showing the structure of a logarithmic amplifying circuit of an embodiment of the present invention.

FIG. 4A is a circuit diagram showing the structure of a MOS coordinate differential pair.

FIG. 4B is a diagram showing a characteristic of a normalized current of the MOS coordinate differential pair shown in FIG. 4A and a characteristic of an approximate expression of the above normalized current.

FIG. 5 is a circuit diagram showing the structure of a pseudologarithmic half-wave rectifier which employs MOS transistors and DC offset voltage sources.

FIG. 6 is a diagram showing a characteristic of the output voltage of the half-wave rectifier shown in FIG. 5.

FIG. 7 is a diagram showing, in a logarithmic scale, a characteristic of the output voltage of the half-wave rectifier shown in FIG. 5.

FIG. 8 is a circuit diagram showing the structure of a pseudologarithmic half-wave rectifier which employs bipolar transistors and DC offset voltage sources.

FIG. 9 is a diagram showing a characteristic of the output voltage of the half-wave rectifier shown in FIG. 8.

FIG. 10 is a diagram showing, in a logarithmic scale, a characteristic of the output voltage of the half-wave rectifier shown in FIG. 8.

FIG. 11 is a circuit diagram showing a bipolar unbalanced differential pair constituted according to emitter degeneration.

FIG. 12 is a diagram showing a characteristic of the output current of the bipolar unbalanced differential pair according to the emitter degeneration.

FIG. 13 is a circuit diagram showing a structure of a pseudo logarithmic half-wave rectifier which employs a DC offset voltage source and bipolar unbalanced differential pairs according to the emitter degeneration.

FIG. 14 is a diagram showing a characteristic of the output voltage of the half-wave rectifier shown in FIG. 13.

FIG. 15 is a diagram showing, in a logarithmic scale, a characteristic of the output voltage of the half-wave rectifier shown in FIG. 13.

FIG. 16 is a circuit diagram showing the structure of a full-wave rectifier which employs MOS transistors and DC offset voltage sources.

FIG. 17 is a diagram showing a characteristic of the output voltage of the full-wave rectifier shown in FIG. 6.

FIG. 18 is a circuit diagram showing the structure of a full-wave rectifier which employs MOS transistors and DC offset voltage sources.

FIG. 19 is a diagram showing a characteristic of the output voltage of the full-wave rectifier shown in FIG. 18.

A preferred embodiment of the present invention will next be described with reference to the drawings. A logarithmic amplifying circuit shown in FIG. 3 is constituted according to a CMOS process comprising a number n of differential amplifiers A.sub.1 to A.sub.n, a number (n+1) of half-wave rectifiers 4.sub.1 to 4.sub.n+1, a number (n+1) of active loads 5.sub.1 to 5.sub.n+1 and an adder 3. In FIG. 3, each of differential amplifiers A.sub.1 to A.sub.n has an identical gain being cascade-connected to each other and a differential output from the differential amplifier of a former stage becomes a differential input to the differential amplifier of a subsequent stage. A differential input of the differential amplifier A.sub.1 of the first stage represents the input to this logarithmic amplifying circuit and V.sub.IN represents the input voltage to the circuit. A differential input voltage of the differential amplifier A.sub.i of the i-th stage is expressed as V.sub.i, and a differential output voltage of a differential amplifier A.sub.n of the last stage is expressed as V.sub.n+1. Therefore, the output voltage V.sub.OUT of the differential amplifier series is expressed as V.sub.n+1.

