Thevenin Formula: Derivation, Rearranged Forms, and Worked Circuit Examples

Thevenin's theorem is the ultimate circuit simplification tool. It states that any linear electrical network containing only voltage sources, current sources, and resistances can be replaced by an equivalent circuit consisting of a single voltage source in series with a single resistor. For bench engineers and hobbyists, the Thevenin formula transforms a tangled web of nodes into a simple two-terminal model, making it trivial to predict how a circuit will behave when you attach a new load.

The Core Thevenin Formula and Symbol Definitions

At its core, the Thevenin equivalent circuit is defined by two primary parameters: the Thevenin voltage (V_TH) and the Thevenin resistance (R_TH). Once you have these, calculating the behavior of any attached load requires only basic Ohm's law and the voltage divider rule.

The foundational equations are:

• Thevenin Voltage: V_TH = V_OC (Open-Circuit Voltage)
• Thevenin Resistance: R_TH = V_OC / I_SC (where I_SC is the short-circuit current)
• Loaded Output Voltage: V_L = V_TH × [R_L / (R_TH + R_L)]

Understanding the realistic magnitude of these variables prevents catastrophic math errors on the bench. A calculated R_TH of 5MΩ for a 12V lead-acid battery circuit is an immediate red flag that your parallel/series reduction failed.

Rearranged Forms and Fatal Unit Mistakes

On the bench, you rarely solve for V_TH and R_TH in isolation. More often, you are reverse-engineering a black-box power supply or calculating the exact load current. Here are the rearranged forms of the Thevenin formula, solving for each critical variable:

• Solving for Short-Circuit Current: I_SC = V_TH / R_TH
• Solving for Load Current: I_L = V_TH / (R_TH + R_L)
• Solving for Load Voltage (Drop Method): V_L = V_TH - (I_L × R_TH)
• Solving for R_TH (from measured load drop): R_TH = (V_TH - V_L) / I_L
• Solving for R_L (to achieve a specific V_L): R_L = (V_L × R_TH) / (V_TH - V_L)

Furthermore, beware of multimeter input impedance. If your circuit has a high R_TH (e.g., 1MΩ) and your DMM has a standard 10MΩ input impedance, your measured V_OC will be artificially low by roughly 9% because the meter itself acts as a load. For high-impedance networks, use an electrometer or buffer the measurement with an op-amp.

Worked Problem 1: DC Resistive Network Reduction

Let's derive the Thevenin equivalent for a loaded voltage divider fed by a non-ideal power supply. This is a standard topology in sensor biasing and transistor base networks.

The Circuit: A 24V DC supply has an internal resistance (R_int) of 10Ω. It feeds a voltage divider consisting of R₁ = 40Ω (top) and R₂ = 60Ω (bottom to ground). We need the Thevenin equivalent at the output node (across R₂).

1. Calculate V_TH (Open-Circuit Voltage):
   With no external load attached, R_int, R₁, and R₂ form a single series loop.
   Total Resistance (R_T) = R_int + R₁ + R₂ = 10Ω + 40Ω + 60Ω = 110Ω.
   Loop Current (I) = V_source / R_T = 24V / 110Ω = 0.21818A.
   V_TH is the voltage dropped across R₂.
   V_TH = I × R₂ = 0.21818A × 60Ω = 13.09V.
2. Calculate R_TH (Thevenin Resistance):
   'Kill' the 24V independent source by replacing it with a short circuit (0Ω wire).
   Look back into the output terminals. R₂ is now in parallel with the series combination of R₁ and R_int.
   R_branch = R₁ + R_int = 40Ω + 10Ω = 50Ω.
   R_TH = R₂ || R_branch = (60Ω × 50Ω) / (60Ω + 50Ω) = 3000 / 110 = 27.27Ω.
3. Calculate Load Behavior (Attach R_L = 100Ω):
   Now we treat the circuit as a 13.09V source in series with 27.27Ω, feeding a 100Ω load.
   I_L = V_TH / (R_TH + R_L) = 13.09V / (27.27Ω + 100Ω) = 13.09V / 127.27Ω = 0.1028A (102.8mA).
   V_L = I_L × R_L = 0.1028A × 100Ω = 10.28V.

Notice how the load voltage (10.28V) sagged significantly from the open-circuit voltage (13.09V) due to the relatively high R_TH of 27.27Ω. According to All About Circuits, recognizing this sag is critical when designing bias networks for high-current loads.

Worked Problem 2: Finding R_TH via the Black Box Method

In field diagnostics, you often cannot 'kill' the source or look inside the circuit to calculate R_TH theoretically. Instead, you use the load-drop method to empirically derive the Thevenin formula parameters.

The Scenario: You have a sealed 12V nominal lead-acid battery bank. You need to know its exact internal Thevenin resistance to size an inverter cable.

1. Measure V_OC (V_TH):
   Using a calibrated Fluke 87V multimeter, measure the terminals with no load.
   Measured V_OC = 13.80V.
2. Apply a Known Load and Measure V_L:
   Connect a precision 10Ω power resistor (rated for 50W) across the terminals.
   Measure the voltage across the resistor while under load.
   Measured V_L = 12.40V.
3. Calculate Load Current (I_L):
   Using Ohm's law strictly on the load resistor:
   I_L = V_L / R_L = 12.40V / 10Ω = 1.24A.
4. Derive R_TH using the Rearranged Formula:
   The voltage 'lost' inside the battery is the drop across R_TH.
   V_drop = V_TH - V_L = 13.80V - 12.40V = 1.40V.
   R_TH = V_drop / I_L = 1.40V / 1.24A = 1.12Ω.

This empirical R_TH of 1.12Ω tells you that if you attempt to pull 100A from this battery bank to run a microwave via an inverter, the internal voltage drop will be 112V, collapsing the terminal voltage to negative (meaning the battery is entirely inadequate for that load). As noted in Electronics Tutorials, empirical derivation is often more accurate than theoretical calculation because it accounts for real-world contact resistances and battery chemistry degradation.

When the Thevenin Formula Applies (and When It Breaks)

Thevenin's theorem is incredibly powerful, but it is bound by strict physical assumptions. Applying it outside these boundaries will yield mathematically correct but physically meaningless results.

The Linearity Assumption

The Thevenin formula only applies to linear, bilateral networks. This means the components inside the network being reduced must obey Ohm's law linearly. Resistors, ideal capacitors, and ideal inductors are linear. Diodes, incandescent bulbs (whose resistance changes with temperature), and transistors are non-linear. You cannot use Thevenin's theorem to reduce a network that includes a diode in the reduction path. However, you can use Thevenin to reduce the linear bias network feeding a diode, and then analyze the diode separately.

AC and Impedance (Z_TH)

In AC circuits, resistance (R) is replaced by complex impedance (Z). The Thevenin formula still applies, but you must use phasor math. V_TH becomes a complex voltage magnitude and phase angle, and R_TH becomes Z_TH (which may include reactive components like jX_L or -jX_C). If you attempt to use scalar DC resistance formulas on an AC circuit containing capacitors, your load voltage calculations will be entirely wrong.

Dependent Sources

If your circuit contains dependent sources (e.g., a voltage-controlled voltage source inside an op-amp model), you cannot simply 'kill' them to find R_TH as you do with independent sources. Instead, you must apply a 1V test source (or 1A test current) to the output terminals with independent sources killed, and calculate R_TH = V_test / I_test. This is a common stumbling block for students transitioning from basic DC theory to active semiconductor modeling.