Lilienfeld 1926 Quantify Available Power Gain
# Quantify Available Power Gain: The Gain Ceiling Was Already Set by Steps 1 Through 8
Step 9 established whether this device modulates at all. Step 10 asks how much useful power gain that modulation can deliver — and the answer is bounded by quantities this construction already measured well before any gain test began. The electrode-glass coupling capacitance, fixed by Step 1's glass properties and Step 3's bond quality, sets the input impedance any signal source must match. The decay time constant from Step 8 sets the frequency above which capacitive feedthrough starts competing with genuine modulated signal. Gain measurement here is not an open-ended search; it is a confirmation of a ceiling the earlier steps already implied.
The coupling capacitance from Step 1 and Step 3 is the input impedance this gain test inherits, not chooses. C_coupling was effectively set the moment the glass thickness, dielectric constant, and electrode bond quality were fixed in construction — Step 10 cannot select a different coupling network; it can only select a source impedance Z_s that matches what already exists. A gain measurement using a poorly matched Z_s does not reveal a weaker device — it reveals a mismatch, and the two must not be confused.
The decay time constant from Step 8 sets where capacitive feedthrough starts to dominate. At signal frequencies well below 1/τ, the modulation response has time to settle and dominates any direct capacitive feedthrough from the source to the output. As frequency approaches and exceeds 1/τ, feedthrough current (which scales with frequency for a fixed capacitance) grows relative to the settled modulation signal, and the apparent gain measurement becomes increasingly contaminated by a path that bypasses the field effect entirely.
Measuring below the 1/τ corner is not a convenience — it is the condition for a valid gain number. Any gain figure extracted from measurements taken at or above the corner frequency conflates genuine field-effect amplification with capacitive feedthrough that would exist even with a dead film and no modulation at all. Step 10's frequency choice for the gain test must be made using Step 8's τ, not chosen independently.
| Step | Process operation | Input | Output | Specification | Constraint |
|---|---|---|---|---|---|
| 10.1 | Retrieve C_coupling estimate implied by Step 1 glass properties and Step 3 bond quality | Records from Steps 1, 3 | C_coupling estimate | Derived from glass dielectric constant, thickness, and bonded contact area | Without this, Z_s matching in 10.2 is guesswork rather than a derived target |
| 10.2 | Select and verify source impedance Z_s matched to C_coupling | C_coupling from 10.1 | Matched signal source | Re{Z_s} chosen to maximize available power transfer into C_coupling at the test frequency | Mismatch understates gain without indicating a device deficiency |
| 10.3 | Select test frequency well below 1/τ (Step 8) | τ from Step 8 | Chosen test frequency | Test frequency ≤ 0.1 × (1/τ) to keep feedthrough contribution small | Testing near or above 1/τ mixes feedthrough into the gain measurement |
| 10.4 | Apply signal at full amplitude, measure P_L into load R_L | Matched source from 10.2, frequency from 10.3 | Raw output power reading | R_L chosen for maximum power transfer per standard matching practice | An unmatched load also understates measured gain independent of the device |
| 10.5 | Halve signal amplitude, re-measure P_L; verify proportional halving of I_L | Setup from 10.4 | Linearity-verified measurement | I_L scales linearly with v_s within measurement noise | Nonlinear scaling indicates Step 9's bias point may be outside this device's linear operating range |
| 10.6 | Measure feedthrough with film biased to zero-conductance point (if achievable) or by comparison to Step 8's Path A model | Linearity-verified data from 10.5 | Feedthrough estimate | Feedthrough subtracted from raw P_L to isolate genuine modulated power | Skipping this step overstates G_avail by the feedthrough contribution |
| 10.7 | Compute G_avail = P_L (feedthrough-subtracted) / P_avail,in | Corrected P_L from 10.6, matched source from 10.2 | Available power gain value | P_avail,in = v_s² / (8 · Re{Z_s}) | Using raw (uncorrected) P_L here reports an inflated gain figure |
| 10.8 | Report G_avail alongside C_coupling, τ, and test frequency used | Gain value from 10.7 | Complete gain characterization record | All supporting quantities reported, not just the final gain number | A gain number without its supporting bandwidth and impedance context cannot be compared meaningfully to other constructions |
Step 10 closes the loop between construction and performance. The gain figure this step reports is not an independent discovery; it is the quantitative consequence of choices already made in Steps 1 through 8 — glass dielectric properties, bond quality, film uniformity, and leakage behavior. Reporting G_avail without its supporting C_coupling and τ values would strip away exactly the information that makes the number interpretable, and exactly the information that would let a different construction's gain be compared fairly against this one.