Circuit simulator
Every circuit is a conversation between voltage and resistance. A battery pushes charge through a loop; resistors slow it down; switches open and close the conversation. Place components on the grid, wire them together, and watch animated electrons trace the current that Ohm’s law predicts and Kirchhoff’s laws constrain. Adjust values in real time and see how the physics responds.
Component
The fundamental relationship
Ohm’s law is the bedrock of circuit analysis: V = IR. Voltage (V, measured in volts) equals current (I, in amperes) multiplied by resistance (R, in ohms). It tells you that if you push harder (more voltage), more current flows; if you add resistance, current decreases. Georg Simon Ohm published this relationship in 1827, and it was initially met with skepticism — the scientific establishment considered it too simple to be useful.
The law can be rearranged three ways: V = IR, I = V/R, R = V/I. Each form answers a different question. Know any two quantities and you can find the third. This works for any ohmic material — one where resistance is constant regardless of current. Metals at constant temperature are ohmic; semiconductors, diodes, and LEDs are not.
Conservation of charge at every junction
Gustav Kirchhoff stated in 1845 that the total current entering any node (junction) in a circuit equals the total current leaving it: ΣIin = ΣIout. This is conservation of electric charge — charge cannot accumulate at a point or vanish from one. If 3 amps flow into a junction and the wire splits in two, the currents in the two branches must sum to 3 amps.
KCL is the basis of nodal analysis, the method this simulator uses to solve circuits. At every node, we write an equation saying the sum of all currents is zero (treating outgoing current as negative). This produces a system of linear equations that, when solved simultaneously, gives the voltage at every node. From the voltages, Ohm’s law gives the current through every component.
Energy conservation around every loop
Kirchhoff’s voltage law says that the sum of all voltage changes around any closed loop in a circuit is zero: ΣV = 0. A battery lifts the voltage; each resistor drops it. If you trace a complete loop back to where you started, you must end at the same voltage you began with — like walking around a hill and returning to the same elevation.
In a simple series circuit with a 9V battery and three resistors, the voltage drops across the resistors must sum to 9V. If the resistors are equal, each drops 3V. If they’re unequal, the drops are proportional to the resistances — this is the principle behind a voltage divider.
Series: resistances add
When resistors are connected end-to-end so that the same current flows through each one, they are in series. The total resistance is the sum: Rtotal = R1 + R2 + R3 + …. The current is the same everywhere, but the voltage divides among the resistors in proportion to their resistance. Two 100Ω resistors in series act like a single 200Ω resistor.
Parallel: conductances add
When resistors are connected between the same two nodes so that the voltage across each is identical, they are in parallel. The total resistance follows: 1/Rtotal = 1/R1 + 1/R2 + …. The voltage is the same across each, but the current divides — more current flows through the smaller resistance. Two 100Ω resistors in parallel give 50Ω. Adding a resistor in parallel always decreases total resistance, because you are giving current an additional path.
Three equivalent formulas
Power is the rate at which electrical energy is converted to another form (heat, light, motion). The basic formula is P = IV — power equals current times voltage. Using Ohm’s law to substitute, you get two more forms: P = I²R and P = V²/R.
The I²R form is especially important: it tells you that power dissipation grows with the square of the current. Double the current and you quadruple the heat. This is why high-power transmission lines use very high voltages — for a given amount of power delivered (P = IV), higher voltage means lower current, and lower current means drastically less energy wasted as heat in the wires (Ploss = I²Rwire).
Capacitors store charge
A capacitor stores energy in an electric field between two conductive plates separated by an insulator. Its capacitance C (measured in farads) relates the stored charge to voltage: Q = CV. When connected to a battery through a resistor, the capacitor charges gradually — the voltage across it rises as charge accumulates on the plates.
The time constant
The rate of charging is governed by the time constant τ = RC. After one time constant, the capacitor reaches about 63% of the battery voltage. After five time constants (5τ), it is essentially fully charged (over 99%). The same time constant governs discharging: a charged capacitor discharging through a resistor loses 63% of its voltage in one τ.
RC circuits are everywhere: the flash on a camera uses a capacitor charged through a resistor; the timing in early computers used RC circuits; every touchscreen relies on measuring capacitance changes. The time constant τ = RC has units of seconds — 1 ohm times 1 farad equals 1 second.