Visual walkthrough — Block diagram algebra

We assume nothing but this: an arrow is a signal travelling, and a block is a box you multiply by. Everything else we build.

---

Step 1 — One arrow, one box: what "multiply" looks like

WHAT: Draw a single signal $X$ entering a box labelled $G$, and one signal $Y$ leaving it.

WHY: Before there is any loop, we must be dead certain what a block does. A block is not a delay, not a filter you must picture as wiggles — for our algebra it is one instruction: multiply the number on the incoming arrow by the number in the box.

PICTURE:

$Y=G\cdot X$

• $X$ — the signal on the input arrow (a value at that point on the wire).
• $G$ — the transfer function inside the box; the multiplier.
• $Y$ — the signal on the output arrow, equal to $G$ times $X$.

---

Step 2 — The summing junction: a circle that adds and subtracts

WHAT: Introduce the small circle that takes two arrows in and puts one arrow out. Each incoming arrow wears a $+$ or a $-$ sign.

WHY: Feedback means "compare the output to the command." Comparison = subtraction. We need a picture-object that subtracts, and that is the summing junction.

PICTURE:

$E=R-B$

• $R$ — the reference (what we command the system to do); enters with a $+$.
• $B$ — the fed-back signal (a copy of what actually happened); enters with a $-$.
• $E$ — the error, how far off we are: command minus reality.

---

Step 3 — The take-off point: copying a signal for free

WHAT: Place a dot on the output wire. From that dot, a second arrow branches off. Copying does not change the value — both arrows downstream of the dot carry exactly $Y$.

WHY: To feed something back, we first need to tap it without disturbing it. The take-off (pickoff) point is that tap.

PICTURE:

• The dot sits on the wire carrying $Y$.
• The straight-through arrow still carries $Y$ (the real output).
• The branch arrow also carries $Y$ — this copy is what we will send into the feedback path.

---

Step 4 — The feedback path: send the copy back through $H$

WHAT: Route the branched copy of $Y$ through a second box $H$, then into the $-$ terminal of the summer as the signal $B$.

WHY: We rarely feed the raw output back — we feed back a measurement of it (a sensor, a scaling). That measurement box is $H$, the feedback transfer function.

PICTURE:

$B=H\cdot Y$

• $Y$ — the copied output entering the feedback box.
• $H$ — the feedback block (e.g. a sensor's gain).
• $B$ — what comes out, headed for the $-$ terminal of the summer.

Now the loop is physically complete: $R$ and $B$ meet at the summer, the error $E$ drives $G$, $G$ produces $Y$, $Y$ is copied and sent through $H$ to become $B$ again.

---

Step 5 — Chase one signal all the way around

WHAT: Write down the three facts the picture already forces, in the order the signal travels.

WHY: The whole diagram is now three equations. If we can eliminate the internal signals $E$ and $B$, only $R$ and $Y$ remain — and their ratio is what we want.

PICTURE: the same loop, now with every wire labelled:

$$\underset{\text{summer (Step 2)}}{\underbrace{E=R-B}}\underset{\text{block (Step 1)}}{\underbrace{Y=GE}}\underset{\text{feedback (Step 4)}}{\underbrace{B=HY}}$$

• First equation: the summer's rule.
• Second: the forward block's rule.
• Third: the feedback block's rule.

We have three equations and three unknown internal signals plus the two external ones. That is exactly enough to solve.

---

Step 6 — Substitute to kill the internal signals

WHAT: Put $B=HY$ into $E=R-B$, then put that $E$ into $Y=GE$.

WHY: $E$ and $B$ are private wires inside the box — an outside user never sees them. Eliminating them leaves a relation purely between what we command ($R$) and what we get ($Y$).

$$Y=GE=G(R-B)=G(R-HY)$$

term by term:

• $G(R-HY)$ — the block multiplies the error.
• $R$ — the command.
• $HY$ — the fed-back measurement being subtracted.

Distribute the $G$:

$$Y=GR-GHY$$

• $GR$ — the open-loop response: what $Y$ would be with the loop cut.
• $GHY$ — the correction the loop keeps subtracting, because $Y$ influences itself once per lap.

