Kirchhoff's laws
Ohm's law describes individual elements; Kirchhoff's laws describe how currents and voltages combine across a whole circuit. Together they allow any circuit to be analysed.
Kirchhoff's current law
Gustav Kirchhoff formulated his laws in 1845. Kirchhoff's current law (KCL) states that the algebraic sum of currents entering a node is zero: what flows in must flow out, \(\sum i = 0\). It follows from conservation of charge, since charge cannot accumulate at an ideal node. If 3 A and 2 A enter a node and one other branch leaves, that branch carries 5 A out.
Kirchhoff's voltage law
Kirchhoff's voltage law (KVL) states that the algebraic sum of voltages around any closed loop is zero: \(\sum v = 0\). It follows from conservation of energy: a charge carried around a loop returns to its starting potential. Going around a loop, voltage rises across sources and drops across resistors must balance.
Sign conventions
To apply the laws consistently, choose a direction for each unknown current and mark each element's voltage polarity. For KCL, count currents entering a node as positive and leaving as negative, or the reverse, but be consistent. For KVL, traverse the loop in one direction, adding a voltage when moving from minus to plus and subtracting when moving from plus to minus. A negative answer just means the true direction is opposite to the guess.
Series elements
Elements are in series when they carry the same current, connected end to end with nothing branching between them. By KVL, the voltages across series elements add up to the total applied voltage. A string of old-style Christmas lights is a series circuit: if one bulb fails open, current stops everywhere.
Parallel elements
Elements are in parallel when they are connected between the same two nodes, so they share the same voltage. By KCL, the currents through parallel branches add up to the total current. Household outlets are wired in parallel, so each appliance receives the full supply voltage and can be switched independently.
Applying the laws together
A circuit with b branches and n nodes has n - 1 independent KCL equations and b - n + 1 independent KVL equations, enough, with Ohm's law for each element, to find every unknown. Systematic methods covered later, nodal and mesh analysis, organise these equations efficiently for larger circuits.
Limits of the laws
Kirchhoff's laws are exact for lumped circuits, those small compared with the wavelength of the signals involved, so that changes propagate effectively instantly. At very high frequencies, or over long transmission lines, circuits must be treated with electromagnetic field theory and transmission-line models, as later units discuss.
Worked example
A 12 V battery drives two resistors in series, \(R_1 = 2\) \(\Omega\) and \(R_2 = 4\) \(\Omega\). KVL around the loop: \(12 - 2i - 4i = 0\), so \(i = 2\) A. The voltages are \(v_1 = 4\) V and \(v_2 = 8\) V, summing to 12 V as KVL requires. Power: the battery supplies 24 W; the resistors absorb 8 W and 16 W, totalling 24 W, confirming conservation of energy.
Key takeaways
Kirchhoff's laws (1845) apply to every lumped circuit. KCL: currents into a node sum to zero, from conservation of charge. KVL: voltages around a loop sum to zero, from conservation of energy. Choose current directions and polarities consistently; negative results mean reversed directions. Series elements share current, and their voltages add; parallel elements share voltage, and their currents add. A circuit with n nodes and b branches gives n - 1 KCL and b - n + 1 KVL independent equations. The laws assume circuits small relative to signal wavelengths.
Flashcard
Currents into a node sum to zero
Check your understanding
Currents of 4 A and 1 A flow into a node, and 2 A flows out through one branch. What is the current in the only remaining branch?