Each of half-wave rectifiers 4.sub.1 to 4.sub.n+1 comprises a MOS coordinate differential pair composed of two N channel MOS transistors whose sources are commonly connected, a constant current source for producing a tail current for this differential pair, and an offset voltage source for superimposing a DC offset voltage on the input voltage to be supplied to the MOS differential pair. In concrete, the j-th (1.ltoreq.j.ltoreq.n+1) half-wave rectifier 4.sub.j comprises, a pair of differential input terminals and a pair of differential output terminals, first and second MOS transistors M1.sub.j, M2.sub.j whose sources are commonly connected, a constant current source for a tail current I.sub.Oj placed between the node of these sources and the ground, and an offset voltage source for applying a DC offset voltage V.sub.osj on the gate voltage of the second transistor M2.sub.j. The drains of transistors M1.sub.j, M2.sub.j are connected to the differential output terminals respectively, the gate of the first transistor M1.sub.j is connected to one input terminal and the gate of the second transistor M2.sub.j is connected to the other input terminal through the offset voltage source V.sub.osj. The differential input terminals of each of half-wave rectifiers 4.sub.1 to 4.sub.n are connected to lines for receiving differential input signals of differential amplifiers A.sub.1 to A.sub.n respectively, and the differential input terminals of the remaining half-wave rectifier 4.sub.n+1 are connected to lines for receiving a differential output signal of a differential amplifier A.sub.n of the last stage. Thus, the DC offset voltage V.sub.osj is superimposed on the differential input voltage to be supplied to the MOS differential pair composing the half-wave rectifier 4.sub.j.

Active loads 5.sub.1 to 5.sub.n+1 are connected to differential output lines of half-wave rectifiers 4.sub.1 to 4.sub.n+1, respectively, for outputting electric current signals based on the difference of differential output currents from corresponding half-wave rectifiers 4.sub.1 to 4.sub.n+1. The j-th active load 5.sub.j comprises a current mirror circuit consisting of two P channel MOS transistors (PMOS transistors) M3.sub.j, M4.sub.j, and a capacitor C.sub.oj which serves for smoothing output signals of the half-wave rectifier 4.sub.j. Drains of transistors M3.sub.j, M4.sub.j are connected in common to the voltage source V.sub.DD, and gates of transistors M3.sub.j, M4.sub.j are connected in common to a source of transistor M3.sub.j. Sources of transistors M3.sub.j, M4.sub.j are connected to differential output pair of the corresponding half-wave rectifier 4.sub.j respectively. The capacitor C.sub.oj is placed between sources of transistor M3.sub.j, M4.sub.j and the source of the other transistor M4.sub.j outputs a current .DELTA.I.sub.j corresponding to the difference between a pair of output currents of the half-wave rectifier 4.sub.j, the current representing an output of the active load 5.sub.j.

The adder 3 comprises a current mirror circuit composed of a pair of PMOS transistors M51, M52 and resistance R.sub.RSSI which serves as a load on the current mirror circuit. The adder 3 is designed to output a voltage V.sub.RSSI which corresponds to the sum of output currents .DELTA.I.sub.1 to .DELTA.I.sub.n+1 from active loads 5.sub.1 to 5.sub.n+1. The output voltage V.sub.RSSI represents the output of the logarithmic amplifying circuit. Drains of transistors M51, M52 are connected in common to a power source of voltage V.sub.DD, and gates of transistors M51, M52 are connected in common to the source of one transistor M51. Outputs from active loads 5.sub.1 to 5.sub.n+1 are also connected in common to the source of the transistor M51. The source of the other transistor M52 is grounded through the resistance R.sub.RSSI and the source voltage of this transistor M52 represents the output voltage V.sub.RSSI of the logarithmic amplifying circuit.

Being constructed in this way, output signals from half-wave rectifiers 4.sub.1 to 4.sub.n+1 are smoothed by capacitors C.sub.o1 to C.sub.o(n+1) and outputted as electric current signals respectively, then synthesized and converted by the adder 3 to a voltage signal to be finally outputted as a logarithmic signal.

The operation principle of the MOS half-wave rectifier will next be described to show that a logarithmic amplifying circuit can be realized according to the above construction.

