Every complex linear circuit can be reduced to a single voltage source in series with a single resistor. This is the foundation of the Thevenin equivalent formula, a critical tool for predicting how a power supply, battery, or signal source will behave when you actually connect a load to it. Instead of analyzing dozens of nodes, you extract two parameters—Thevenin voltage and Thevenin resistance—and use them to calculate exact load voltages and currents.

The Core Thevenin Equivalent Formula and Symbol Definitions

The Thevenin equivalent model relies on three primary equations. The first defines the internal resistance of the source network based on its open-circuit and short-circuit limits. The second and third predict the voltage and current delivered to an arbitrary load.

1. Internal Resistance Definition:
RTh = VOC / ISC

2. Load Voltage Prediction (Voltage Divider):
VL = VTh × [ RL / (RTh + RL) ]

3. Load Current Prediction (Ohm's Law):
IL = VTh / (RTh + RL)

Symbol Definition Table

Symbol Parameter Unit Definition
VTh Thevenin Voltage Volts (V) The open-circuit voltage at the output terminals (identical to VOC).
RTh Thevenin Resistance Ohms (Ω) The equivalent internal resistance looking back into the source network with independent sources zeroed.
VOC Open-Circuit Voltage Volts (V) Voltage measured across terminals when no load is connected (infinite resistance).
ISC Short-Circuit Current Amperes (A) Current flowing when output terminals are shorted together (zero resistance).
VL Load Voltage Volts (V) The actual voltage measured across the load resistor when connected.
RL Load Resistance Ohms (Ω) The resistance of the external component or circuit being powered.
IL Load Current Amperes (A) The actual current flowing through the load resistor.

Real-World Thevenin Parameters by Source Type

Theory is useless without context. The magnitude of RTh varies wildly depending on whether you are designing a power delivery network or an audio signal chain. Below is a data-dense reference table of real-world sources measured on the bench.

Source Device Typical VTh (Nominal) Typical RTh (Internal) Max Safe ISC Primary Application
Duracell AA (MN1500) Alkaline 1.50 V 0.15 Ω (150 mΩ) ~10 A (pulsed) Low-drain consumer electronics
Samsung 25R 18650 Li-ion 3.60 V 0.013 Ω (13 mΩ) ~277 A (theoretical) High-drain battery packs, vapes, EVs
Rigol DP832 Bench Supply (5V Rail) 5.00 V (set) 0.005 Ω (5 mΩ) Limited to 3 A by OCP Prototyping, bench testing
Shure SM58 Dynamic Microphone ~2 mV (acoustic signal) 300 Ω N/A (signal level) Audio signal transmission
Honeywell HIH-4000 Humidity Sensor 0.8 V to 3.9 V ~8,000 Ω (8 kΩ) < 1 mA Analog sensor interfacing to ADC

Rearranged Forms and Algebraic Manipulation

On the workbench, you rarely know all the variables upfront. Often, you need to reverse-engineer a battery's internal resistance or determine what load will pull a specific voltage down to a brownout threshold. By rearranging the load voltage formula VL = VTh × [ RL / (RTh + RL) ], you can solve for any missing variable.

  • Solve for Thevenin Voltage (VTh):
    VTh = VL × [ (RTh + RL) / RL ]
    Use case: You measure a loaded voltage and know the internal resistance, but your DMM couldn't capture the exact open-circuit voltage due to surface charge settling.
  • Solve for Thevenin Resistance (RTh):
    RTh = RL × [ (VTh / VL) - 1 ]
    Use case: The standard bench method for finding a battery's internal resistance without a specialized milliohm meter. Measure VTh, apply a known RL, measure VL, and calculate.
  • Solve for Load Resistance (RL):
    RL = (RTh × VL) / (VTh - VL)
    Use case: Sizing a pull-up or load resistor to ensure a sensor's output voltage doesn't sag below the logic-high threshold of a microcontroller GPIO.

Worked Examples with Unit Tracking

Let's apply these formulas to two common bench scenarios. Tracking units through every step prevents the scaling errors that frequently ruin prototype designs.

Problem 1: Sizing a Load for a Solar Panel

Scenario: You have a small 12V nominal solar panel. Your DMM reads an open-circuit voltage (VOC) of 21.5 V. You briefly short the panel through a clamp meter and read a short-circuit current (ISC) of 5.8 A. You want to connect a 4.0 Ω power resistor as a dummy load. What will the load voltage (VL) and load current (IL) be?

