Lesson 2 — Process Gain

Lesson 2 · Objective

What Is Gain?

Before we can talk about how a PID controller actually behaves, there's a piece of vocabulary you need first: gain. It shows up constantly once you start tuning loops, so it's worth nailing down properly right now, before the next lesson leans on it.

At its core, gain is just a multiplier — a ratio that describes how much an output changes for a given change in an input. Think of it as the "bang for your buck" number for any input/output relationship: push the input a little, and gain tells you how much the output moves in response.

The General DefinitionGain = ΔOutput ⁄ ΔInput

That triangle symbol, Δ ("delta"), just means "the change in" — so ΔOutput is simply how much the output changed, and ΔInput is how much the input changed to cause it.

INPUT
changes

PROCESS
(has a gain)

OUTPUT
responds

Every input/output relationship in this section — a controller's output driving a valve, a valve driving a flow, a flow driving a temperature — has its own gain. Over the next few pages you'll see that gain can show up two different ways depending on what's being measured, and then why it matters so much once we get to tuning a real controller.

Form 1 of 2

Gain Without Units — Percent of Span

Sometimes both the input and the output are expressed the same way: as a percentage of their own span. A controller's output is reported as 0–100% of its full range, and a valve's position is reported as 0–100% of fully closed to fully open. When input and output are both percentages like this, the percent signs cancel in the division — so the resulting gain has no units at all, just a plain number.

Worked Example — Controller Output Driving a Valve
Signal	Before	After	Δ (change)
Controller Output (CO)	40%	60%	+20%
Valve Position	40%	60%	+20%

Gain = ΔValve ⁄ ΔCO = 20% ⁄ 20% = 1.0 (unitless)

Why no units? Percent-of-span divided by percent-of-span leaves nothing behind — the "%" on top cancels the "%" on the bottom. A gain of exactly 1.0 here means the valve tracks the controller's output move-for-move, as a fraction of its own range.

Form 2 of 2

Gain With Units — One Case Worth Seeing

Sometimes the input and output are two genuinely different kinds of measurement — not comparable percentages of the same kind of span. A quick example: a 4–20 mA control signal drives a valve, the valve changes a flow, and that flow changes a downstream temperature. Milliamps and degrees aren't the same kind of thing, so when you divide one by the other, the units don't cancel — they stick around, and the gain carries them.

Worked Example — 4–20 mA Signal Driving a Downstream Temperature
Signal	Before	After	Δ (change)
Control Signal	8 mA	14 mA	+6 mA
Downstream Temp	150°F	168°F	+18°F

Gain = ΔTemp ⁄ ΔSignal = 18°F ⁄ 6 mA = 3°F per mA

You'll hear a process gain described this way in the field — "3°F per mA." That's a real, correct way to describe how strongly the physical process responds.

But here's the twist: a real PID controller never actually does its own gain math in units like these. A few pages from now you'll see why the controller's own gain setting is always just a plain number — no units attached — no matter what units the process itself happens to use.

A Property of the Process

Every Process Has Its Own Inherent Gain

Here's the important part: process gain isn't something an instructor picks, and it isn't something a controller invents. It's a physical property of that specific process — how much a real tank's level actually moves for a given valve change, how much a real room's temperature actually moves for a given furnace output change. Two different processes, even ones doing "the same job," can have wildly different gains.

Processes that swing a lot for a small input change are called "twitchy" — a small nudge produces a big output response, which means high gain. Processes that barely move even for a large input change are called "sluggish" — a big push produces only a small output response, which means low gain.

High Gain

Process A — Small Pressure Vessel ("Twitchy")

CO change +5%

Pressure +40 psi

Gain = 40 psi ⁄ 5% = 8 psi per %

A tiny valve nudge swings this vessel's pressure hard.

Low Gain

Process B — Large Storage Tank ("Sluggish")

CO change +50%

Level +2 in

Gain = 2 in ⁄ 50% = 0.04 in per %

Even a huge valve move barely budges this tank's level.

Bar lengths above are illustrative only, to show relative size of the response at a glance — read the numbers for the actual values.

Practice

More Worked Examples

The arithmetic is always the same — find ΔOutput, find ΔInput, divide. Notice that all three examples below use percent of span on both sides. That's not a coincidence, and it's not just to keep the numbers tidy — it's exactly how a real controller sees the world, and it's why its gain always comes out unitless. You'll see why on the next page.

1. Level Tank (unitless)
Signal	Before	After	Δ
Controller Output	35%	50%	+15%
Level (% of span)	35%	65%	+30%

Gain = 30% ⁄ 15% = 2.0 (unitless) — a fairly twitchy tank.

2. Temperature Loop With a Heater (unitless — % of span)
Signal	Before	After	Δ
Heater Output	20%	45%	+25%
Temperature (% of span)	30%	55%	+25%

Gain = 25% ⁄ 25% = 1.0 (unitless)

3. Pressure Vessel (unitless — % of span)
Signal	Before	After	Δ
Control Signal (% of span)	20%	30%	+10%
Pressure (% of span)	20%	60%	+40%

Gain = 40% ⁄ 10% = 4.0 (unitless) — another twitchy vessel.