---

Step 7 — Collect $Y$: the loop-gain appears

WHAT: Move every $Y$ to the left side and factor.

WHY: $Y$ appears on both sides because the output feeds itself. Gathering the $Y$ terms turns a self-referential loop into one clean solvable line.

$$Y+GHY=GR⟹Y(1+GH)=GR$$

• $1$ — the signal going straight down once with no feedback trip (the "$Y$ by itself").
• $GH$ — the loop gain: what a signal is multiplied by after one full lap ($G$ forward $\times$ $H$ back).
• $1+GH$ — passthrough plus one round trip.

Divide:

$$\boxed{\frac{Y}{R}=\frac{G}{1+GH}}(\text{negative feedback})$$

---

Step 8 — Every case: signs, unity, and degenerate limits

WHAT: We must never leave a scenario undrawn. Walk each special case.

WHY: A formula you can't stress-test isn't understood. Here is every sign and every extreme.

Positive feedback. If the summer adds $B$ ($E=R+B$), redo Step 6: $Y=G(R+HY)=GR+GHY$, so $Y(1-GH)=GR$: $\frac{Y}{R}=\frac{G}{1-GH}(\text{positive feedback}).$ The only change is the sign in the denominator — because the round-trip now reinforces instead of opposes.

Unity feedback ($H=1$): the sensor is a plain wire, $B=Y$. Then $\frac{Y}{R}=\frac{G}{1+G}.$

No loop ($H=0$): the feedback path is cut, $B=0$, so $E=R$ and $Y=GR$. Formula agrees: $\frac{G}{1+0}=G$. ✔ Sanity: with the loop open you just get the plain block from Step 1.

Huge forward gain ($G\to \infty$): $\frac{Y}{R}=\frac{G}{1+GH}\stackrel{G\to \infty }{\to }\frac{1}{H}.$ The output stops caring about $G$ and depends only on $H$. This is exactly why an op-amp's enormous, sloppy open-loop gain becomes a precise closed-loop gain set by the feedback resistors.

Denominator hits zero ($1+GH=0$): the ratio blows up — a finite command drives infinite output. That equation, $1+GH=0$, is the characteristic equation; its roots are the loop's poles. This is the boundary between a controller that settles and one that runs away.

PICTURE — the four faces of one formula:

---

The one-picture summary

One diagram, one substitution chain, one formula. Read it top to bottom: signal $R$ enters, gets its error, is multiplied by $G$, is copied, is measured by $H$, comes back as $B$ — and the algebra of "$Y$ feeds itself once per lap" is exactly the $1+GH$ on the bottom.

Recall Feynman: the whole walkthrough in plain words

Picture a thermostat. You command a temperature ($R$). The room's real temperature gets measured and sent back ($B$). The summer does the arguing: "command minus reality" is the error ($E$). The heater is the block $G$ — it turns error into actual heat ($Y=GE$). Now the sneaky part: the heat you just made becomes the new reality, gets measured again, and comes back to shrink the error. So $Y$ secretly depends on itself. When you write that dependence down honestly — $Y$ equals its open-loop value $GR$ minus one lap of feedback $GHY$ — and gather the $Y$'s, you get $Y(1+GH)=GR$. The "$1$" is you going straight through once; the "$GH$" is one full trip around. Divide, and the loop is tamed into a single number $\frac{G}{1+GH}$. Crank $G$ huge and the machine forgets its own strength and obeys only the sensor $1/H$ — that's the whole trick of feedback.

---

Connections

• Parent: Block diagram algebra — the rules this page derives.
• Transfer functions — what lives in each box.
• Laplace transform — why a box means "multiply."
• Feedback control loops — where $\frac{G}{1+GH}$ is used.
• Signal flow graphs & Mason's gain formula — the same result without diagram surgery.
• Op-amp gain — the $G\to \infty \Rightarrow 1/H$ limit in hardware.
• Stability & characteristic equation — the denominator $1+GH=0$.

---

Quick self-check