Foundations — Block diagram algebra

This page assumes nothing. Before you can "series-combine blocks" or "apply the feedback formula," you must know exactly what a block is, what the letter $s$ means, what an arrow carries, and why a box multiplies. We build each symbol from the ground up, each one earned before the next.

0. The picture we are decoding

Everything in the parent topic is drawn from just four kinds of marks on paper: an arrow, a box, a circle, and a dot. Let us meet them one at a time.

Look at the figure. Trace the flow left to right: a signal enters from the left arrow, passes through the box, and leaves as a new signal on the right. That is the whole story of one block. Now we name each piece.

1. The arrow — a signal

The picture: a wire with a direction. Water in a pipe, current in a cable, a voltage on a line — the arrow does not care what it is physically; it only carries a number that can change.

Why the topic needs it: every rule in the parent ("output $=G\cdot$input", "$E=R-B$") is a statement about signals. Without arrows there is nothing to multiply, add, or feed back.

We will meet several named signals. The parent uses these letters:

Each is just an arrow with a job. Hold onto them; they reappear the moment we draw a loop.

2. The variable $s$ — "the world of Laplace"

Before we can say what a box does, we must decode the letter inside it, because the box holds "$G(s)$" and that $s$ is doing heavy lifting.

Why this tool and not another? In the time world, sending a signal through a physical system means performing convolution, an integral that smears the input over time — genuinely painful arithmetic. The Laplace transform answers the question "is there a viewpoint where that smearing becomes ordinary multiplication?" and the answer is yes: that viewpoint is the variable $s$. We choose $s$ precisely because it converts the ugly operation into $\times$.

You do not need to compute a Laplace transform to use this topic. You only need to accept the one fact it grants us: through-a-system $=$ multiply-by-$G(s)$.

3. The box — a transfer function $G(s)$

Now the box makes sense.

The picture: a gain-machine. Whatever value flows in on the left arrow gets multiplied by $G(s)$ and flows out on the right arrow. In the figure of Section 0, if the input is $X$ then the output is $G(s)X$.

Why $G(s)$ and not just a number $G$? A plain number would mean "always multiply by 3, no matter how fast the signal wiggles." Real systems respond differently to slow vs. fast changes (a heavy motor lags fast commands). Making the multiplier depend on $s$ lets one box encode all those speed-dependent behaviours at once. The details of what lives inside $G(s)$ are the job of Transfer functions.

The figure shows the box as a multiplier: input $X$ (cyan) enters, the amber label $G(s)$ scales it, and the output $G(s)X$ (white) leaves. This single multiplication is the atom of the whole topic.

4. The circle — a summing junction

The picture: a meeting-of-wires that outputs a running total. The parent's key equation $E=R-B$ is read straight off such a circle: the reference $R$ comes in with a $+$, the fed-back signal $B$ comes in with a $-$, and the circle spits out their difference $E$.

Why the topic needs it: subtraction here is the heart of control. We want the error $E$ = "how far off are we?" A summing junction with a $-$ on the feedback is what measures the mistake so the system can correct it.

Study the signs in the figure. The same circle can add ($E=R+B$) or subtract ($E=R-B$) — only the little symbol beside the arrow changes. Every "positive vs. negative feedback" distinction in the parent lives entirely in that one sign.

5. The dot — a take-off (pickoff) point

The picture: a splitter. Water in one pipe fans out into two identical pipes; both carry the same value that arrived at the dot. If the signal at the dot is $A$, then every wire leaving the dot also carries $A$.

Why the topic needs it: feedback and parallel paths both start with "the same signal goes two ways." The take-off is where a signal is reused. Crucially — and this is a favourite trap — the copy reads the value at that exact location. Move the dot to a different point and the copy reads a different value. That single fact is the reason "moving a take-off past a block costs a $\frac{1}{G}$" in the parent's Rule 4.

6. Putting the marks together — a full loop

Now every symbol in the parent's big formula is defined. Let us assemble them into the single-loop diagram the parent calls "the big one" and read the equation off the picture.

• Circle gives $E=R-B$ (Section 4).
• Forward box $G$ gives $Y=GE$ (Section 3).
• Feedback box $H$ gives $B=HY$ — same rule, a different box.
• The wire from $Y$ back to the circle starts at a take-off dot (Section 5): it copies $Y$ so the output can both leave and be fed back.

Substituting exactly as the parent does yields the closed-loop transfer function $\frac{Y}{R}=\frac{G}{1+GH}$. Notice we invented no new symbol to get there — arrow, box, circle, dot, and the multiplier fact from $s$ were enough. That is the point of this foundations page.

7. Prerequisite map

Equipment checklist

Connections

• Parent topic — full algebra & rules
• Laplace transform — grants the "through-a-block $=$ multiply" fact.
• Transfer functions — what actually lives inside each box $G(s)$.
• Feedback control loops — where the assembled loop is used.
• Signal flow graphs & Mason's gain formula — an alternative to redrawing these marks.
• Op-amp gain — same closed-loop $G/(1+GH)$ story.
• Stability & characteristic equation — the denominator $1+GH$ we just built sets the poles.