Stephen Woodward EDN Perhaps the most elementary rule of controlloop design theory is that feedbackloop performance is fundamentally linked to the careful choice – and stability – of loop gain. Insufficient loop gain leads to poor setpoint accuracy. Too much gain can induce feedback instabilities, such as overshoot, ringing, and, ultimately, oscillation. Therefore, the greater the accuracy you expect from a control system, the more critical maintaining nearoptimal loop gain becomes. Precision temperaturecontrol loops are no exception.
Given the aforementioned truisms, it's surprising that the following rule of designing highprecision thermostats receives so little notice: The thermal output (which is power, the primary feedback parameter) from a resistive heater is proportional to the current squared. In Figure 1, Curve A illustrates this elementary relationship.
Therefore, the overall thermostat loop gain is not constant but is instead proportional to heater input current. It consequently varies wildly in response to changes in ambient temperature and other factors that impact heat demand. The result is that it becomes more difficult to choose suitable loop parameters. The circuit in Figure 2 remedies these difficulties by inserting an analog squareroot circuit ahead of the heaterdrive circuit.
The circuit stabilizes the temperature of liquidnitrogencooled solidstate infrared lasers in an airborne spectrometer. The cryosensor diode, D_{1} (2 mV/K), senses laser temperature and drives the PI (proportionalintegral) control circuit comprising error amplifier IC_{1C} and error integrator IC_{1B}. Q_{1} converts the resulting feedback correction voltage to a currentmode signal and applies the signal to the LM3146 transistor array, IC_{4}. The array generates the squareroot function. Analog aficionados will be quick to point out that using IC_{4} is not the most accurate way to approximate a squareroot curve. However, this method is adequate for making the feedback linear and stabilizing the loop gain. In operation, array transistors IC_{4A} through IC_{4C} combine the current, I_{1}, from Q_{1} with the reference current (I_{REF}) to produce a logarithmic control voltage proportional to The inherent matching of transistor parameters in the IC_{4} monolithic array results in an IC_{4E} collector current of approximately The IC_{3B}Q_{2} heaterdriver circuit subsequently amplifies IC_{4E}'s output current by a factor of 8450 and applies the amplified current to the lasercryostat heater. Figure 1 shows five relevant curves. A is the uncompensated I^{2}R heater transfer function. B is the ideal squareroot function. C is the squareroot approximation from the IC_{4} array, which you calculate assuming transistor betas of approximately 100. D is the product of A and B and represents the ideal compensated (linear) loopgain linearization with constant loop gain. E is the product of A and C and is the achieved loopgain linearization. The net result is a linear relationship between the control circuit and heater outputs and a consequent optimization of the cryostat's steadystate and dynamic stabilities over a range of ambientheat loading. Without IC_{4}, a gain setting adequate for setpoint stability at low heater powers is likely to be excessive and produce overshoot or oscillation at high heater powers. Materials on the topic 

