Circuits and Kirchhoff's laws
Trace current paths and node voltages to decide how loads should be wired and to localize opens or shorts systematically.
- Solve a small DC resistor network with Kirchhoff's current and voltage laws and use the result to diagnose a wiring fault.
Understand
The working idea
A circuit needs a closed path. Schematics emphasize electrical connections rather than physical layout: every ideal wire joined to a node shares one voltage, and crossing lines are not connected unless the drawing marks a junction. Series components share the same branch current. Parallel components share the same voltage. These statements follow from conservation rather than visual resemblance on a breadboard.
Kirchhoff's current law (KCL) says the algebraic sum of currents at a node is zero: current entering equals current leaving. It expresses charge conservation when the node is not accumulating charge. Kirchhoff's voltage law (KVL) says signed voltage changes around a closed loop sum to zero: energy gained from a source matches energy transferred in components. Choose current arrows and loop directions arbitrarily but keep signs consistent. A negative solved current simply means reality opposes the original arrow.
Series resistances add: R_eq = R1 + R2. For parallel branches, 1/R_eq = Σ1/R_i. A voltage divider uses two series resistors: with output across R2, V_out = V_in R2/(R1+R2), provided the next circuit does not load it significantly. That condition matters when scaling a battery voltage for a sensor input; a low-resistance load changes the ratio.
A frequent misconception is that current is consumed by the first component, leaving less for the second series component. The same steady branch current passes both; their voltage drops differ. Another is assuming two parts drawn beside each other are parallel without checking both endpoints. Label nodes, trace every closed path, and predict measurements before probing. An open circuit often leaves zero branch current but can still leave full supply voltage across the break; that clue is more informative than replacing parts at random.
Follow the reasoning
Worked example
A 9 V source feeds 1 kΩ and 2 kΩ resistors in series. Find current and the node voltage across the 2 kΩ resistor.
- 1
Equivalent resistance is 1 kΩ + 2 kΩ = 3 kΩ.
- 2
Loop current is 9 V / 3000 Ω = 0.003 A = 3 mA.
- 3
Voltage across 2 kΩ is 3 mA × 2 kΩ = 6 V.
- 4
The 1 kΩ drop is 3 V; KVL gives +9 − 3 − 6 = 0 V.
The series current is 3 mA and the divider output across the 2 kΩ resistor is 6 V unloaded.
Sanity checkThe larger resistor receives the larger share of voltage, and the two drops sum exactly to the source voltage.
Try it yourself
Independent check
Move to the bench
Verify KCL and use node voltages to distinguish an open branch from a working parallel branch.
- 3×AA battery holder
- two 1 kΩ resistors
- breadboard
- multimeter
- jumper wires
Experiment
- With batteries removed, build two 1 kΩ branches in parallel and continuity-check the intended common nodes.
- Power the circuit and measure source voltage plus the voltage across each branch.
- Calculate each branch current from its actual resistance and voltage, then add them for predicted source current without moving the meter to current mode.
- Remove power and deliberately open one branch; repower and measure node voltages on both sides of the break to localize it.
- Restore the branch and confirm the original voltage pattern.
Observe
- Expected: both intact branches have approximately the source voltage; actual computed currents may differ because resistor values and source voltage differ.
- Expected fault clue: the intact branch remains powered while current in the open branch is zero; a voltage can appear across the open connection.
Draw a two-load diagram for a low-voltage screen and controller that must each receive 5 V, and explain why placing them in series is the wrong decision.
- Use only the battery holder; remove it before continuity or resistance tests.
- Do not short the supply nodes with a jumper or a meter in current mode.
Keep evidence
Bench note
Record what you expected, what you actually measured or observed, and one difference worth investigating. This is self-reported evidence, not a certification.