FIG. 4A is a circuit diagram of a MOS coordinate differential pair which does not include an offset voltage source for a gate voltage. This differential pair is composed of a pair of NMOS transistors M1, M2, and a current source I.sub.O for a tail current commonly provided for transistors M1, M2. If a differential voltage V.sub.i is applied between gates of transistors M1, M2, and resulted drain currents of transistors M1, M2 are represented as I.sub.D1, I.sub.D2, respectively, then the differential output current .DELTA.I.sub.D, i.e., the difference between the drain currents is expressed by equation (1). ##EQU1## where .beta. is a transconductance parameter of the transistor to be expressed by the following equation (2). ##EQU2## where .mu. is the effective mobility of a carrier, C.sub.ox is a capacity of a gate oxide film per unit area, W is the gate width, and L is the gate length.

Therefore, if the characteristic of the differential output current of the MOS coordinate differential pair is normalized and denoted as f.sub.M (x), the characteristic of the normalized output current of the MOS coordinate differential pair is expressed as shown in FIG. 4B. Here, x is a normalized input voltage and f.sub.M (x) is given by equation (3). ##EQU3##

Accordingly, when a DC offset voltage V.sub.os is superimposed on the input voltage to be supplied to the MOS coordinate differential pair, the characteristic curve of the normalized output current shown in FIG. 4B shifts by the offset voltage V.sub.os in the direction of the axis of the input voltage. If the DC offset voltage V.sub.os is expressed as shown in equation (4), ##EQU4## it is possible to modify the MOS coordinate differential pair to be almost equivalent to a half-wave rectifier by superimposing the DC offset voltage V.sub.os. The half-wave rectifier thus obtained is each of the half-wave rectifiers 4.sub.1 to 4.sub.n+1 shown in FIG. 3.

By summing up output currents of thus obtained half-wave rectifiers 4.sub.1 to 4.sub.n+1 and converting them into a voltage, as shown in FIG. 3, an output voltage V.sub.RSSI which is approximate to the logarithmic value of the input voltage V.sub.IN can be obtained. The voltage V.sub.RSSI is given by equation (5).

In this case, the logarithmic input dynamic range of each half-wave rectifier is 8 dB, and the logarithmic accuracy is secured for each stage for the range of about 8 dB. Therefore, in order to secure the logarithmic accuracy of the whole logarithmic amplifying circuit, it is necessary and sufficient to arrange the gain of each of differential amplifiers A.sub.1 to A.sub.n to about 8 dB. Since this circuit requires one constant current source for each half-wave rectifier, it is possible to reduce the number of constant current sources used in a logarithmic amplifying circuit compared with a case employing the conventional circuit described above.

In the above example, inasmuch as each half-wave rectifier is composed of one MOS coordinate differential pair, the gain of a differential amplifier of each stage is arranged to about 8 dB. However, with a general differential amplifier of a standard structure, it is easy to have a gain of about 20 dB or more. Therefore, if the logarithmic dynamic range per half-wave rectifier can be expanded, it becomes possible to reduce the number of differential amplifiers to be cascade-connected while maintaining the total logarithmic dynamic range as a logarithmic amplifying circuit.

The logarithmic dynamic range per half-wave rectifier can be expanded by providing a plurality of parallel-connected MOS differential pairs for every half-wave rectifier, each differential pair having a different characteristic and operational conditions. FIG. 5 shows the structure of a pseudologarithmic half-wave rectifier composed of parallel-connected N pairs (N.gtoreq.2) of MOS differential pairs. The pseudologarithmic half-wave rectifier comprises a differential input terminal pair T.sub.i1, T.sub.i2 provided common to a differential pair and a differential output terminal pair T.sub.o1, T.sub.o2 similarly provided common to the differential pair. A constant current source for the tail current and an offset voltage source for superimposing a DC offset voltage on the difference input voltage are provided for every differential pair. Specifically, the j-th differential pair is composed of two MOS transistors M11.sub.j, M12.sub.j with a transconductance parameter .beta., and sources of these MOS transistors M11.sub.j, M12.sub.j are connected in common and grounded through the tail current source I.sub.Oj '. The gate of one transistor M11.sub.j is connected to one input terminal T.sub.i1 through the offset voltage source V.sub.kj, and the gate of the other transistor M12.sub.j is directly connected to the other input terminal T.sub.i2, while drains of transistors M11.sub. j, M12.sub.j are connected to output terminals T.sub.o1, T.sub.o2, respectively.

