ELECTENG291 · Fundamentals of Electrical Engineering
Resistive Circuits: Ohm's & Kirchhoff's Laws
This chapter of Module 1 in University of Auckland ELECTENG 291 is the workhorse toolkit: Ohm's law (v = R i), Kirchhoff's voltage and current laws (KVL and KCL), combining resistors in series and parallel, and the voltage- and current-divider rules. Every later technique — nodal, mesh, Thévenin, transients, AC — is built on these four ideas, so they underpin the Module 1 online assignment, both in-person tests and the final exam.
What this chapter covers
- 01Ohm's law v = R i (equivalently i = v/R, R = v/i) under the passive sign convention
- 02KVL: the sum of voltages around any closed loop is zero
- 03KCL: the sum of currents into any node is zero
- 04Resistors in series add: R_eq = R1 + R2 + …
- 05Resistors in parallel combine by reciprocals: 1/R_eq = 1/R1 + 1/R2 + … (two resistors: R_eq = R1R2/(R1+R2))
- 06Voltage divider: V_out = [R2/(R1 + R2)]·V_s across R2 in a series pair
- 07Current divider: the current splits in inverse proportion to the branch resistances
Series solve with Ohm's law, KVL and the voltage divider
- +1Series equivalent resistance: R_eq = R1 + R2 = 2 kΩ + 4 kΩ = 6 kΩ.
- +1Loop current from KVL and Ohm's law: going round the loop, 12 − i·R1 − i·R2 = 0, so i = 12 / R_eq = 12 / 6 kΩ = 2 mA.
- +1Voltage across R2 by Ohm's law: V_2 = i·R2 = 2 mA × 4 kΩ = 8 V.
- +1Voltage-divider cross-check: V_2 = [R2/(R1 + R2)]·V_s = [4/(2 + 4)]·12 = (2/3)·12 = 8 V ✓. And V_1 = [2/6]·12 = 4 V, with V_1 + V_2 = 4 + 8 = 12 V, satisfying KVL.
Key terms
- Ohm's law
- The resistor terminal law v = R i (equivalently i = v/R), written under the passive sign convention. R is the resistance in ohms [Ω]; it converts a current into a proportional voltage drop.
- Kirchhoff's voltage law (KVL)
- The sum of the voltage rises and drops around any closed loop is zero — a statement of energy conservation. Assign a loop direction and be consistent with the passive sign convention as you go around.
- Kirchhoff's current law (KCL)
- The sum of currents entering any node (or closed region) is zero — a statement of charge conservation. It is the basis of node-voltage analysis.
- Series resistors
- Resistors carrying the same current; their resistances add: R_eq = R1 + R2 + …. A series string forms a voltage divider.
- Parallel resistors
- Resistors sharing the same voltage; their conductances add, so 1/R_eq = 1/R1 + 1/R2 + …. For two, R_eq = R1R2/(R1+R2), always smaller than the smallest resistor.
- Voltage / current divider
- Shortcut rules for series/parallel resistors: a series pair splits the source voltage in proportion to each resistance (V_out = [R2/(R1+R2)]·V_s); a parallel pair splits the current in inverse proportion to each resistance.
Resistive Circuits: Ohm's & Kirchhoff's Laws FAQ
What is the difference between KVL and KCL?
KVL is about loops: the voltages around any closed loop sum to zero (energy conservation). KCL is about nodes: the currents into any node sum to zero (charge conservation). You use KCL as the backbone of nodal analysis and KVL as the backbone of mesh analysis, but both, together with Ohm's law, appear in almost every solve.
When can I use the voltage-divider rule?
Only when the resistors form a true series pair carrying the same current — no other branch tapping current off between them. Then V_out = [R2/(R1+R2)]·V_s. If a load or another branch draws current at the midpoint, the divider assumption breaks and you should solve with KVL/KCL (or first find the Thévenin equivalent).
How do I combine resistors in series and parallel quickly?
Series resistances add directly. Parallel resistances add as reciprocals (1/R_eq = Σ 1/R_i); for exactly two, use the product-over-sum shortcut R1R2/(R1+R2). Reduce the network step by step, collapsing series and parallel groups until one equivalent resistor remains, then work back to find branch currents and voltages.
Can Sia help me solve resistive circuits in ELECTENG 291?
Yes, as a study aid. Sia can apply Ohm's law, KVL and KCL step by step, reduce a series-parallel network, and check your divider results and units. It explains the method and drills you on fresh numbers; it does not do graded assessment for you, and University of Auckland academic-integrity rules apply — confirm what is allowed on Canvas.
Exam move
Drill the four core moves until they are automatic: Ohm's law, KVL round a loop, KCL at a node, and series/parallel reduction. Always draw the circuit and mark the passive sign convention first, then write the law you are using before the numbers — examiners award method marks for a clearly stated KVL/KCL equation even if arithmetic slips. Practise keeping units tidy (V ÷ kΩ = mA) so magnitudes never come out orders of magnitude wrong. Use the voltage and current dividers as fast shortcuts, but check the series/parallel assumption each time. A reliable self-check is that your branch voltages must close KVL round every loop and your branch currents must satisfy KCL at every node. Confirm assessment details on Canvas.
Working through Resistive Circuits: Ohm's & Kirchhoff's Laws in ELECTENG 291? Sia is AskSia’s AI Electrical Engineering tutor — ask any ELECTENG 291 Resistive Circuits: Ohm's & Kirchhoff's Laws question and get a clear, step-by-step explanation grounded in how ELECTENG 291 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
