Norton's Theorem Calculator
Norton equivalent of a linear circuit: a current source IN = VTH / RTH in parallel with a resistor RN = RTH.
Inputs
Enter the Thévenin voltage and resistance of the circuit seen from the terminals. Add a load resistance to get the load current. Drag the handle (or focus it and press ↑/↓) to reorder. Scientific notation such as 4.7e3 is accepted, and any unit converts automatically.
Results
Enter values and press Calculate. Your answer will appear here.
Load power vs. load resistance
The power delivered to the load rises, peaks and then falls as RL grows. The peak is at RL = RN (maximum power transfer). The dark dot is your load.
What is being calculated
Reference values
| Thévenin VTH | RTH | Norton IN = VTH / RTH | RN |
|---|
Source transformation: the resistance stays the same, a voltage source in series becomes a current source in parallel.
Load current for a 1 A, 10 Ω Norton source
| Load RL | IL | VL | PL |
|---|
The load power is largest when RL equals RN (10 Ω here).
Thévenin vs. Norton
| Property | Thévenin | Norton |
|---|---|---|
| Source | Voltage source VTH in series with RTH | Current source IN in parallel with RN |
| Open-circuit voltage | VTH | IN · RN |
| Short-circuit current | VTH / RTH | IN |
| Resistance | RTH | RN = RTH |
| Load current | VTH / (RTH + RL) | IN · RN / (RN + RL) |
Unit conversions
| Quantity | Conversion |
|---|---|
| Current | 1 kA = 1,000 A · 1 A = 1,000 mA · 1 mA = 1,000 µA · 1 µA = 1,000 nA |
| Voltage | 1 kV = 1,000 V · 1 V = 1,000 mV · 1 mV = 1,000 µV |
| Resistance | 1 kΩ = 1,000 Ω · 1 MΩ = 10⁶ Ω · 1 GΩ = 10⁹ Ω · 1 Ω = 1,000 mΩ |
| Power | 1 kW = 1,000 W · 1 W = 1,000 mW · P = I²R = V·I |
Formula
- IN: Norton current, equal to the short-circuit current at the terminals (A)
- VTH: Thévenin (open-circuit) voltage at the terminals (V)
- RTH: Thévenin resistance seen from the terminals (Ω), and RN = RTH
- IL = IN · RN / (RN + RL): current in a load RL (current divider)
Example: VTH = 12 V and RTH = 4 Ω give IN = 12 ÷ 4 = 3 A with RN = 4 Ω. A load of 8 Ω then draws IL = 3 × 4 ÷ (4 + 8) = 1 A and has VL = 8 V.
Rearranged: VTH = IN · RN and RTH = VTH / IN. To find the equivalent of a circuit, short the terminals to get IN, and find RN as the resistance seen from the terminals with independent voltage sources shorted and independent current sources opened. The load voltage is VL = IL · RL = IN · (RN ∥ RL), and the load power is PL = IL² · RL. It peaks at RL = RN with Pmax = IN² · RN / 4. The equivalent matches the original circuit only at its terminals; the power lost inside RN is not the power lost inside the original circuit. The theorem holds for linear circuits. A negative value means the polarity or direction is reversed.
References
- Wikipedia, Norton's theorem: statement, proof outline and conversion to Thévenin
- Wikipedia, Thévenin's theorem: the dual voltage-source equivalent
- All About Circuits, Norton's Theorem: worked examples
- Wikipedia, Maximum power transfer theorem: why power peaks at RL = RN
How to Use This Tool
- Pick a preset, or enter the Thévenin voltage VTH and resistance RTH with their units.
- Optionally enter a load resistance RL to see the load current and voltage.
- In Advanced, solve for VTH from IN and RN, or for the resistance from VTH and IN, and see power and the Thévenin-versus-Norton comparison.
- Click Calculate.
- Read IN in the highlighted result, then check the table and the chart.
What This Tool Offers
- Basic and Advanced modes on one calculation engine
- Voltage, current and resistance units from micro to kilo with automatic conversion
- Norton current and resistance, open-circuit voltage and short-circuit current
- Load current, voltage and power, with maximum power transfer (Advanced)
- Load-power chart and a labelled Norton equivalent circuit
- Reorderable inputs, presets and a mobile-friendly layout
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| I_N | Norton equivalent current | A |
| I_SC | Short-circuit current | A |
What it means
Norton’s theorem states that any linear two‑terminal circuit can be replaced by an equivalent current source (I_N) in parallel with a resistance (R_N). The Norton current I_N is the short‑circuit current that would flow between the two terminals when they are shorted. The Norton resistance R_N is the equivalent resistance seen looking back into the terminals with all independent sources deactivated (voltage sources shorted, current sources opened). This theorem is the dual of Thévenin’s theorem and is especially useful for parallel‑connected loads. It simplifies the analysis of circuits where the load is variable, as the equivalent circuit remains fixed. Norton’s theorem is used in power systems for fault analysis, in amplifier design for output impedance calculations, and in sensor interfacing. The relationship between Norton and Thévenin equivalents is V_TH = I_N * R_N and R_TH = R_N. Example: If the short‑circuit current measured at the terminals is 2A, then I_N = 2A. The Norton equivalent is a 2A current source in parallel with R_N.
Worked example
Norton's Theorem – Practical Example
Real‑World| Parameter | Value |
|---|---|
| Short‑circuit current (ISC) | 2 A |
| Formula | IN = ISC |
Common mistakes
- Norton current I_N: The short‑circuit current at the terminals – not the open‑circuit voltage.
- Resistance R_N: The equivalent resistance seen from the terminals with all independent sources turned off (voltage sources shorted, current sources open).
- Dependent sources: Keep them active when computing R_N – use a test source method.
- Polarity: The current source direction is defined by the short‑circuit current direction.
- Source transformation: Norton and Thevenin equivalents are interchangeable via V_TH = I_N·R_N.
Applications
Norton's theorem states that any linear two‑terminal circuit can be replaced by a current source (I_N) in parallel with a resistance (R_N). I_N is the short‑circuit current, and R_N is the equivalent resistance seen from the terminals. This theorem simplifies circuit analysis, especially when dealing with variable loads. Engineers use it to simplify power source models, to analyse parallel circuits, and to design maximum power transfer. By converting a network to its Norton equivalent, they can quickly determine load current and voltage. Norton's theorem is complementary to Thévenin's theorem and is widely used in circuit simulation and teaching. Understanding Norton's theorem is essential for efficient circuit analysis and design.
- Simplification of complex networks for analysis
- Source modelling for current sources and sensors
- Maximum power transfer and load matching
- Circuit simulation and symbolic analysis
- Educational foundation of network theorems
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