When it is structured in this way, the DC offset voltage V.sub.kj required for the j-th differential pair in order to be a half-wave rectifier is given by equation (6). ##EQU5##

The short-circuit transconductance g.sub.mj of each MOS transistor composing the j-th differential pair is given by equation (7). ##EQU6##

The pseudologarithmic characteristic of the parallel-connected differential pairs can be obtained by arranging transconductance g.sub.m of MOS transistors in such a manner that g.sub.m varies in accordance with the geometric progression as shown in equation (8). ##EQU7##

FIG. 6 is a diagram showing a characteristic of output currents of the pseudologarithmic half-wave rectifier shown in FIG. 5 when the number of differential pairs is 3 (N=3), .beta..sub.2 /.beta..sub.1 =.beta..sub.3 /.beta..sub.2 =10, and I.sub.O1 '=I.sub.O2 '=I.sub.O3 '=I.sub.O. The vertical axis gives the one of differential output currents, and the horizontal axis gives the normalized input voltages. In the diagram, drain currents of MOS transistors M12.sub.1, M12.sub.2, M12.sub.3 are represented as I.sub.D1, I.sub.D2, I.sub.D3 respectively, and a combined current of these drain currents is represented as I.sub.2. FIG. 7 represents a characteristic of the output current when the input voltage is given in a dB scale. From FIGS. 6 and 7, it is understood that the logarithmic input dynamic range of a half-wave rectifier is expanded by connecting MOS differential pairs in parallel. Therefore, by using the pseudologarithmic half-wave rectifier shown in FIG. 5 in place of half-wave rectifiers 4.sub.1 to 4.sub.n+1 shown in FIG. 3, the number of cascade-connected differential amplifiers can be reduced while maintaining the logarithmic input dynamic range as a logarithmic amplifying circuit.

A pseudologarithmic half-wave rectifier composed of parallel-connected differential pairs can also be realized by using bipolar transistors. FIG. 8 is a circuit diagram showing the structure of a pseudologarithmic half-wave rectifier which employs a plurality of differential pairs each using bipolar transistors. The pseudologarithmic half-wave rectifier comprises N pairs of bipolar differential pairs. It is composed of NPN transistors Q11.sub.1, Q12.sub.1, Q11.sub.2, . . . , Q11.sub.N, Q12.sub.N instead of MOS transistors M11.sub.1, M12.sub.1, M11.sub.2, . . . , M11.sub.N, M12.sub.N of the MOS pseudologarithmic half-wave rectifier shown in FIG. 5. However, in order to change the input-output characteristic of each differential pair, emitter resistance is inserted in the circuit for changing the emitter degeneration value (product of emitter resistance and tail current). In other word, between the emitter of each transistor M11.sub.j, M12.sub.j of the j-th differential pair and its tail current source I.sub.oj ', resistance R.sub.j of the identical value is inserted, however, the first differential pair has no resistance inserted because R.sub.1 is set to 0.

In order to obtain, with the parallel-connected differential pairs on which DC offset voltage are impressed, a half-wave rectification characteristic, it is necessary that the relation given in equation (9) is established between the emitter degeneration value R.sub.j .multidot.I.sub.Oj of the j-th differential pair and the DC offset voltage V.sub.kj to be applied on the j-th differential pair.

where V.sub.T represents a thermovoltage which is expressed as V.sub.T =kT/q. Here, k represents a Boltzmann's constant, T an absolute temperature, and q the unit electron charge. In order to produce a half-wave rectifier structured as shown in FIG. 8 while having a pseudologarithmic characteristic, it is necessary to change the input voltage range of each parallel-connected differential pair in accordance with an inverse logarithmic function as shown in equation (10), i.e., an exponential function. ##EQU8##

