Kirchhoff Laws Simulator — Series-Parallel Resistor Circuit

Kirchhoff's laws supply two conservation constraints — loop voltage (KVL) and node current (KCL) — that close any linear network of resistors, voltage sources and current sources. The simulator handles the simplest non-trivial case: a single loop with one node, a source $V$ driving a series resistor $R_1$ followed by the parallel combination $R_2$ // $R_3$.

The equivalent of the parallel section is $R_{par}=R_2 R_3/(R_2+R_3)$ and the total resistance is $R_{total}=R_1+R_{par}$. KVL around the loop reads $V = I_{total} R_1 + V_{par}$, immediately giving $I_{total}=V/R_{total}$, and the parallel-section voltage is $V_{par}=V-I_{total} R_1$. The branch currents follow from Ohm's law, $I_2=V_{par}/R_2$ and $I_3=V_{par}/R_3$, and the KCL identity $I_{total}=I_2+I_3$ is satisfied automatically at node a.

With the defaults $V=12$ V, $R_1=100$ Ω, $R_2=200$ Ω and $R_3=300$ Ω, the equations yield $R_{par}=120$ Ω, $R_{total}=220$ Ω, $I_{total}=12/220\approx 54.5$ mA, $V_{par}=6.55$ V, $I_2\approx 32.7$ mA and $I_3\approx 21.8$ mA, so that $I_2+I_3=54.5$ mA $=I_{total}$. Drag any of the four sliders to see all of these quantities update live.

Ammeter shunts: The classic way to extend an ammeter's range is to wire a small shunt resistor $R_s$ in parallel with the meter movement $R_m$. Most of the current flows through $R_s$, while only $I_m=I\cdot R_s/(R_s+R_m)$ reaches the movement. Power meters, clamp meters and automotive battery monitors still use this design today.

Parallel LED drive imbalance: Wiring two or more LEDs directly in parallel is a textbook mistake. Small differences in forward voltage $V_f$ cause the LED with the lowest $V_f$ to hog the current and overheat. The cure is to add a small ballast resistor in series with each LED so that the voltage drop absorbs the $V_f$ tolerance — a direct application of KVL and KCL.

Power distribution with parallel loads: Residential AC panels and data-center DC busbars feed many parallel loads, each pulling its share according to KCL. When the cable resistance $R_w$ is non-negligible — long EV charging cables, sensor power runs, busbars — each branch sees a voltage drop and conductor cross-sections must be sized accordingly. KVL is the bookkeeping that ensures the end-of-line voltage still meets spec.

Wheatstone bridges: A bridge of four resistors with a galvanometer across the middle is the most elegant application of Kirchhoff's laws. The null condition $R_1/R_2=R_3/R_4$ is detected as zero current through the galvanometer, enabling precise unknown-resistance measurements and the differential read-out used in strain gauges and temperature sensors.

The most frequent mistake is the assumption that parallel branches share current equally. The shared quantity is voltage, not current; the currents split inversely to resistance, so equal currents only occur when the resistors are equal. With the defaults R_2 = 200 and R_3 = 300, the ratio is 3:2 in favor of R_2. The bar chart in the simulator makes this immediately visible.

Another classic error is adding parallel resistances as if they were in series. Series resistances add directly, but parallel resistances combine as a sum of reciprocals, $1/R_{par}=1/R_2+1/R_3$, and the result is always smaller than the smallest branch. With R_2 = 200 and R_3 = 300 the equivalent is 120 Ω, which is the safety net to check whether your design calculation is consistent.

Finally, ignoring wiring resistance when writing KVL produces real-world surprises. Lead wires, connectors and PCB traces all carry a small but non-zero resistance, so the "12 V at the source" can become "11 V at the load". EV battery wiring, long sensor runs and DC distribution panels must always include wiring as series elements in the KVL/KCL model to make end-of-line voltages predictable.