Step 1: Calculate RTh
RTh = VOC / ISC
RTh = 21.5 V / 5.8 A
RTh = 3.707 Ω

Step 2: Calculate Load Current (IL)
IL = VTh / (RTh + RL)
IL = 21.5 V / (3.707 Ω + 4.0 Ω)
IL = 21.5 V / 7.707 Ω
IL = 2.79 A

Step 3: Calculate Load Voltage (VL)
Using Ohm's law across the load resistor:
VL = IL × RL
VL = 2.79 A × 4.0 Ω
VL = 11.16 V
Result: The panel sags from 21.5 V down to 11.16 V under this specific load, delivering 2.79 A.

Problem 2: Finding the Internal Resistance of a LiFePO4 Pack

Scenario: A 4S LiFePO4 battery pack has a fully charged open-circuit voltage (VTh) of 14.4 V. You connect a motor controller that draws a steady 15.0 A. Under this load, your multimeter reads a terminal voltage (VL) of 13.8 V. What is the Thevenin resistance (RTh) of the pack, including the wiring?

Step 1: Identify knowns and select the rearranged formula
VTh = 14.4 V
VL = 13.8 V
IL = 15.0 A
We need RTh. While we could calculate RL first (13.8 V / 15.0 A = 0.92 Ω) and use the standard rearranged formula, it is faster to use the voltage drop method derived from Kirchhoff's Voltage Law: Vdrop = IL × RTh.

Step 2: Calculate the voltage dropped across RTh
Vdrop = VTh - VL
Vdrop = 14.4 V - 13.8 V = 0.6 V

Step 3: Solve for RTh
RTh = Vdrop / IL
RTh = 0.6 V / 15.0 A
RTh = 0.04 Ω (or 40 mΩ)
Result: The pack and its interconnecting wires have a total Thevenin resistance of 40 milliohms. If this exceeds the manufacturer's spec, you likely have a high-resistance crimp or a failing cell.

Assumptions, Applicability, and Common Unit Traps

The Thevenin equivalent formula is a powerful simplification, but it is bound by strict physical and mathematical assumptions. Misapplying it or mishandling units will yield designs that fail on the bench.

When the Formula Applies (and When it Doesn't)

Thevenin's theorem strictly applies only to linear, bilateral, time-invariant networks.

  • Linear: The components inside the "black box" you are simplifying must obey Ohm's law. Resistors, standard capacitors, and inductors are fine. If your source network contains diodes, transistors operating in their active region, or saturated inductors, the equivalent resistance changes with the load current. You cannot use a single static RTh value for non-linear source networks.
  • The Load Can Be Non-Linear: A common misconception is that the entire circuit must be linear. Thevenin simplifies the source. The load (RL) can absolutely be a non-linear device like an LED, a motor, or a microcontroller entering sleep mode. You simply use the Thevenin equivalent to plot the source's I-V curve and intersect it with the load's I-V curve to find the operating point.
  • Frequency Dependence: In AC circuits or fast-switching DC transients, RTh becomes a complex impedance (ZTh). At high frequencies, parasitic capacitance and inductance mean your "resistor" is actually a frequency-dependent network.

Unit Mistakes That Break the Math

The most frequent error in Thevenin calculations occurs when mixing base SI units with scaled prefixes.

  • The Milliamp Trap: If you measure VOC in Volts (e.g., 5V) and ISC in milliamps (e.g., 20 mA), dividing 5 by 20 yields 0.25. If you blindly label this as Ohms, you are wrong by a factor of 1,000. The actual RTh is 250 Ω. Rule: Always convert mA to A (0.020 A) before dividing.
  • The Milliohm Blindspot: When calculating RTh for lithium cells or bench supplies, the result is often 0.015 Ω. When plugging this into the load voltage formula alongside a 4.0 Ω load, failing to carry the decimal places accurately will make RTh look mathematically insignificant, leading you to ignore voltage sag that will actually cause a brownout reset in a high-current pulse scenario.

Realistic Answer Magnitudes

Developing an intuition for realistic RTh magnitudes acts as a sanity check for your calculations. If your calculated Thevenin resistance falls outside these typical ranges, re-check your measurements and math:

  • Grid-Tied Power & Bench Supplies: 1 mΩ to 50 mΩ. (Stiff sources, minimal sag).
  • Battery Cells (Li-ion, LiPo, Lead-Acid): 5 mΩ to 200 mΩ depending on chemistry, age, and temperature.
  • Audio & Signal Sources: 50 Ω (RF coax), 150 Ω to 600 Ω (pro audio microphones).
  • Analog Sensors & Transducers: 1 kΩ to 100 kΩ. (Requires high-impedance buffer op-amps to prevent loading effects).

For deeper theoretical proofs and network reduction techniques, refer to the foundational circuit analysis materials provided by All About Circuits and the DC circuit tutorials on Electronics Tutorials. Mastering the Thevenin equivalent formula transforms you from a builder who guesses at power supply sizing into an engineer who mathematically guarantees system stability under load.