In other words, the relation between values of the short-circuit transconductance g.sub.m of parallel-connected differential pairs is given by equation (11). ##EQU9##

FIG. 9 is a diagram showing a characteristic of the output current of the pseudologarithmic half-wave rectifier shown in FIG. 8 when the number of differential pairs is 3 (N=3), I.sub.O1 '=I.sub.O2 '=I.sub.O3 '=I.sub.O. V.sub.k1 =2 V.sub.T, V.sub.k2 =6.4 V.sub.T, and V.sub.k3 =20.4 V.sub.T. In this diagram, the vertical axis gives the output currents, and the horizontal axis gives the normalized input voltage. In the diagram, collector currents of bipolar transistors Q12.sub.1, Q12.sub.2, Q12.sub.3 are represented as I.sub.C1, I.sub.C2, I.sub.C3 respectively, and a combined current of these collector currents is represented as I.sub.2. FIG. 10 represents characteristics of the output currents when the input voltage is given in a dB scale. From FIGS. 9 and 10, it is understood that the logarithmic input dynamic range of a half-wave rectifier is expanded by connecting bipolar differential pairs in parallel. Therefore, by using the pseudologarithmic half-wave rectifier shown in FIG. 8 in place of half-wave rectifiers 4.sub.1 to 4.sub.n+1 shown in FIG. 3, the number of cascade-connected differential amplifiers can be reduced while maintaining the logarithmic input dynamic range as a logarithmic amplifying circuit.

When the bipolar process is applied, by arranging the value of the emitter degeneration within the differential pair so that it may be unbalanced, it also becomes possible to superimpose a DC offset voltage on the differential input voltage without providing an offset voltage source. Unbalanced emitter degeneration is achieved by inserting the emitter resistance only to one transistor of the bipolar differential pair. FIG. 11 is a circuit diagram showing a half-wave rectifier composed of a bipolar unbalanced differential pair. This half-wave rectifier can be used in the logarithmic amplifying circuit shown in FIG. 3 in place of each of half-wave rectifiers 4.sub.1 to 4.sub.n+1.

In FIG. 11, there are provided two bipolar transistors Q.sub.1, Q.sub.2 of almost similar characteristic. Gates of each transistor Q.sub.1, Q.sub.2 are connected to differential input terminals T.sub.i1, T.sub.i2, respectively, and collectors of each transistor Q.sub.1, Q.sub.2 are connected to differential output terminals T.sub.o1, T.sub.o2, respectively. Resistance R is inserted only in the emitter of one transistor Q.sub.1, the other end of the resistance R is grounded through a constant current source for a tail current I.sub.O. The emitter of a second transistor Q.sub.2 is connected to the node between resistance R and the constant current source. If the base-to-emitter voltages of the first and second transistors Q.sub.1, Q.sub.2 are assumed as V.sub.BE1, V.sub.BE2, and collector currents thereof are assumed as I.sub.C1, I.sub.C2, equation (12) is established for a differential input voltage V.sub.i. ##EQU10## where .alpha..sub.F is the DC common-base current amplification factor. The equation (13) is established for V.sub.BE1, V.sub.BE2. ##EQU11## where I.sub.S1, I.sub.S2 are saturation currents of the bipolar transistors Q.sub.1, Q.sub.2. Now, it is assumed that equation (14) stands.

