Learn the foundations of DC circuit analysis — Ohm's law, series and parallel resistance, Kirchhoff's current and voltage laws, and the node-voltage (nodal analysis) method with a fully worked example you can verify in a circuit simulator.
Why Circuit Analysis Matters
Circuit analysis is the skill of predicting the voltage at every node and the current through every branch of a network before you build it. Master a handful of laws — Ohm's law, the series/parallel rules, and Kirchhoff's two laws — and you can solve almost any linear resistive circuit. Nodal analysis then packages those laws into a systematic procedure that scales to complex networks. As you read, build each circuit in the Electronic Circuit Simulator and confirm the math against the meter readings — nothing cements these ideas faster.
Ohm's Law: The Foundation
Ohm's law relates the three core quantities in a resistor:
V = I x R
Voltage (volts) equals current (amperes) times resistance (ohms). Rearranged, I = V / R and R = V / I. Power follows directly: P = V x I = I²R = V²/R. If you push 2 A through a 10 Ω resistor, it drops 20 V and dissipates 40 W. Ohm's law is the single most-used relationship in all of electronics — every other technique reduces to applying it correctly at the right place.
Series and Parallel Resistance
Before reaching for heavy machinery, simplify. Two reduction rules handle most ladder networks:
- Series: resistors in series carry the same current and their resistances add. R(total) = R1 + R2 + R3. The voltage divides in proportion to each resistance (the voltage-divider rule).
- Parallel: resistors in parallel share the same voltage and their conductances add. 1/R(total) = 1/R1 + 1/R2 + 1/R3. For just two resistors, R(total) = (R1 x R2) / (R1 + R2). The current divides in inverse proportion to resistance.
Collapsing series and parallel groups often reduces a daunting circuit to a single equivalent resistance, after which Ohm's law gives the source current immediately. When the network is non-trivial (bridges, multiple sources), you graduate to Kirchhoff's laws.
Kirchhoff's Current Law (KCL)
KCL states that the algebraic sum of currents entering a node equals the sum leaving it — charge is conserved, so what flows in must flow out. Sum of currents in = Sum of currents out. At a node where 3 A and 2 A enter and one branch leaves, that branch carries 5 A. KCL is the basis of nodal analysis.
Kirchhoff's Voltage Law (KVL)
KVL states that the algebraic sum of voltages around any closed loop is zero — energy is conserved, so the rises (sources) and drops (resistors) around a loop must cancel. Sum of voltage rises = Sum of voltage drops. Walk a loop, add each EMF and subtract each I x R drop, set the total to zero, and you have an equation. KVL is the basis of mesh (loop) analysis.
The Node-Voltage Method (Nodal Analysis)
Nodal analysis is a systematic way to apply KCL that turns any resistive circuit into a small set of linear equations. The procedure:
- 1. Pick a reference node (ground), usually the one with the most connections, and call its voltage 0 V.
- 2. Label the remaining node voltages V1, V2, … relative to ground.
- 3. Write a KCL equation at each non-reference node, expressing every branch current as a node-voltage difference divided by its resistance: current from node A to node B is (Va − Vb) / R.
- 4. Solve the simultaneous equations for the node voltages.
- 5. Back-substitute with Ohm's law to get any branch current you need.
A Worked Example
Take a single unknown node V1 fed by a 10 V source through a 2 Ω resistor, with V1 also connected to ground through a 4 Ω resistor and through a 4 Ω resistor (two 4 Ω paths to ground). Writing KCL at V1 — current in from the source equals current out to ground:
(10 − V1) / 2 = V1 / 4 + V1 / 4
Multiply through by 4: 2(10 − V1) = V1 + V1, so 20 − 2V1 = 2V1, giving 20 = 4V1 and V1 = 5 V. Now back-substitute: the current from the source is (10 − 5) / 2 = 2.5 A, and each 4 Ω branch carries 5 / 4 = 1.25 A. KCL checks: 1.25 + 1.25 = 2.5 A in equals 2.5 A out. The node-voltage method delivered every branch current from one equation.
Verify It in the Simulator
Hand calculations are only trustworthy once you confirm them. Rebuild that exact circuit in the Electronic Circuit Simulator — place the 10 V source, the 2 Ω feed resistor, and the two 4 Ω resistors to ground — and read the node voltage and branch currents. You should see V1 settle at 5 V and 2.5 A from the source, matching the nodal solution exactly. The simulator is the fastest feedback loop for catching a flipped sign or a misplaced resistor, and it lets you sweep values to see how the node voltage responds. For longer conductor runs where the wire resistance itself causes a meaningful drop, pair the analysis with the Voltage Drop Calculator to confirm the delivered voltage at the load.
Choosing a Method
Reach for series/parallel reduction first — it is fastest when the topology allows it. Use nodal analysis when you have several nodes and want voltages (it shines with many current sources and grounded networks); use mesh analysis when loops and voltage sources dominate. All three rest on the same two conservation laws, and all three should agree. When they do — and when the Electronic Circuit Simulator confirms it — you can trust the result.