Thévenin & Norton Equivalents#

Any linear circuit, no matter how complex, can be replaced by a simple equivalent as seen from two terminals: a voltage source in series with a resistance (Thévenin) or a current source in parallel with a resistance (Norton). This is one of the most powerful simplification tools in circuit analysis.

Thévenin Equivalent#

From any two terminals, a linear circuit looks like:

V_th — The open-circuit voltage (voltage at the terminals with nothing connected)
R_th — The equivalent resistance seen looking back into the circuit (with all independent sources turned off: voltage sources replaced by short circuits, current sources replaced by open circuits)

The load sees V_th in series with R_th. That’s it.

Finding V_th#

Remove the load from the two terminals
Calculate (or measure) the open-circuit voltage between the terminals
That’s V_th

Finding R_th#

Method 1 (Source deactivation): Turn off all independent sources. Replace voltage sources with short circuits, current sources with open circuits. Calculate the equivalent resistance seen from the terminals.

Method 2 (Short-circuit current): Find the short-circuit current I_sc at the terminals (wire the terminals together and calculate the current). Then R_th = V_th / I_sc.

Method 3 (Test source): Apply a test voltage V_test at the terminals and calculate the resulting current I_test. Then R_th = V_test / I_test. (This is the standard approach when dependent sources are present, since they can’t just be “turned off.”)

Norton Equivalent#

Same circuit, different representation:

I_n — The short-circuit current (current between the terminals when shorted)
R_n — Same as R_th (the equivalent resistance is identical)

I_n = V_th / R_th. The two representations are completely interchangeable.

When to Use Which#

Thévenin — When the source is “voltage-like” (stiff voltage, significant source impedance). Power supplies, signal sources, sensor outputs
Norton — When the source is “current-like” (stiff current, high parallel impedance). Current mirrors, photodiodes, some sensor types

In practice, Thévenin is more commonly used because most sources in electronics are voltage sources.

Practical Applications#

Understanding Source-Load Interaction#

Once the Thévenin equivalent is known:

Loaded output voltage: V_out = V_th × R_load / (R_th + R_load)
Load current: I_load = V_th / (R_th + R_load)
How much the output drops under load: The ratio R_th / R_load indicates the regulation. Small ratio = stiff source = small drop

Maximum Power Transfer#

Maximum power to the load occurs when R_load = R_th. The load gets V_th / 2 volts and power = V_th² / (4 × R_th).

This is important in RF (50 Ω matching) and audio (impedance matching). It’s generally NOT the goal in power delivery, where maximum efficiency (R_load » R_th) is preferred over maximum power transfer.

Output Impedance Measurement#

To find the output impedance of a real circuit (amplifier, power supply, etc.):

Measure the open-circuit output voltage (no load): V_oc
Connect a known load R_load and measure the loaded voltage: V_loaded
Calculate: R_out = R_load × (V_oc - V_loaded) / V_loaded

This is the practical, bench-measurable version of finding R_th.

Linearity Requirement#

Thévenin and Norton equivalents only apply to linear circuits. The circuit must obey superposition — output is proportional to input, and the effect of multiple sources is the sum of individual effects.

Nonlinear elements (diodes, transistors in large-signal operation, saturated transformers) break linearity. Thévenin/Norton equivalents can still be used:

At a specific operating point (small-signal analysis)
For the linear portion of a circuit, treating nonlinear elements as separate
As an approximation when the operating range is small enough to be approximately linear

Limits of Abstraction#

The Thévenin/Norton equivalent captures the DC or single-frequency behavior perfectly. But:

Frequency dependence — Real circuits have impedance that varies with frequency. R_th becomes Z_th(f), and a single number doesn’t capture the full picture. At DC or a single frequency, it works. Over a range of frequencies, the full impedance vs. frequency characteristic is needed
Noise — R_th generates thermal noise. The equivalent circuit’s noise behavior matches the real circuit’s only if the noise source is included (V_noise = √(4kTR_th × B))
Nonlinearity — As noted above, the equivalent is only valid in the linear operating region
Dynamic behavior — Thévenin gives the steady-state equivalent. Transient behavior (capacitive and inductive storage elements) requires more than a single R and V

Tips#

Use Thévenin equivalents to simplify complex networks before analyzing load behavior
The method works for finding the source impedance of any circuit — treat it as a black box and measure V_oc and V_loaded
For frequency-dependent analysis, calculate Z_th at each frequency of interest

Caveats#

Dependent sources can’t be “turned off” — When finding R_th, only deactivate independent sources. Dependent sources stay active. Use the test-source method instead
R_th is not always resistive — In AC circuits, Z_th can be complex (resistive + reactive). The “resistance” is actually an impedance
Negative R_th means instability — Some active circuits present negative output impedance in certain frequency ranges. This can cause oscillation when connected to certain loads. If measured R_th comes out negative, investigate stability
Don’t confuse open-circuit voltage with nominal voltage — A 9 V battery’s V_th is 9 V only when fresh and unloaded. Under load, the terminal voltage drops by I × R_internal. The “9 V” on the label is the nominal (approximately open-circuit) voltage

In Practice#

Output voltage that drops significantly under load indicates high source impedance — calculate R_th from the voltage drop
A circuit that works unloaded but fails under load often has excessive output impedance for the intended load
Measured output impedance higher than expected suggests additional series resistance in the circuit (wiring, contact resistance, component degradation)
Two circuits that should have the same output impedance but measure differently indicate a fault or difference in one of them