Since the circuit is driven by the tail current I.sub.O, equation (15) is satisfied. ##EQU12##

From the above equations (12) through (15), the following equation (16) is obtained. ##EQU13##

If I.sub.c1 =I.sub.c2 =.alpha..sub.F .multidot.I.sub.O /2 is satisfied in equation (16), an input offset voltage V.sub.1 is given by equation (17). ##EQU14##

The equation (18) is obtained by differentiating equation (15) with respect to the collector current I.sub.c1. ##EQU15##

Therefore, when V.sub.i =V.sub.1 (I.sub.c1 =I.sub.c2 =.alpha..sub.F .multidot.I.sub.O /2), the inclination is maximum. This inclination is given in equation (19). Since the tangent at the point with the maximum inclination (V.sub.1, .alpha..sub.F .multidot.I.sub.O /2) passes the point (-2V.sub.T, O), by applying a bias offset of 2 V.sub.T to the input signal, this bipolar unbalanced differential pair becomes almost equivalent to the half-wave rectifier. Accordingly, this half-wave rectifier can be used in place of each of half-wave rectifiers 4.sub.1 to 4.sub.n+1 shown in FIG. 3. ##EQU16##

FIG. 12 shows a characteristic of the output currents of bipolar unbalanced differential pairs with various emitter degeneration values structured as above.

As described above with reference to FIG. 5 or 8, the logarithmic input dynamic range can be expanded by means of a plurality of parallel-connected bipolar unbalanced differential pairs. FIG. 13 is a circuit diagram showing the structure of a pseudologarithmic half-wave rectifier composed of a plurality of bipolar unbalanced differential pairs connected in parallel. The pseudologarithmic half-wave rectifier includes a pair of differential input terminals T.sub.i1, T.sub.i2, a pair of differential output terminals T.sub.o1, T.sub.o2, and one offset voltage source. These terminals and offset voltage source are commonly provided to all the differential pairs. The offset voltage V.sub.k to be generated by the offset voltage source can be set to 2V.sub.T for any differential pair regardless of the position of the differential pair. The j-th differential pair is composed of two bipolar transistors Q21.sub.j, Q22.sub.j, an emitter resistance R.sub.j inserted to one transistor Q21.sub.j, and a tail current source I.sub.Oj '. The base of one transistor Q21.sub.j is connected to one input terminal T.sub.i1 through the offset voltage source V.sub.k, and the base of the other transistor Q22.sub.j is directly connected to the other input terminal T.sub.i2. Collectors of transistors Q21.sub.j, Q22.sub.j are connected to output terminals T.sub.o1, T.sub.o2, respectively. However, in the example illustrated, only the first differential pair is a balanced differential pair because of its emitter resistance R.sub.1 which is zero.

This pseudologarithmic half-wave rectifier is different from the pseudologarithmic half-wave rectifier shown in FIG. 8 in that the former rectifier has an unbalanced bipolar differential pair and has only one offset voltage source. Therefore, by comparing equation (9) with the above bias offset voltage V.sub.k =2V.sub.T, it can be understood that the voltage V.sub.k of the offset voltage source required for approximating the characteristic of an unbalanced differential pair to a half-wave rectifying characteristic is less than the voltage shown in FIG. 8 by R.sub.j .multidot.I.sub.Oj. Further, an input range of the operational voltage as a half-wave rectifier is narrowed by the range corresponding to the value of R.sub.j .multidot.I.sub.Oj.

FIG. 14 is a diagram showing a characteristic of the output current of the pseudologarithmic half-wave rectifier shown in FIG. 13 when the number of unbalanced differential pairs is 3 (N=3), I.sub.O1 '=I.sub.O2 '=I.sub.O3 '=I.sub.O, and V.sub.k =2 V.sub.T, In this diagram, the vertical axis gives the output currents and the horizontal axis gives the input voltages. In the diagram, collector currents of bipolar transistors Q21.sub.1, Q21.sub.2, Q21.sub.3 are represented as I.sub.C1, I.sub.C2, I.sub.C3 respectively, and a combined current of these collector currents is represented as I.sub.1. FIG. 15 represents a characteristic of the output current when the input voltage is given in a dB scale. From FIGS. 14 and 15, it is understood that the logarithmic input dynamic range of a half-wave rectifier is expanded by connecting a bipolar unbalanced differential pair in parallel. Therefore, by using the pseudologarithmic half-wave rectifier shown in FIG. 13 in place of half-wave rectifiers 4.sub.1 to 4.sub.n+1 shown in FIG. 3, the number of cascade-connected differential amplifiers can be reduced while maintaining the logarithmic input dynamic range as a logarithmic amplifying circuit.

