Case Study: Medical Laser System; Part 2: Photodiode Amplifier continued coverage of some problems with a medical laser design. In this third part, the goal is to design the dynamic compensation of a NdYAG laser power control loop to achieve stability within the static error (accuracy) requirements. The loop dynamic response requirements were a 20 Hz maximum pulse rate and a 50 ms minimum pulse duration. Then for a chosen transition (rise and fall) time of less than 5 ms, incremental single-pole bandwidth must be

The single-pole time constant associated with this bandwidth is

The loop is repeated in block-diagram (system-level) form below, redrawn in more detail than the original notebook copy.

The loop gain required for *static* stability (that is, no drift) and accuracy conflicts with dynamic stability. The higher the loop gain is made, the harder it is to keep the feedback loop from oscillating. Reducing loop gain stabilizes the feedback loop but it also causes the low-frequency error to increase. In the original loop without the square-root circuit, *T*_{SQRT}, the variation in loop gain from zero-scale (zs) to full-scale (fs) caused error to be excessive at zs and instability at fs. The square-root circuit resulted in a constant gain over the output power range.

This design exercise goes around the loop, characterizing the transfer functions of the various blocks (subsystems) before determining whether the loop is stable, and by how much.

**Forward-Path Operating-Point Analysis**

The forward path of the loop, or *G*, has the nonlinear flashlamp-laser block. Its incremental (small-signal) gain can be found by differentiating the expression for power, *P*_{O}(*i*). At the zs of 0.1 W out, *dP*_{O}/di is 0.30 W/A, but at the fs output of 5 W, it increases by over seven times, to 2.15 W/A. A 7-to-1 loop-gain variation makes feedback design difficult. A square-root circuit was inserted upstream from the flashlamp-laser to compensate this nonlinearity.

The block diagram of the forward path has a summing block inserted between the square-root circuit and flashlamp transconductance power driver, *G*_{m}. The additional summing block would affect a *linear* system by resulting in two closed-loop contributions to the output;

where *G*_{2} = *P*_{O}/*v*_{3}. The closer to the output the *G*-path summing block is inserted, the less closed-loop gain *V*_{os} has. Consequently, the less sensitive the output is to *V*_{os}. In the original design, this offset was inserted at the input summer, and consequently affected the output as much as the commanding input, *v*_{i}. In the refinement, it was moved into the forward path so that drift or error in *V*_{os} would not be as detrimental.

This offset is used to compensate for the offset of *I*_{os} in the flashlamp-laser block. To compensate the nonlinearity, an inverse nonlinearity, the square-root circuit, was inserted earlier in the path. However, both the addition of *V*_{os} and the *G*_{m} block are in-between the two nonlinear blocks. Instead of cascading transfer functions, which multiply as they do in linear systems, the successive blocks combine as composite functions, as *P*_{O}(*i*(*v*_{2}(*v*_{e}))) for large-signal (total-variable) analysis. Like transistor circuit analysis, the large-signal analysis must be done first to establish operating points for nonlinear elements so that linearization can occur around them, resulting in incremental variables. These small changes in quantities around the operating-point are then treated as linear. With a linearized system, linear feedback control analysis can then be applied.

Thus, we start by thinking total-variable, and begin with the offending nonlinearity at the output of *G*:

where *v*_{e} is the loop error voltage, the input to the error amplifier. Then set

and substituting,

The parameters of *K*_{SQRT} come from the square-root circuit components, as covered in Part 1. When calculated, *K*_{SQRT} = 5.832 V. *A*_{e} = 10 (from the Part 1 circuit diagram of *G*_{1}), and substituting values, the quasistatic (or 0+ Hz) gain of *G* is

The feedback path, *H*, is scaled at the error-summing block as 0.5 V/W. Then

and the quasistatic loop gain is

This gain is high enough to maintain sub-percent accuracy of *P*_{O}. However, can we maintain this value while achieving a dynamically acceptable response?

** Incremental ***G*_{1} and *G*_{2}

The incremental gains - the transfer functions - can be derived for *G*_{1} and *G*_{2} as follows. First, *G*_{2}:

Because of flashlamp-laser nonlinearity, *G*_{2} depends on the total-variable operating-point value of *v*_{2}, which is

Solving for *v*_{2},

Then expressed as a function of *v*_{e},

For *G*_{1},

Then the incremental, linearized forward-path gain at the op-pt is

The nonlinearities of *G*_{1} and *G*_{2} have cancelled incrementally through multiplication, leaving a linear *G*. Substituting values,

Throughout the power range, the incremental gains of *G*_{1} and *G*_{2} change, but their product remains constant, at *G*. This is illustrated in tabular form.

The *G*_{1} x *G*_{2} product for both zs and fs is equal to *G*; the quasistatic gain remains constant over the forward path, and the nonlinearity compensation succeeds in producing a linearized forward path.

The dynamic analysis will continue - and conclude - in the next part of this case study.