MOSFET 1962 Measure Square Law Drain Current Characteristic

# Measure the Square-Law Drain Current Characteristic: Controlled by a Voltage, Not a Current

## 1. Why This Device's Output Curves Don't Look Like Anything Bipolar

This step sweeps drain-to-source voltage at several fixed gate voltages and records the resulting drain current, producing this project's first full current-voltage family of curves for a device whose output current is set entirely by a voltage rather than by a current — a genuine departure from every bipolar device this project has characterized since 1954, whose collector current has always been controlled by a base current flowing into a forward-biased junction. At small drain voltage, the inversion channel Steps 5 and 6 established behaves like an ordinary resistor whose value depends on how far the gate voltage sits above threshold, and drain current rises roughly in proportion to drain voltage — the triode region. As drain voltage keeps increasing, though, the voltage actually available to sustain inversion narrows toward the drain end of the channel, until at some point the channel there pinches to essentially zero depth; beyond that point, additional drain voltage no longer pulls meaningfully more current through, and the curve bends over into a region where drain current stays almost flat regardless of further drain voltage — saturation, but saturation for a reason that has nothing to do with how a bipolar transistor saturates.

$$I_{D,\text{sat}} = \frac{\mu_n C_{\text{ox}}}{2}\,\frac{W}{L}\,(V_{GS} - V_T)^2$$

where $\mu_n$ is the channel's own carrier mobility, $C_{\text{ox}}$ the oxide capacitance Step 2 fixed, $W$ and $L$ the channel's width and effective length as Step 3 and Step 4 defined them, and $V_{GS} - V_T$ the gate voltage's margin above the threshold Step 5 identified — every quantity this equation needs was already established by an earlier step in this series, and the current it predicts grows with the *square* of that margin, not linearly with any current fed into the gate, because no current flows into this device's gate at all.

A Family of Curves, Each Rising Then Flattening triode region, then saturation, at each of several fixed gate voltages drain voltage, VDS → drain current, ID highest VGS medium VGS lowest VGS above threshold pinch-off boundary — triode ends, saturation begins ID,sat = (μnCox/2)(W/L)(VGS − VT)² a square-law dependence on gate overdrive, not a linear dependence on any current

## 2. Real Diagram: The Channel, Shrinking to a Point Right Where It Meets the Drain

The cross-section below shows the channel at the onset of saturation — full depth near the source, where the full gate overdrive is still available, narrowing continuously toward the drain, where the local voltage has eaten away nearly all of the margin above threshold, until the channel pinches to essentially zero depth right at the drain's own edge.

The Channel Narrows to a Point at the Drain full gate overdrive near the source, almost none left right at the drain source drain channel — wide near the source, pinched near the drain pinch-off point — channel depth goes to zero here beyond pinch-off, more drain voltage drops across a depleted region, not across the channel itself

## 3. The Qualitative Fact Every Later Section of This Project Will Assume

Every bipolar device this project has documented produces an output current controlled by an input current — a base current, amplified by the transistor's own current gain, sets the collector current almost linearly, and the output characteristic curves cluster together in roughly evenly-spaced steps as base current steps evenly. This step's output characteristic looks nothing like that. The controlling terminal here draws essentially no steady current at all; the gate simply holds a voltage, and that voltage alone, through the square-law relation this step has now measured directly, sets how much drain current flows. This is not a minor variation on the bipolar output characteristic's shape — it is a different control variable entirely, voltage instead of current, producing curves that step unevenly with equal increments of gate voltage because the underlying relation is quadratic rather than linear. Every later step and every later series that compares this device family to a bipolar transistor will be comparing a voltage-controlled device against a current-controlled one, and this step is where that distinction first becomes a measured curve rather than a claim.

Step 8 does not merely add one more characterization to this series' record; it measures the one curve that makes explicit, for the first time in this project's history, what it actually means for a device's output current to answer to a voltage instead of a current.

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