This exponential converter is used in a voltage-controlled oscillator to tailor the linear voltage control to the accepted log scale of frequency variation. Its full circuit diagram is depicted in Fig.1.

Fig.1. Exponential converter.

     The exponential converter itself is built on two op amps (U1, U2) and matched transistors Q1, Q2 from the LM3086 device (U5). Operation of this transistor pair can be described by the following relationships:

     After expressing Vbe1 and Vbe2 from the first two equations of the system and substituting them to the third, we obtain:

     Keeping both transistors Q1 and Q2 at constant equal temperature (T1 = T2 = Tc) ensures the equality Is1 = Is2 , that greatly simplifies the last relationship whose further exponentiation leads to

     As can be seen from the circuit of Fig.1, the pair’s collector currents Ic1=VA/R1 and Ic2=VOUT/R5 , the first one being fixed while the second being varied and determining the converter output.
     Finally, the transfer characteristic of the exponential converter:

     The initial point VIN=0 gives VOUT=VAR5/R1=4mV (the desired range of converter output is 4mV-4V).
For the taken resistors’ values R1=910kOhm and R5=22kOhm the calculated voltage VA is:

     This voltage is derived from a Zener diode D1 reference VC=9,12V with the help of network consisting of resistors R2, R6 and adjustment potentiometer R15 (see Fig.1). Assuming that R2=30kOhm and R15=5kOhm, the value of R6 for establishing VA=166mV can be found from the following relationship:

     The op amp U4 buffer stage ensures the converter high input impedance and drives a voltage divider R3, R4 which allows to set the exponential 4mV-4V output voltage range to be exactly corresponding to the linear 0-3V input range. Finding the logarithm of both parts of (1) leads to:

     The unknown value of R3 will be determined after substituting into the last formula VOUT=4V (maximum), VIN=3V (maximum), R4=0,24kOhm, TC=85°C=353K, k=8.617×10-5.

     To check the calculated data, the converter circuit is simulated in Multisim 10, the chosen interactive mode providing the results shown in Fig.2.

Fig.2. Simulation of the exponential converter circuit.

     The virtual measurements confirm the above calculations, the program has tested the converter at the default temperature (+27°C) with the correctly made setting of R3. The converter transfer characteristic is obtained in these conditions too, the necessary for that sweeping input voltage being varied by the program linearly from 0 to 3V and each its 0,5V increment producing also the registered output reading (see Fig.3). The characteristic’s 0,5% deviation from the ideal can be further reduced by a more accurate adjustment.

Fig.3. Exponential converter – transfer characteristic.

     The main factors influencing the converter accuracy are the temperature dependence of the Q1, Q2 transistors’ parameters and the degree of their matching. The LM3086N device is optimal in this respect, especially because it contains three other transistors which in this design are used in an original way: Q4, Q5 behave as a heater and Q3 performs the role of a thermo-sensing element. Together with op amp U3 they constitute a thermostat to keep the Q1, Q2 pair at the stable elevated temperature Tc =+85°C.
     The thermostat setup is simple – to define the potential Vbe3 (transistor Q3) corresponding to this temperature and applied to the non-inverting input of the U3 and then to set the same potential at the op amp inverting input, with the help of VC and resistors R13, R20. The rest will be carried out automatically by the servo system and the introduced temperature TC will be maintained after that with high accuracy. The conducted by Multisim 10 temperature sweep analysis of the potential Vbe3 (the circuit net 5) produces the table Fig.4 from which we choose Vbe3=483mV for TC=+85°C.

Fig.4. Transistor Q3 base-emitter potential - temperature characteristic.

     The op amps U1, U2 handle very small DC currents therefore their type was a critical choice in achieving high conversion accuracy and circuit stability. The OP97 device with its 0,1nA bias current and 25µV offset voltage has proved to be indispensable just in this application. I have left also the possibility of offset trimming in the U1 op amp whose minimum output voltage is 4mV. The trimmer potentiometers R3, R15, R17 are of precision multi turn type. The exponential converter features better than 0,5% accuracy within the whole 4mV-4V range of converted outputs.
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