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Kirchhoff's Laws · Loop Voltage and Node Current

DC circuit (dual mesh) | Adjust any resistor value and observe that KVL and KCL always hold
\(U_1 = 12\text{V}\) (fixed)  |  Three loops · Four resistors · Two mesh currents

Circuit Parameter Settings

100 Ω
10 Ω1000 Ω
200 Ω
10 Ω1000 Ω
300 Ω
10 Ω1000 Ω
100 Ω
10 Ω1000 Ω
12.00 V

Fixed voltage source in the left-hand branch

Circuit Structure Diagram

Dual Mesh DC Circuit
Dual mesh DC circuit topology
Mesh currents: \(I_1\) (left, blue)   \(I_2\) (right, orange)
Branch currents: \(I_{R1}=I_1,\; I_{R4}=I_1,\; I_{R2}=I_1-I_2,\; I_{R3}=I_2\)

Branch Voltage Distribution · Bar Chart

Left Mesh: \(U_1 - U_{R1} - U_{R2} - U_{R4} = 0.000\text{ V}\)
Right Mesh: \(U_{R2} - U_{R3} = 0.000\text{ V}\)
Voltage drops across the four resistors  |  Voltage across each resistor \(U_{R1} \sim U_{R4}\)

Current Calculation Results

Mesh Current \(I_1\) (blue) — mA
Mesh Current \(I_2\) (orange) — mA
\(I_{R1}\) — mA
\(I_{R2} = I_1 - I_2\) — mA
\(I_{R3}\) — mA
\(I_{R4}\) — mA

KVL & KCL Verification

Loop 1 (left): \(U_1 - U_{R1} - U_{R2} - U_{R4}\) = 0.000 V
Loop 2 (right): \(U_{R2} - U_{R3}\) = 0.000 V
Loop 3 (outer): \(U_1 - U_{R1} - U_{R3} - U_{R4}\) = 0.000 V
KCL (Node B): \(I_{R1} = I_{R2} + I_{R3}\)
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Kirchhoff's Voltage Law (KVL)

\(\sum V = 0\) Around any closed loop, the algebraic sum of voltage rises equals the algebraic sum of voltage drops. Regardless of how component parameters change, KVL always holds strictly. This simulation demonstrates KVL verification for 3 loops: left loop (\(U_1\)-\(R_1\)-\(R_2\)-\(R_4\)), right loop (\(R_2\)-\(R_3\)), and outer loop (\(U_1\)-\(R_1\)-\(R_3\)-\(R_4\)).

Kirchhoff's Current Law (KCL)

\(\sum I = 0\) At any node, the total current flowing into the node equals the total current flowing out. In this circuit, at the top middle node B: \(I_{R1}\) (inflow) = \(I_{R2} + I_{R3}\) (outflow). KCL always holds exactly. Adjusting any resistor value verifies this relationship.