2. The Two Conservation Laws: Kirchhoff's Voltage and Current Laws

These two laws are the foundations of circuit analysis. Everything else (Thevenin, Norton, mesh, nodal) is just systematic ways to apply KVL and KCL.

2.1 KCL: current is conserved at every node

Kirchhoff's Current Law (KCL): the algebraic sum of currents entering any node is zero. Equivalently: the current flowing in equals the current flowing out.

Why? Because charge is conserved. Charges cannot accumulate at a single point in a wire (well, except very briefly through a capacitor, but in steady state, no). So every electron that flows into a node must flow out somewhere.

Plumbing analogy. Water flowing into a pipe junction must flow out the other branches. Otherwise the junction would be filling up, and a wire is not a tank. KCL is just "water that flows in must flow out."

Practical use: pick any node in a circuit. Sum the currents flowing in (give them positive sign) and the currents flowing out (negative sign). Set the sum to zero. That is one equation. Do this at every node and you get $n-1$ independent equations (one is always redundant; it is the sum of the others).

2.2 KVL: voltage drops sum to zero around any loop

Kirchhoff's Voltage Law (KVL): the algebraic sum of voltage drops around any closed loop is zero.

Why? Because voltage is potential energy per unit charge, and energy is conserved. If you take a charge around a closed path and back to where you started, its potential energy is back where it started, total change zero.

Hiking analogy. Hike a closed trail starting and ending at your car. Add up all the elevation changes along the way (positive when going up, negative when going down). The total must be zero. You ended where you started. Same for voltage around any loop.

Practical use: pick any closed loop in the circuit. Walk around it in one direction. For each element you cross, add (or subtract) the voltage drop, with sign chosen by your direction of travel. Set the sum to zero. One equation per loop.

2.3 Worked example: a simple resistor network

Three resistors in a configuration with one voltage source. Pick the values: $V_s = 10$ V, $R_1 = 1$ kΩ, $R_2 = 2$ kΩ, $R_3 = 3$ kΩ. Wire them with $R_2$ and $R_3$ in parallel, then the parallel combination in series with $R_1$ across the source.

Step 1: combine $R_2 \| R_3 = (2 \times 3)/(2+3) = 1.2$ kΩ. Step 2: total resistance $R_{total} = R_1 + 1.2 = 2.2$ kΩ. Step 3: total current from source: $I = V_s/R_{total} = 10/2200 = 4.5$ mA. Step 4: voltage across the parallel combo: $V_{||} = I \cdot 1200 = 5.45$ V. Step 5: current through each parallel branch: $I_2 = V_{||}/R_2 = 2.73$ mA, $I_3 = 1.82$ mA. (Check: $I_2 + I_3 = 4.55$ mA, matches the 4.5 mA from before to within rounding.)

KCL is what lets us write step 5 (the input current to the parallel combo equals the sum of the branch currents). KVL is what lets us write step 4 (the voltage across $R_1$ plus the voltage across the parallel combo equals the source voltage). The whole analysis is two laws applied repeatedly.

For a circuit this simple you can do it by hand. For anything more complex, we use a systematic method, and that is what the next two sections introduce.

2.4 KCL and KVL at higher frequencies

A subtle warning. KCL holds when every wire is short compared to the wavelength of the signals it carries. At microwave frequencies (and inside multi-gigahertz digital signals) what looks like a single node really is not a single equipotential. Chapter 9 on transmission lines treats the wire itself as a distributed network. For this chapter, treat KCL and KVL as ironclad, but remember the lumped approximation eventually breaks down. An attacker analyzing a chip's RF emanations is looking precisely at the part of physics that lumped models miss.