Setting Up Proportional Control

Why This Actually Matters

The Controller Has to Match the Process's Gain

Here's the bridge to everything coming next. A PID controller has its own tunable gain setting — the P (Proportional) term, often written Kp — and that setting has to be chosen appropriately relative to how twitchy or sluggish the actual process is. The controller's job is to counteract the process's own gain, not ignore it.

Get that match wrong, and control suffers in one of two predictable ways:

Too aggressive on a twitchy (high-gain) process — the controller keeps overreacting to a process that already overreacts on its own, and you get overshoot, oscillation, even instability.
Too gentle on a sluggish (low-gain) process — the controller barely pushes a process that already barely responds, and you get painfully slow, sleepy control.

That's exactly what Proportional control is built around. Its whole mechanism is a tunable gain setting for precisely this reason — so you, the technician, can dial the controller's own gain up or down to counteract whatever gain the process happens to have. We won't fully unpack how Kp works here — that's Proportional Control's job, a couple lessons ahead — but now you know why a proportional controller needs a gain setting in the first place.

Why Gain Has No Units

Kp Itself Never Carries Units — Here's Why

Look back at the last two pages. Every unitless example used percent of span on both sides, and the one example with real engineering units (3°F per mA) was describing the process's gain — not the controller's. A real PID controller's proportional gain, Kp, is always a plain, unitless number — even when the process variable (PV) and the control output (CO) are two completely different physical quantities from each other.

Here's why that works. Before a signal ever reaches the controller's math, the instrumentation scales it into percent of span — 0–100% of whatever range that instrument is configured for. A transmitter reading flow as a 4–20 mA signal ranged for 0–500 GPM doesn't hand the controller "312 GPM" or "13.6 mA" — internally, it's already treated as a percentage of that 500 GPM span. The valve's output is already a percentage by nature (0–100% open). So by the time the controller does its proportional math, both sides of the equation are percentages — no matter what the PV and CO physically represent.

Example — PV (Flow) and CO (Valve Position) Are Physically Different, but Kp Isn't
Signal	Real-World Value	Scaled to % of Span
PV — Flow (4–20 mA, ranged 0–500 GPM)	250 GPM (12 mA)	50%
CO — Valve Position	—	62%

Kp = ΔCO ⁄ ΔPV = %CO change ⁄ %PV change

GPM and "percent open" never actually meet inside that formula — the controller only ever sees the two percent-of-span numbers, and a percentage divided by a percentage leaves nothing behind. That's why Kp comes out unitless no matter what the PV and CO physically are — gallons per minute vs. percent open, degrees vs. psi, it doesn't matter.

The takeaway: the units are implied, not calculated — the controller only ever sees percent of span, so its gain is always just a number, no matter what real-world units the process itself uses. When a technician says "this loop is tuned with a gain of 2," there's no unit missing — that's exactly correct.

One More Property of the Process

Self-Regulating vs. Integrating Processes

Twitchy vs. sluggish describes how much a process responds to a change in output. There's a second, independent question worth knowing before you go further: does the process ever stop moving on its own after that change, or does it just keep going forever unless you undo the change? That's the difference between self-regulating and integrating processes.

Self-Regulating

Temperature and Flow Processes

Bump the heater output up a step and hold it there — the room's temperature rises, but it doesn't rise forever. It climbs toward a new steady value and settles there on its own, because the process naturally loses heat back to its surroundings faster the hotter it gets, until gains and losses balance out. Flow processes behave the same way: open a valve a fixed amount and the flow rate settles at a new steady number almost immediately. The process finds its own new equilibrium without you touching anything else.

Integrating

Level Processes

Bump a tank's inlet valve open a step and hold it there — if outflow doesn't also change, the level doesn't settle at a new value. It just keeps climbing, steadily, for as long as the imbalance lasts — because level is simply the running total (the integral) of inflow minus outflow over time. The only way to make the level stop moving is to change something again — close the valve back down, or increase the outflow to match. The process never finds its own new equilibrium; it keeps accumulating until you intervene.

Nice to know — why "integrating"? The word comes straight from calculus. Integration is the math operation that takes a rate and accumulates it over time into a running total — that's literally what an integral is. A tank's level at any moment is that running total: the integral of inflow rate minus outflow rate, added up over time. It's not just a name that sounds sciency — "level = the integral of net flow" is the literal equation, which is exactly why this kind of process is called integrating.

Why this matters going forward: the demos and examples you'll see with level in the next few lessons behave differently from the ones with temperature or flow, precisely because of this. Keep this distinction in mind — it explains behavior that would otherwise look like a mistake in the simulator.

Review

Next up — Lesson 3: Loops and Acting Direction — open-loop vs. closed-loop control, and how to tell whether a controller needs to be direct-acting or reverse-acting. Proportional Control, where you'll finally see Kp in action, comes after that.