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.

**Dynamic Analysis**

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.

**Error Amplifier**

The error amplifier is a one-op-amp differential amplifier with a LM358, which has a GBW = *f*_{T} = 1 MHz. Then the transfer function of this inherently linear block is

where *v*_{i} is the incremental commanded output power, and *v*_{b} 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 *f*_{bw} with a –1 slope, from a quasistatic, open-loop voltage gain of K, until it intersects the unity-gain axis at *f*_{T}. Then with a single-pole rolloff of –1,

For a typical *K* = 10^{5} and *f*_{T} = 1 MHz, then the open-loop op-amp bandwidth is quite low; *f*_{bw} = 10 Hz.

The quasistatic closed-loop gain value, *A*_{cl} << *K*, and the resulting bandwidth, *f*_{bwcl} is proportionately higher; in the case of *A*_{e}(*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 *A*_{cl}, 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 *f*_{T} = 1 MHz, * τ*_{T} ≅ 159 ns.