“The unfortunate relationship between servo systems and oscillators is very apparent in thermal-control systems,” said Linear Technology's Jim Williams (Reference 1). Although high-performance temperature control looks simple in theory, it proves to be anything but simple in practice. Over the years, designers have devised a long laundry list of feedback techniques and control strategies to tame the dynamic-stability gremlins that inhabit temperature-control servo loops. Many of these designs integrate the temperature-control error term T_{S} – T in an attempt to force the control-loop error to converge toward zero (Reference 2).

One tempting and “simple” alternative approach makes the heater power proportional to the integrated temperature error alone. This “straight-integration” algorithm samples the temperature, T, and subtracts it from the setpoint, T_{S}. Then, on each cycle through the loop, the loop gain, F, multiplies the difference, T_{S} – T, and adds it as a cumulative adjustment to the heater-power setting, H. Consequently,

H = H + F × (T_{S} – T).

The resulting servo loop offers many desirable properties that include simplicity and zero steady-state error. Unfortunately, as Figure 1 shows, it also exhibits an undesirable property: an oscillation that never allows final convergence to T_{S}. Persistent oscillation is all but inevitable because, by the time that the system's temperature corrects from a deviation and struggles back to T = T_{S}, the heater power inevitably gets grossly overcorrected. In fact, the resulting overshoot of H is likely to grow as large as the original perturbation. Later in the cycle, H's opposite undershoot grows as large as the initial overshoot, and so on.

Figure 1. |
A simple integrating control algorithm virtually guarantees that the system’stemperature oscillates and never converges to the setpoint temperature, T _{S}. |

Acting on intuition, you might attempt to fix the problem by adopting a better estimate of H whenever the system's temperature crosses the setpoint, T = T_{S}). This Design Idea outlines a TBH (take-back-half) method that takes deliberate advantage of the approximate equality of straight-integration's undamped overshoots and undershoots. To do so, you introduce variable H0 and run the modified servo loop, except for the instant when the sampled temperature, T, passes through the setpoint, T = T_{S}. Whenever a setpoint crossing occurs, the bisecting value (H + H_{O})/2 replaces both H and H0. As a result, at each setpoint crossing, H and HO are midway between the values corresponding to the current (H) and previous (H_{0}) crossings. This action takes back half of the adjustment applied to the heater setting between crossings. Figure 2 shows how a simulated TBH algorithm forces rapid half-cycle convergence.

Figure 2. |
In this simulation, applying the “take-back-half” algorithm forces convergenceto the setpoint value in a single half-cycle. |

Successful applications of the TBH algorithm range from precision temperature control of miniaturized scientific instrumentation to managing HVAC (heating/ventilation/air-conditioning) settings for crew rest areas in Boeing's 777 airliner. Experience with TBH applications shows that, with a reasonable choice for loop gain, F, the algorithm exhibits robust stability.

In general, a TBH system's natural cycle time is proportional to the square root of the ratio of the thermal time constant to F. Based on both simulations and experiments, a cycle time that's at least eight times longer than the heater-sensor time delay ensures convergence. Therefore, setting loop gain F low always achieves convergence, and the steady-state error, T_{S} – T, remains equal to zero.

Figure 3 shows a practical example of a TBH controller that's suitable for managing large thermal loads. Thermistor RT_{1} senses heater temperature. The output of error-signal integrator IC_{5A} ramps negative when T_{S} < T and ramps positive when T_{S} > T, producing a control signal that's applied to comparator IC_{5C}, which in turn drives a solid-state relay, IC_{3}, which is rated for 10 A loads.

Figure 3. |
For safety, this version of a TBH heater controller features full isolation of the ac-line and control circuits. |

Comparator IC_{5D} and the reverse-parallel diodes formed by the collector-base junctions of Q_{6 }and Q_{7}, and the CMOS switches of IC_{1} perform the TBH zero-crossing convergence function.

In most temperature-control circuits, it's advantageous to apply a reasonably linear feedforward term that represents the actual ac voltage applied to the heater; the need for complete galvanic isolation between the control and the power-handling circuits complicates this requirement. In this example, a linear isolation circuit comprising a PS2501-2 dual-LED/phototransistor optoisolator (IC_{2A} and IC_{2B}) and op-amp IC_{5B} delivers feedback current to C_{15} and IC_{5C} that's proportional to the averaged ac heater current. As a bonus, the feedback circuit provides partial instantaneous compensation for ac-line voltage fluctuations.

## References

- Williams, Jim, Linear Applications Handbook, Linear Technology, 1990.
- “Hybrid Digital-Analog Proportional-Integral Temperature Controller“