Lilienfeld 1925 Apply Control Electrode Bias

# Apply Control-Electrode Bias: There Is No Single Film Potential to Stay Above

Step 11's main bias does not sit at one potential across the film — it drives current through a resistive path, and a resistive path carrying current develops its own internal voltage drop from one end to the other. The roadmap's own instruction to "maintain the foil above the local film potential" is easy to misread as a single comparison between two fixed numbers. It is not. The film's potential varies continuously along its own length, from whichever terminal sits at the higher voltage down to the lower one, and the foil sits above some stretch of that varying potential rather than above one fixed point. Satisfying this step correctly means satisfying it everywhere along the foil's footprint at once, not just somewhere convenient to measure.

## 1. The Film's Own Potential Falls Steadily Along Its Length

Current flowing through the film's resistance produces a position-dependent voltage drop, following directly from the same resistivity and geometry Step 5 and Step 9 already established:

where $V_{T1}$ is the potential at the terminal the position $x$ is measured from, and the potential falls steadily toward $V_{T2}$ at the far terminal. This is the same quantity a much later generation of field-effect devices would call the channel potential, and the same basic fact applies here, decades earlier and in a cruder form: a resistive conducting path carrying current is never at one potential along its own length, and any control electrode sitting above part of that path has to be referenced against whichever value of $V_{\text{film}}(x)$ actually sits beneath it at each point.

## 2. Real Diagram: The Potential Falls, but the Foil Sits at One Fixed Value

## 3. The Foil's Bias Has to Clear the Most Demanding Point, Not an Average One

Because the foil sits above a whole stretch of varying film potential at once, "maintaining the foil above the local film potential" really means satisfying the inequality at the single worst point along that stretch — the one point where $V_{\text{film}}(x)$ happens to be highest beneath the foil, since that is the point closest to failing the requirement first:

Choosing $V_{\text{foil}}$ to satisfy this only at the average film potential beneath the foil, rather than at the actual maximum, leaves part of the foil's own footprint sitting below the local film potential there — inverting the intended sense of the field at exactly that stretch, even while the rest of the foil's footprint still has the field pointed the right way. A bias chosen against the worst point, rather than a convenient average, is the only choice that keeps the control electrode's influence consistent across its entire length.

## Real Diagram: The Average Looks Safe, the Peak Is What Matters

## Apply Control-Electrode Bias's Place in the Process Lineage

Apply Control-Electrode Bias follows Step 11, Apply the Main DC Bias, whose current through the film is the actual source of the position-dependent potential this step now has to stay above everywhere; it precedes Step 13, Superimpose an Input Signal, which will vary the foil's potential around whatever fixed offset this step establishes. It is the twelfth step of this concept's sequence and the point where Step 11's bias, chosen for its effect on current through the film, turns out to also set a second, less obvious requirement: a floor on the control electrode's own potential, determined not by a single number but by the single highest point along a sloping voltage this step's own bias was responsible for creating in the first place.