Kirchhoff's Laws

Series and Parallel ended at the edge of its own method: collapsing works while a circuit stays tidy, in clean blocks of one path or side-by-side branches. A circuit with two sources and a shared branch is not tidy, and no amount of equivalent resistance will crack it. Two rules crack it anyway, and they are not even new physics. They are bookkeeping for the two things this track already established: charge does not vanish, and energy does not appear from nowhere. Gustav Kirchhoff wrote them down in 1845 as a university student, and every circuit solved since then, by hand or by software, runs on them.

Charge never piles up

The junction rule says that the current flowing into any node equals the current flowing out. A node where three or more paths meet is a junction, and the accounting has to balance there. If 12 milliamps arrives and one branch leaves with 5 milliamps, the remaining branches must carry the other 7. No amount of resistor cleverness changes the ledger.

The reason is physical, not mathematical. Electrons repel each other, so a node cannot quietly accumulate extra charge: any surplus pushes back against more charge arriving within nanoseconds. Steady circuits have no charge warehouse at the junctions. What comes in has to leave, split among the branches in proportion to what each branch will pass. A branch is the path between two nodes, and a two-part circuit has one junction worth checking: the node between the resistor and the LED in the LED loop. One path in, one path out, so the two parts carry the same current, which is exactly the series rule from series-and-parallel. The junction rule is what that rule becomes when a third path joins.

Written as one statement, currents into a junction sum to zero, with arrivals counted as positive and departures as negative. Sign conventions like this one are half of what the next section needs, so it is worth fixing the habit now: pick a direction, call it positive, and let the arithmetic keep the books.

Voltage sums to zero around a loop

The loop rule says that around any closed path, the voltage rises and drops add to zero. Walk a full lap and the walk ends where it started, at the same potential, so every climb must be paid for by every descent. A battery climb of 12 volts has to be balanced by 12 volts of drops somewhere along the same loop. That is energy conservation wearing circuit clothes: a coulomb of charge returns from its lap with exactly the energy it started with.

your-first-circuit already measured this law without naming it. The meter read about 3 volts across the resistor and about 2 volts across the LED, and the two had to total the 5 volt supply. The lesson called that sum the fastest sanity check there is. Kirchhoff's loop rule is the reason the check works, and the reason it catches faults: if the measured drops refuse to sum to the source, some of the story is missing.

The craft is in the sign discipline. Pick a direction to walk the loop, clockwise or counterclockwise, and count a voltage rise as positive and a drop as negative, or the reverse, as long as the choice holds for the whole loop. Traversing a battery from its minus terminal to its plus terminal is a rise. Traversing a resistor in the direction the current flows is a drop, sized by Ohm's law. Walking the loop the other way flips every sign and multiplies the whole equation by -1, which changes nothing. What ruins a solution is switching conventions halfway around the loop, so pick once and keep it.

One full pass with numbers makes the bookkeeping concrete. Take a 9 volt battery feeding a 470 ohm resistor in series with a blue LED, which drops about 3.2 volts while conducting, as your-first-circuit noted for blue parts. The current is the leftover voltage over the resistance: 9 minus 3.2 is 5.8, and 5.8 / 470 is about 12.3 milliamps. Now walk the loop clockwise, starting below the resistor: a rise of 9 at the battery, a drop of 12.3 milliamps times 470 ohms, which is 5.8 volts, and a drop of 3.2 at the LED. The sum is 9 - 5.8 - 3.2 = 0. The loop rule closes the books, and any measured reading that refuses to close them points at a missing piece of the story.

Cracking a two-loop circuit

Here is a circuit no divider can touch. Three resistors form an upside-down U: a 100 ohm resistor R1 along the top left, a 200 ohm resistor R3 along the top right, and a 100 ohm resistor R2 down the middle, shared by both halves. A 12 volt source stands on the left edge and a 6 volt source on the right edge, flipped so that both sources push current clockwise around their own loops. The bottom rail is common to everything.

