Case Study: Medical Laser System; Part 3: Laser Loop Dynamic Design in this series began investigation of the problem of power control for a 5 W fs output NdYAG laser. In this fourth part, the design of the whole laser loop is completed, beginning with the dynamics. The loop block diagram is repeated from the previous part, below.
We now proceed with dynamic analysis, one block at a time, until we have traversed the loop. Then it will be put together into a loop-gain frequency-response plot from which to predict behavior. The G 1 circuitry is repeated below, from the project notebook, and redrawn more neatly.
The error amplifier is a one-op-amp differential amplifier with a LM358, which has a GBW = fT = 1 MHz. Then the transfer function of this inherently linear block is
where vi is the incremental commanded output power, and vb the incremental H -path feedback out of the photodiode (PD) amplifier (PDA). When using low-cost (and low-speed) op-amps, it is generally a good practice to include the speed of the op-amp in the feedback equations. This at first appears to be a complicating nuisance leading to unwieldy algebra, but it is actually not that hard to include. The frequency response of the typical op-amp is single-pole, as shown below.
This plot is the asymptotic approximation to the magnitude of the gain, on a log-log plot. The frequency response decreases from op-amp open-loop bandwidth at fbw with a –1 slope, from a quasistatic, open-loop voltage gain of K, until it intersects the unity-gain axis at fT . Then with a single-pole rolloff of –1,
For a typical K = 105 and fT = 1 MHz, then the open-loop op-amp bandwidth is quite low; fbw = 10 Hz.
The quasistatic closed-loop gain value, Acl << K , and the resulting bandwidth, fbwcl is proportionately higher; in the case of Ae (f ),
Consequently, the error amplifier has contributed a single pole at 100 kHz and no zeros.
Here is the simplifying technique for including op-amp speed. Because K is so large relative to Acl , for dynamic analysis, we can let K go to infinity. When this occurs, the op-amp transfer function reduces to
Whenever an op-amp feedback circuit is analyzed, use 1/(s x τ T ) instead of K to obtain the infinite-K approximation of the dynamic response, where
For an op-amp with fT = 1 MHz, τT ≅ 159 ns.
The input stage of the SQRT circuit is the op-amp log-amp stage, with its BJT from input to output. It also has a capacitor from op-amp output to inverting input. The dynamic analysis of this circuit produces the following approximate transfer function:
The approximation assumes the op-amp pole is well beyond (around a decade higher than) the one given above.
The problem with this circuit is that the 50-to-1 dynamic range corresponds with a 50-to-1 range in input and BJT current. Consequently, re also varies by 50-to-1. In retrospect, the current range should be higher. It is 5.83μA fs and a measly 117 nA zs. Although the circuit works in the application, I would in retrospect change Ri from 10 kΩ to 1.0kΩ (or less) and Rf from 100kΩ to 10kΩ or less, thus boosting both v1 (from 58.3 mV to 583 mV fs) and SQRT input current by a decade. Perhaps then, Cf = 150 pF and Ro = 2.0kΩ could also be eliminated.
The wide variation in re causes the quasistatic gain and pole location to vary, from a gain of –22.3 and pole at 4.71 kHz at zs to a much-reduced –0.65 and 163 kHz at fs. Two loop analyses will be needed for the two ends of the range. The values of re are 221kΩ zs and 4.43kΩ fs.
The SQRT translinear BJTs are fast, with fT = 550 MHz, and circuit poles well into multiple MHz, beyond the need for consideration. (Any poles or zeros more than a decade above the fT of a feedback loop contribute negligible phase to the loop that affects stability.) The squaring stack of two BJTs forms a CB stage, which is inherently fast. The output current is applied to the output op-amp, with Rf = 100kΩ that develops the output voltage. The non-inverting input is also offset by Vos (or as close as 5 % resistors will allow – another place to refine the design, with 1 % parts).
The output-stage op-amp is also part of an LM358. Its transfer function, using an inverting op-amp G = –1/(s x τT ), is
Consequently, the SQRT output stage has a pole at the op-amp fT of around 1 MHz. We will find later that this is of some significance.
The laser flashlamp current driver was a purchased motor-drive power amplifier, adapted to the application. It outputs over 13 A. I did not want to modify it, causing this purchased subsystem to become customized, because any changes made within it by its supplier were outside our control and would force us to modify our own customization of it. The circuit diagram of it was available, but its complexity did not encourage the kind of analysis of it that we are doing here of the laser loop. Its behavior was investigated on the bench. Measurements produced an approximation of its frequency response: a pole at 1.8 kHz and a zero at 7.1 kHz. The manufacturer's specification gave a 2-kHz bandwidth.
The flashlamp-laser subsystem dynamics were unknown but believed to be relatively fast, being optical. The bench measurements of the driver included flashlamp loading. The assumption for this analysis is that no appreciable affect is contributed by this subsystem to the loop response.
