EXPONENTIAL CONVERTER |

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 (U _{1},
U_{2}) and matched transistors Q_{1}, Q_{2}
from the LM3086 device (U_{5}). Operation of this transistor
pair can be described by the following relationships:
After expressing V _{be1} and V_{be2} from the first
two equations of the system and substituting them to the third, we
obtain:
Keeping both transistors Q _{1} and Q_{2} at constant
equal temperature (T_{1} = T_{2} = T_{c})
ensures the equality I_{s1} = I_{s2} , that greatly
simplifies the last relationship whose further exponentiation leads
toAs can be seen from the circuit of Fig.1, the pair’s collector currents I _{c1}=V_{A}/R_{1} and
I_{c2}=V_{OUT}/R_{5} , 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 V _{IN}=0 gives
V_{OUT}=V_{A}R_{5}/R_{1}=4mV (the
desired range of converter output is 4mV-4V).For the taken resistors’ values R _{1}=910kOhm and
R_{5}=22kOhm the calculated voltage V_{A} is:
This voltage is derived from a Zener diode D _{1} reference
V_{C}=9,12V with the help of network consisting of resistors
R_{2}, R_{6} and adjustment potentiometer R_{15}
(see Fig.1). Assuming that R_{2}=30kOhm and R_{15}=5kOhm,
the value of R_{6} for establishing V_{A}=166mV can be
found from the following relationship:
The op amp U _{4} buffer stage ensures the converter high input
impedance and drives a voltage divider R_{3}, R_{4}
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 R _{3} will be determined after
substituting into the last formula V_{OUT}=4V (maximum),
V_{IN}=3V (maximum), R_{4}=0,24kOhm,
T_{C}=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 R _{3}. 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 Q _{1}, Q_{2} 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: Q_{4}, Q_{5}
behave as a heater and Q_{3} performs the role of a
thermo-sensing element. Together with op amp U_{3} they
constitute a thermostat to keep the Q_{1}, Q_{2} pair at
the stable elevated temperature Tc =+85°C.The thermostat setup is simple – to define the potential Vbe3 (transistor Q _{3}) corresponding to this temperature and applied to the
non-inverting input of the U_{3} and then to set the same
potential at the op amp inverting input, with the help of V_{C}
and resistors R_{13}, R_{20}. The rest will be carried
out automatically by the servo system and the introduced temperature
T_{C} will be maintained after that with high accuracy. The
conducted by Multisim 10 temperature sweep analysis of the potential
V_{be3} (the circuit net 5) produces the table Fig.4 from which
we choose V_{be3}=483mV for T_{C}=+85°C.Fig.4. Transistor Q _{3} base-emitter potential - temperature
characteristic.The op amps U _{1}, U_{2} 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 U_{1} op amp whose minimum output voltage is 4mV.
The trimmer potentiometers R_{3}, R_{15}, R_{17}
are of precision multi turn type. The exponential converter features
better than 0,5% accuracy within the whole 4mV-4V range of converted
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