Assign a loop current to each loop: I1 clockwise around the left loop, I2 clockwise around the right loop. The middle resistor carries both, in opposite directions, so its net downward current is I1 minus I2. Drawing an imagined current around a loop is the step that looks illegal, and the junction rule is what makes it legal: where two imagined loop currents share a branch, their contributions add to produce the real branch current, and everywhere else a loop current is the real one. The alternative method, nodal analysis, assigns an unknown voltage to each node instead and writes one junction equation per node; same two laws, different unknowns, and the choice is about which bookkeeping is shorter. Simulators prefer nodal analysis for exactly that reason. Now write the loop rule once per loop, drops matching rises:

Loop 1:  12 = I1*100 + (I1 - I2)*100
Loop 2:  (I2 - I1)*100 + I2*200 = 6

Loop 1 says the 12 volt source is spent across R1 and across the middle resistor. Loop 2 says the 200 ohm resistor is paid for by the 6 volt source plus whatever arrives, in reverse, through the middle. Simplify both and solve by substitution, the way any two simultaneous equations fall:

Loop 1:  200*I1 - 100*I2 = 12
Loop 2:  -100*I1 + 300*I2 = 6
From loop 1:  I1 = (12 + 100*I2) / 200
Substitute into loop 2:
  -100*(12 + 100*I2)/200 + 300*I2 = 6
  -6 - 50*I2 + 300*I2 = 6
  250*I2 = 12

So I2 = 12 / 250 = 0.048 amps, and back-substituting gives I1 = (12 + 4.8) / 200 = 0.084 amps. The middle resistor carries the difference, 36 milliamps downward. Check both loops with Ohm's law and the books balance: loop 1 spends 8.4 volts on R1 and 3.6 on the middle resistor, totaling the source's 12. Loop 2 spends 9.6 volts across the 200 ohm resistor, covered by 6 from its own source and 3.6 arriving backward through the middle branch. One check more, from the nodes: the top of the middle branch sits 12 volts minus 8.4, that is 3.6 volts above the bottom rail, which is exactly the drop the middle resistor must show. Every number agrees with every other number, which is what a correct solution looks like.

# Loop 1: 200*I1 - 100*I2 = 12
# Loop 2: -100*I1 + 300*I2 = 6
i2 = 12 / 250                    # from -6 - 50*I2 + 300*I2 = 6
i1 = (12 + 100 * i2) / 200       # back-substitute into loop 1
print(i1 * 1000, i2 * 1000, (i1 - i2) * 1000)

The print shows the three branch currents in milliamps: 84.0, 48.0, and 36.0. The same substitution would work for ten loops and a hundred resistors; the equations just stack higher, and nobody solves a hundred of them by hand.

One sign convention deserves its own warning, because it alarms everyone the first time. If a solved current comes out negative, nothing failed. The minus sign means the actual current flows opposite the assumed direction. The magnitude is correct, the arrow was not, and flipping the arrow finishes the job. Negative results are information, not errors.

Why this is machine work

Look at what the last section actually did. Label currents, write one loop equation per loop with a fixed sign convention, solve the stack of equations. Nothing in the recipe needs judgment, which makes it a job for a program. Circuit simulators do exactly that, at scale: SPICE, the Simulation Program with Integrated Circuit Emphasis first released from Berkeley in the early 1970s, reads a netlist, the machine-readable list of every part and connection, stamps out one equation per unknown using the nodal form, and solves the whole system numerically. Every simulator since, from LTspice on a laptop to the browser tools used in this track's projects, is running Kirchhoff's two laws thousands of times per analysis, with matrix algebra doing the substitution step at machine speed.

That mechanical nature is also the honest limit of hand analysis. Two loops solve on one page. Ten loops solve in a simulator in milliseconds and by hand in an afternoon nobody wants to spend. The skill worth building is knowing which three or four loops matter for a design decision, and letting software handle the rest.

The two laws also set up the shortcut this course takes next. Most design questions do not involve a whole network; they involve what one load will see at one pair of terminals. Kirchhoff's laws guarantee that the entire mess behind those two terminals can be summarized completely, and Sources and Thevenin Equivalents shows the summary that makes a loaded divider a one-glance problem.