The photodiode amplifier has four stages, including the laser-beam pickoff mirror. The simplified PDA model is shown below.
The mirror is essentially an optical power divider. Its transfer function or transmittance is
where TPD is the photodetector sensitivity (optical power in to current out transfer function). The mirror passes only a tiny fraction of the total incident laser beam. Very small fractions can be potential trouble indicators, yet except for the problem that the divider ratio is different depending upon light polarization, it has worked acceptably in practice as an optical divider.
The overall transresistance of the PDA is
Transresistance of the first amplifier stage is
The output voltage amplifier stage has a gain of
A magnitude frequency-response plot for the PDA shows a dominant pole rolling off at 106 Hz. At 150 kHz, another pole is encountered and the pole slope changes from –1 to –2. It encounters a zero at 796 kHz, changing the slope back to –1, but not for long. At 813 kHz, another pole changes the slope to –2, near the unity-gain crossover frequency, fc . Way out at 14.88 MHz a final pole within sight bends the slope to a precipitous –3, but well beyond f;c where any reduction in phase margin can be contributed.
Now all the poles and zeros from the loop blocks are put together into a Bode plot. We will do this twice, at each end of the operating range, because of that shifting pole between zs and fs. The following table, in ascending frequency, is what we must plot. In addition, the quasistatic loop gain, G0 x H0 = 429.
The full-scale response is plotted below.
The interesting part occurs around fc , where it crosses a gain of one. Generally, if the slope around fc is too negative, excess phase will have accumulated, causing the phase margin, a measure of stability, to be too small.
In plotting the magnitude of G x H , we can determine f c . As the magnitude slope, m, changes, the decrease in loop gain from frequency f 1 to f 2 is
For the laser loop, the gain is 429 at the first pole, at 106 Hz. The second pole occurs at 1.8 kHz. What is the gain at the frequency of the second pole? Apply the formula;
This can be seen marked on the fs plot.
Now that the magnitude plot has been calculated, the phase is next. We will also use asymptotic linear approximations for phase on its semi-log plot. A pole or zero will affect phase a decade each side of it, and will shift phase from a decade below to a decade above by Δ 90o , –90o for a pole and +90o for a zero. The linearized slope of phase for a pole or zero has a magnitude of 45o /decade. Linear interpolation of frequencies on a log-log plot produces the phase contribution at a given frequency. In general, if a pole is at fp , then the phase contribution of this pole at frequency f is
At fp /10, there is no significant phase lag. (Actually, it is about –6o .) At the pole frequency, fp , it is –45o . And at 10 x fp , the phase contribution is (nearly) the full –90o . It works the same with zeros, except their phase contribution is positive. Hence, we use zeros to cancel the phase lags of poles, to keep the phase from descending to near –180o before fc is reached. The indicator of how much stability the loop has is the amount that the phase is above –180o at fc , which is the phase margin . It is the indicator of dynamic behavior being sought. It is –45o at the first pole and being more than a decade below fc , contributes a full –90o at fc .
The 1.8 kHz pole contributes
The 7.1 kHz zero contributes
The 100 kHz pole contributes –2.7o . The poles at higher frequencies can be considered out of range and have no appreciable effect. Accumulating the phase contributions from the contributing poles and zeros at fc , the resulting phase is –120o , and the phase margin, PM, is –120o – (–180o ) = 60o . Is this sufficient?
A two-pole feedback system with a phase margin between zero and 64o will have a closed-loop damping of
and a closed-loop fractional overshoot to a step input, for PM > 20o , of
The closed-loop pole angle is thus
For our case, the closed-loop pole angle is around 53o , an angle in which overshoot of a step input will ring through little more than one cycle before being damped. The overshoot, which might be more of a concern since it involves peak power, is 15 %. While the laser loop is not a two-pole loop, the simpler two-pole case at least gives us a clue as to what the response might be.
The zs case is more of a stability concern because of the additional 4.71 kHz pole contributed by the log-amp of the SQRT circuit. At low current, the pole moved from its 163 kHz fs value to well within the frequency range of the loop dynamics. When the zs frequency-response analysis is carried out, fc is found to be 7.38 kHz. The log-amp pole causes a brief –3 slope near fc – definitely a bad sign – but is quickly corrected by the zero at 7.1 kHz. The proximity in frequency of the zero to the pole essentially causes the zero to cancel it, not allowing it to add much phase lag to an already lagging loop. When the phase at fc is calculated, it is not much worse than the fs case: φ = –126o , and PM = 54o .
Although this loop could be better optimized for dynamic response, beginning perhaps with a new, improved log-amp that doesn't shift its pole with input current, it functions acceptably in the application with the above modifications to the original, essentially hopeless, loop. The key was the SQRT linearization.
The lesson in this case study of a laser controller design applies as a general principle: linearize open-loop blocks first before depending upon feedback to do the rest. This applies as much to laser power controllers as to low-distortion audio amplifiers or feedback circuits generally.