Lesson DC Circuits · Advanced analysis and integration
Kirchhoff’s laws
When the circuit is not a clean series/parallel, or there is more than one source, Ohm alone is not enough. Kirchhoff: around any closed loop, algebraic sum of voltages is zero (KVL); at any node, current in equals current out (KCL). Write equations and solve.
Why you need Kirchhoff
So far: one source, clear topology, reduction by series/parallel. In the field and on advanced prints you see:
- two batteries or sources in different loops,
- bridges where no R is "obviously" series or parallel,
- currents splitting in non-obvious ways.
Kirchhoff gives accounting rules that always apply.
KVL — voltage law
The algebraic sum of voltage rises and drops around any closed path is zero.
In practice: pick a walk direction around a loop; add sources with one sign and resistor drops (\(I×R\)) with the other according to your polarity convention. Complete the loop → total is zero.
Informal series rule \(E_T = E_1 + E_2 + \ldots\) is KVL on one loop with a single source.
KCL — current law
The algebraic sum of currents entering and leaving a node is zero.
Informal: what goes in comes out. If 2 A and 1.5 A enter a junction and only one wire leaves, that wire carries 3.5 A (steady DC).
KCL is how you catch missing return paths and clamp meter mistakes.
Sign convention (pick one)
Typical loop walk:
- Crossing a voltage source from − to + → add \(+E\).
- Crossing a resistor in the direction of assumed current → add \(-IR\) (drop).
Other conventions work if consistent. Fatal error: mixing conventions mid-problem.
Problem-solving workflow
- Label branch or mesh currents (assume directions).
- Write KCL at key nodes.
- Write KVL for enough independent loops.
- Substitute \(V = IR\) with assumed directions.
- Solve simultaneous equations.
- Negative current → actual direction opposite; magnitude still valid.
Yes—simultaneous equations. That is the price of multiple unknowns.
Minimal KCL example
Node: \(I_1 = 2\,\mathrm{A}\) and \(I_2 = 1.5\,\mathrm{A}\) enter; one exit \(I_3\): \[ I_3 = 2 + 1.5 = 3.5\,\mathrm{A} \] Clamp reads 2 A leaving? A path is missing or the clamp is on the wrong conductor.
Minimal KVL example
Loop: 24 V source, drops 10 V + 14 V across two resistors at same current. Sum accounts for 24 V. If you measure 12 V on the second with unchanged current, hidden drop (wire) or bad measurement.
Relation to earlier tools
| Situation | Enough tool |
|---|---|
| Single source, clear series/parallel | Ohm + reduction |
| Multiple sources / bridge | Kirchhoff (or Thévenin/Norton/superposition) |
| Verify field measurements | KCL/KVL as sanity check |
In diagnosis, KCL/KVL are lie detectors: 10 A in, 6 A out on two wires—find the third path.
Field case
Situation. A test bench has two DC supplies in loops that share a load resistor. A tech calculates load current with Ohm as if only one battery existed—and is wrong by nearly 2×.
How to think with this lesson.
- Two sources → coupled loops.
- Write KVL for each mesh (or use superposition later).
- Shared resistor current is a combination of mesh currents, not one supply ÷ R.
Conclusion: more than one source → stop averaging by eye; use loop/node laws.
In the field
Symptom
Currents or voltages impossible for one simple loop
Where to look
Second sources, common returns, bridge connections
Likely causes
- Ignored a source
- wrong assumed direction
- shared neutrals
What to measure
- I in every wire at a node
- V around a closed walk
What not to do
- Force the circuit into one series string on paper
Checklist
- I state KVL and KCL in my own words
- I label currents and polarities before equations
- I keep one sign convention throughout
- I interpret negative currents as reversed direction
- I use KCL/KVL to verify measurements
- I know when Ohm + reduction is no longer enough