A preferable embodiment of the present invention has been described above with reference to examples in which logarithmic amplifying circuits are composed of half-wave rectifiers, however, the logarithmic amplifying circuit can be composed by using full-wave rectifiers in place of the half-wave rectifiers. It is possible to construct a full-wave rectifier by cross-coupling an even number of differential pairs. Each pair of every two differential pairs has identical characteristic and a DC offset voltage is applied on each pair. The polarity of the DC offset voltage is alternately changed for every two pairs.

It will be described below that a characteristic of the full-wave rectifier thus structured can be approximate to a square-law characteristic.

The case according to the MOS process will first be described. A normalized differential output current f.sub.M (x) (refer to equation (3)) of a coordinate differential pair, i.e., .DELTA.I.sub.D /.DELTA.I.sub.O1 can be obtained approximately by equation (20). ##EQU17##

In equation (20), if it is calculated to the third degree of x, the approximation error of the solution is 3% or less. Here, a function .PHI.(V.sub.i) which depends on the input voltage is defined as shown in equation (21). ##EQU18##

The above equation (21) can be approximated to equation (22).

When equation (23) is satisfied, equation (1) coincides with equation (21). ##EQU19##

In the above FIG. 4B, a chain line expresses an approximate characteristic according to equation (20). It is nearly coincident with the curve of .DELTA.I.sub.D of a solid line with the error within 3%. Consequently, equation (19) provides good approximation to the differential output current. Further, by normalizing the function .PHI.(V.sub.i) given by equation (20), equation (24) is obtained. ##EQU20##

FIG. 16 is a circuit diagram showing the structure of a pseudologarithmic full-wave rectifier composed of two pairs of MOS coordinate differential pairs and two offset voltage source. A first differential pair is composed of MOS transistors M31, M32, and a second differential pair is composed of MOS transistors M33, M34. A constant current source is provided for each differential pair to cause a tail current I.sub.O of the same strength to flow. Transistors M31 to M34 are each of the same characteristic. Drains of transistors M31, M34 are connected to one output terminal T.sub.o1, and drains of the remaining transistors M32, M33 are connected to the other output terminal T.sub.o2. The gate of the MOS transistor 31 is connected to one input terminal T.sub.i1 through the offset voltage source with a voltage V.sub.k. At this time, the polarity of the offset voltage source V.sub.k is positive on the gate side of the transistor 31. The gate of the transistor M33 is directly connected to the input terminal T.sub.i1. On the other hand, the gate of the transistor M32 is directly connected to the other input terminal T.sub.i2, and the gate of the transistor M34 is also connected to the other input terminal T.sub.i2 through the other offset voltage source with the voltage V.sub.k. Here, the polarity of the offset voltage source V.sub.k is positive on the gate side of the transistor 34. Consequently, with reference to the differential input voltage V.sub.i, DC offset voltages of mutually reverse polarity are applied to the input voltage of each differential pair.

The differential output current .DELTA.I.sub.OSM of two source-coupled differential pairs shown in FIG. 16 is given by equation (25) as follows. ##EQU21##

By normalizing equation (25), equation (26) is obtained. ##EQU22## where b is given by equation (27). ##EQU23##

Further, equation (28) is obtained when equation (25) is rewritten according to the approximate expression defined by equation (21). ##EQU24##

