Successful design of precision temperature-control loops, like all high-performance servo systems, depends on careful management of the usual feedback gain and bandwidth tradeoffs. While always tricky, these interactions can become unmanageable if the relevant thermal "constants" are unknown, or worse, not constant at all. Discussed in this article is a thermal design that arose from just such a challenging scenario. The design involves a system complicated by the nonlinear temperature-dependent parameters of radiant heat transfer.

The application requires accurate thermostasis of a silicon (Si) wafer over a 100°C to 1000°C setpoint range in an evacuated chemical-vapor-deposition (CVD) rig. Temperature control is accomplished via radiant heating from a large (250 W or larger) low-voltage dc incandescent lamp.

It was the radiant heating feature that made this control problem extra "interesting." So-called "Newtonian" heat exchange which occurs via conduction or convection tends to be nicely linear with temperature. But radiation, alas, is proportional to the fourth power of absolute temperature.

Consequently, the thermal time "constants" (degree-sec/degree) of radiation-coupled systems aren't linearly independent of temperature as they are in Newtonian systems. In-stead, they're inversely proportional to T^{3}. This causes the thermal time-constant of the vacuum chamber's Si wafer to vary by a factor of 40 over the 100°C (373K) to 1000°C (1273K) setpoint range!

Such variation adds substantially to the difficulty of designing an accurate yet non-oscillatory control loop. Coping with this complication required use of a robust integrating convergence-by-bisection feedback control algorithm *(Fig. 1)*. This algorithm is used as the basis for the thermostat circuit shown in Figure 2. It's described in detail in *"Take-Back-Half: A Novel Integrating Temperature-Control Algorithm," *Electronic Design, *Dec. 4, 2000, p. 132.*

The Si wafer's temperature is sensed by a thermocouple. Next, it's cold-junction compensated, amplified, linearized, and repeated as a 1-mV/deg. analog output, V_{T}, by the digital panel thermometer. V_{T} is compared to the setpoint voltage, V_{S}. The V_{S }− V_{T} difference is then integrated by A1, buffered by A4, and applied the control input of the programmable lamp supply.

Therefore, whenever V_{T }< V_{S}, the lamp voltage (and, therefore, the heat radiated onto the wafer) will ramp up, warming the wafer. Conversely, if V_{T }> V_{S}, the wafer will be cooled. Of course, if this simplistic error integration comprised the entire control algorithm, stable convergence to the setpoint wouldn't be likely. Instead, persistent oscillation above and below the setpoint would be virtually inevitable.

The "Take-Back-Half" (TBH) algorithm damps oscillations and stabilizes the servo loop. It does so by revising the estimate of the optimum steady-state lamp voltage at each setpoint (V_{T} = V_{S}) crossing. To make TBH action possible, some means for detecting setpoint crossings must exist. Crossed-diode-connected transistors Q1 and Q2 and comparator A2 accomplish this task by continuously tracking the polarity of the (V_{S }− V_{T})/R1 error current. A2 goes high when V_{T }< V_{S} and low when V_{T }> V_{S, }while inverter A3 generates the complementary logic term. Positive feedback around A2 keeps the logic transitions snappy. Meanwhile, the roles of TBH variables H_{O} and H are served by sample-and-hold capacitor C1 and integrator cap C2, respectively.

CMOS switches S1, S2, and S3 are arranged so that whenever V_{T }< V_{S}, S2 turns on and connects S1's control input to A3's logic-zero. This shuts off S1, which in turn isolates C1 and holds H_{O}. Alternatively, when V_{T }> V_{S}, S2 turns off, allowing R2 to pull S1's input to A2's logic-zero. Again, S1 turns off and C1 is isolated.

The fun begins whenever V_{T} = V_{S}. When V_{T} < V_{S} flips to V_{T} = V_{S}, A3 switches from zero to one, which turns on S1. As a result, C1 (H_{O}) and C2 (H) are connected in parallel and set to (H + H_{O})/2. This state persists for the time-out set by R3C3 (approximately 70 ms). After this period, S3, S2, and S1 all turn off and isolate C1 to await the next setpoint crossing. A similar cascade follows any toggle from V_{T }> V_{S} to V_{T} = V_{S}. A2 turns S1 on via R2 until R3C3 times out and turns S3 and S2 on and S1 back off.

Optimization of overall servo-loop dynamics is easy since selected-at-test R1 is the only variable involved in the tuning process.