Intuition The ONE core ideaA tangle of resistors between two wires always behaves like a single resistor — because voltage and current relate through one straight-line rule. Everything else in this topic is just bookkeeping: we merge little groups of resistors, two at a time, until only that one imaginary resistor is left.
Before you can "collapse a network inward" you must own every word and squiggle the parent note throws at you. Below, each symbol is built from nothing, drawn as a picture, and justified — why does the topic need this? Read top to bottom; each item leans on the one above it.
Definition Wire, node, terminal
A wire is a perfect path: charge slides along it losing zero energy. On a diagram it's just a line.
A node is a dot where two or more wires meet — a junction. Every point on the same unbroken wire is the same node.
A terminal is one of the two special nodes we poke to measure the whole box: call them A B
Look at the figure. The two amber dots are terminals A B one node even though it stretches across — no resistor breaks it.
Intuition Why we obsess over nodesThe entire "series vs parallel" decision is a node-counting question. You cannot say two resistors are in series until you can point at the shared node and confirm nothing else taps it . So nodes are the unit of thought for this whole topic.
Definition Current, symbolI Current is how much charge flows past a point each second . Picture water in a pipe: current is litres-per-second. Symbol I amperes (A). An arrow on the wire shows its direction.
Intuition Why a single letterI series path the current is the same everywhere — one number describes the whole path. That single-number-ness is exactly what lets the parent note write "same current I R 1 R 2
Definition Voltage, symbolV Voltage across a component is the energy difference per unit charge between its two ends — the "push" that drives current through it. Picture the height drop of a waterfall: the taller the drop, the harder the water is shoved. Symbol V volts (V).
Voltage is always measured between two nodes (a difference), never at a single point alone.
In the figure, current (blue arrow) flows through a resistor and the voltage is the height of the drop (amber) from the high node to the low node. This "drop" picture is why the parent can add voltages along a path — see §7.
Definition Resistance, symbolR Ω Resistance measures how much a component fights the current for a given push. Picture a narrow section of pipe : narrower = more resistance = less flow for the same push. Symbol R ohm , written Ω
Intuition Why "linear" is the secret engine of this whole topicBecause V = I R straight line (no squares, no bends), any two-terminal box of resistors also obeys a straight line V = I ⋅ R e q Ohm's Law — is why a whole tangle can be replaced by one number R e q R e q
Definition Equivalent resistanceR e q R e q equivalent ) is the single resistance that, placed between A B exact same current for the exact same voltage as the whole messy box. It's the slope of the box's straight line from §4.
We also write R A B which two terminals we measured between.
The topic's entire goal is one sentence: find R e q
Definition Notation you'll meet constantly
R 1 , R 2 , R 3 , … names for individual resistors (the little number is a label, not a power or multiplication).R 23 after merging them into one.R 1 ∥ R 2 R 1 in parallel with R 2 …
Common mistake The subscript is not mathsR 12 not mean R 1 × 2 labels .
The two "atomic rules" of the parent note come from two conservation laws. You don't need to master them here — just recognise the pictures.
Definition Kirchhoff's Voltage Law (KVL) — the pictureWalk any loop of the circuit and the voltage drops add up to the voltage you put in — because you return to the same height you started. In a series path this means V = V 1 + V 2 Kirchhoff's Voltage Law and it justifies series addition .
Definition Kirchhoff's Current Law (KCL) — the pictureAt any node, current in = current out — no charge piles up at a junction. When one wire splits into two branches, the branch currents add back up: I = I 1 + I 2 Kirchhoff's Current Law and it justifies parallel addition .
Left panel: one current threading a series path (KVL — heights add). Right panel: one current splitting at a node into two branches (KCL — flows add). These two pictures are the series and parallel rules.
Definition Conductance, symbolG = 1/ R Conductance is the upside-down of resistance: how easily current flows, not how hard it's fought. G = R 1 R G G
Intuition Why the topic needs the reciprocal at allIn parallel , extra branches add more ease of flow , so it's the conductances 1/ R R e q 1 = R 1 1 + R 2 1 + … Conductance and Admittance .
Common mistake The classic reciprocal slipAfter computing R 1 1 + R 2 1 = 0.5 R e q 1 not R e q R e q = 0.5 1 = 2 Ω
Definition When series/parallel runs outSome networks have a resistor "bridging" across the middle so that no pair is purely series or parallel. These are bridge networks. Picture the Greek letters: Δ Y Delta-Wye Transformation swaps one shape for the other to unlock the collapse. You only need to recognise you're stuck — the fix lives in its own note.
Equivalent resistance Req
Parallel rule reciprocals
Reveal each — if you can't answer instantly, reread that section.
A node is… a junction dot where wires meet; every point on the same unbroken wire is one node.
Two resistors are in series only when their shared node… connects to nothing else, so they carry the same current.
Current I charge flowing past a point per second (litres-per-second of the "water"); unit ampere.
Voltage V energy per charge / the "push" between two nodes — the height of the drop; unit volt.
Resistance R how hard a component fights current for a given push (pipe narrowness); unit ohm Ω
Ohm's Law states… V = I ⋅ R
Why can a whole box equal one resistor? Because V = I R V = I R e q R e q
R e q the single resistor giving the same current for the same voltage as the whole network.
The subscript in R 12 a label for "resistors 1 and 2 merged into one," not multiplication.
KVL in one line… around a loop, voltage drops add up (same height returned) → justifies series sums.
KCL in one line… current in equals current out at a node → justifies parallel addition.
Conductance G 1/ R easily current flows; conductances add in parallel.
The ∥ "in parallel with" — combine the two resistors using the reciprocal rule.
A bridge network is… one with no purely series or parallel pair; needs a Y Δ
