5.4 Thévenin and Norton equivalent circuits

Thévenin and Norton equivalent circuits let you replace a complex network with a simple single-source model. Instead of re-analyzing an entire circuit every time you swap out a load, you reduce everything to one source and one resistor. This makes it far easier to study how voltage, current, and power change as the load changes.

Thévenin and Norton Equivalent Circuits

Thévenin's Theorem and Equivalent Voltage Source

Thévenin's theorem says that any linear circuit (one with resistors and independent sources) can be replaced at a pair of terminals by a single voltage source $V_{Th}$ in series with a single resistance $R_{Th}$. The load "sees" exactly the same voltage and current as it would from the original circuit.

• $V_{Th}$ is the open-circuit voltage at the terminals: the voltage you'd measure if nothing were connected.
• $R_{Th}$ is the equivalent resistance looking back into the terminals after you turn off all independent sources (replace voltage sources with short circuits, current sources with open circuits).

Why is this useful? Suppose you have a complicated network and you want to know what happens when you connect different loads. Rather than re-solving the whole circuit each time, you find $V_{Th}$ and $R_{Th}$ once, and then every load calculation becomes a simple series circuit.

Norton's Theorem and Equivalent Current Source

Norton's theorem is the dual of Thévenin's. It says the same linear circuit can be replaced by a single current source $I_N$ in parallel with a single resistance $R_N$.

• $I_N$ is the short-circuit current at the terminals: the current that flows if you connect the two terminals directly with a wire.
• $R_N$ is the same equivalent resistance as $R_{Th}$, found the same way.

Norton equivalents are especially handy when you're working with circuits that naturally involve parallel elements or when the quantity you care about is current rather than voltage.

Relationship Between Thévenin and Norton Equivalents

The two representations describe the exact same behavior at the terminals, so you can always convert between them:

• $V_{Th} = I_N \times R_N$
• $R_{Th} = R_N$

This conversion is called source transformation. Which form you choose depends on what's more convenient. If you need the voltage across a load, Thévenin is often easier. If you need the current through a load, Norton may be the better pick.

Circuit Analysis Techniques

Determining Open-Circuit Voltage

The open-circuit voltage $V_{oc}$ is the voltage across the terminals when no load is connected (the terminals are left open).

Steps to find $V_{oc}$:

1. Identify the two terminals of interest.
2. Remove the load resistance so the terminals are open.
3. Use any circuit analysis method (KVL, node voltage, mesh current, superposition) to find the voltage across those open terminals.
4. The result is your Thévenin voltage: $V_{Th} = V_{oc}$.

You could also measure this directly in a lab with a voltmeter across the open terminals, since an ideal voltmeter draws no current.

Determining Short-Circuit Current

The short-circuit current $I_{sc}$ is the current that flows when the two terminals are connected directly by a wire.

Steps to find $I_{sc}$:

1. Identify the two terminals of interest.
2. Replace the load with a short circuit (a wire connecting the terminals).
3. Calculate the current flowing through that wire using any standard analysis method.
4. The result is your Norton current: $I_N = I_{sc}$.

Once you have both $V_{oc}$ and $I_{sc}$, you can also find the equivalent resistance directly:

$R_{Th} = R_N = \frac{V_{oc}}{I_{sc}}$

This gives you a useful cross-check if you've already calculated $R_{Th}$ by turning off sources.

Source Transformation

Source transformation converts between Thévenin and Norton forms. It works in both directions:

Thévenin → Norton:

1. $I_N = \frac{V_{Th}}{R_{Th}}$
2. $R_N = R_{Th}$ (now placed in parallel with $I_N$)

Norton → Thévenin:

1. $V_{Th} = I_N \times R_N$
2. $R_{Th} = R_N$ (now placed in series with $V_{Th}$)

Beyond just converting a final equivalent, source transformation is a circuit simplification technique on its own. You can apply it repeatedly to interior parts of a circuit to collapse series and parallel combinations, step by step, until the network is simple enough to solve directly.

Power Considerations

Maximum Power Transfer Theorem

The maximum power transfer theorem states that a load receives the greatest possible power from a linear source when the load resistance equals the Thévenin (or Norton) equivalent resistance:

$R_{load} = R_{Th}$

Under this condition, the maximum power delivered to the load is:

$P_{max} = \frac{V_{Th}^2}{4R_{Th}}$

Equivalently, using Norton parameters: $P_{max} = \frac{I_N^2 \, R_N}{4}$

There's an important trade-off here: at maximum power transfer, the efficiency is only 50%. Half the total power is dissipated inside the source's equivalent resistance $R_{Th}$, and the other half goes to the load. That's fine in low-power applications like audio systems or communications, where getting the most signal power to the load matters more than wasting energy.

In high-power systems (like the utility grid delivering electricity to your house), engineers deliberately make $R_{load}$ much larger than $R_{Th}$. This reduces the total power delivered but pushes efficiency well above 50%, which matters a lot when you're dealing with kilowatts or megawatts. So maximum power transfer and maximum efficiency are competing goals, and the design choice depends on the application.