Therefore, if the input voltage range is limited according to expression (29), the circuit shown in FIG. 16 becomes nearly equivalent to the square-law circuit. ##EQU25## FIG. 17 represents the output rectifying characteristic of the MOS full-wave rectifiers shown in FIG. 16. The input-output characteristic which is approximate to the square-law characteristic can be obtained by superimposing a DC offset voltage on the differential input voltage. As it is clear from FIG. 17 or expression (27), by employing a full-wave rectifier which is shown in FIG. 16 and arranging so that the product of the tail current I.sub.O and the transconductance parameter .beta. is different for each full-wave rectifier and further multiplying the product by an appropriate constant less than 1 to obtain the DC offset voltage of the full-wave rectifier, full-wave rectifiers each having a different characteristic can be obtained. By parallelly connecting a plurality of full-wave rectifiers of this like each having a different characteristic, it becomes possible to obtain the pseudo logarithmic full-wave rectification characteristic. Therefore, it is possible to construct a logarithmic amplifying circuit by using the pseudologarithmic full-wave rectifiers shown in FIG. 16 in place of half-wave rectifiers 4.sub.1 to 4.sub.n+1 shown in FIG. 3.

In the same way as the above case which employs the MOS process, it is also possible to realize the full-wave rectifier by using the bipolar process. FIG. 18 shows the structure of a pseudologarithmic full-wave rectifier composed of two pairs of bipolar MOS differential pairs and two offset voltage source. A first differential pair is composed of bipolar transistors Q31, Q32, a second differential pair is composed of bipolar transistors Q33, Q34. A constant current source is provided for each differential pair to cause the tail current I.sub.O of the identical strength to flow. Transistors Q31 through Q34 have each the identical characteristic. Collectors of transistors Q31, Q34 are connected to one output terminal T.sub.o1, and collectors of the remaining transistors Q32, Q33 are connected to the other output terminal T.sub.o2. The base of the transistor Q31 is connected to one input terminal T.sub.i1 through the offset voltage source with a voltage V.sub.k. At this time, the polarity of the offset voltage source V.sub.k is positive on the base side of the transistor 31. Further, the base of the transistor M33 is directly connected to the input terminal T.sub.i1. On the other hand, the base of the transistor M32 is directly connected to the other input terminal T.sub.i2, and the base of the transistor M34 is also connected to the other input terminal T.sub.i2 through the other offset voltage source with the voltage V.sub.k. Here, the polarity of the offset voltage source V.sub.k is positive on the base side of the transistor 34. Consequently, with reference to the differential input voltage V.sub.i, DC offset voltages of mutually reverse polarity are applied respectively on the input voltage of each differential pair.

The differential output current .DELTA.I.sub.OSB of two emitter-coupled differential pairs shown in FIG. 18 is given by equation (30) as follows. ##EQU26##

By normalizing equation (30), equation (31) is obtained. ##EQU27## where f.sub.B (x) is a normalized differential output current of the bipolar coordinate differential pair to be driven by the tail current I.sub.O and given by equation (32). ##EQU28##

FIG. 19 represents the output rectifying characteristic of the bipolar MOS full-wave rectifiers shown in FIG. 18. As it is clear from FIG. 19, the input-output characteristic which is approximate to the square-law characteristic can be obtained by superimposing a DC offset voltage on the differential input voltage of the bipolar full-wave rectifier. If an emitter resistance is inserted to each differential pair of the bipolar full-wave rectifier, the slope of the curve representing the input-output characteristic becomes gentle causing the input voltage range to expand while maintaining the characteristic of the full-wave rectifier. Therefore, by inserting emitter resistances with different values to the bipolar differential pairs, fill-wave rectifiers each having a different characteristic can be obtained. By parallelly connecting the full-wave rectifiers each having a different characteristic, it become possible to construct a pseudologarithmic full-wave rectifier. It is possible to construct a logarithmic amplifying circuit by using the pseudologarithmic full-wave rectifiers shown in FIG. 16 in place of half-wave rectifiers 4.sub.1 to 4.sub.n+1 shown in FIG. 3.

It is to be understood that variations and modifications of the logarithmic amplifying circuit disclosed herein will be evident to those skilled in the art. It is intended that all such modifications and variations be included within the scope of the appended claims.