Table of Contents
Why Thevenin and Norton Equivalents Matter 🎯
In practical circuit analysis, you often care about what a complex network does at just two terminals, where a load is connected. The internal details of the network can be very complicated, with many sources and resistors. Thevenin and Norton equivalents provide a way to replace that complex part with a simple, standard form that behaves exactly the same at those two terminals.
A Thevenin equivalent is a single voltage source in series with a single resistor.
A Norton equivalent is a single current source in parallel with a single resistor.
At the chosen pair of terminals, both equivalents produce the same current and voltage as the original network for any load connected there.
These equivalents are especially useful when you want to see how different loads affect a circuit, or when you want to simplify analysis before applying other methods.
A two terminal linear network can always be replaced, as seen from those two terminals, by:
- a Thevenin equivalent: a voltage source $V_\text{Th}$ in series with a resistance $R_\text{Th}$, or
- a Norton equivalent: a current source $I_\text{N}$ in parallel with a resistance $R_\text{N}$,
where $R_\text{Th} = R_\text{N}$ and $V_\text{Th} = I_\text{N} R_\text{Th}$.
Thevenin Equivalent: Concept and Structure 🔌
The Thevenin equivalent of a circuit at two terminals is a very simple model: one ideal voltage source and one resistor in series.
Imagine you have a complicated left side of a circuit, and on the right you only see two terminals A and B. Everything on the left can be replaced, for the purpose of what happens at A and B, by:
- an ideal voltage source of value $V_\text{Th}$,
- in series with a resistor $R_\text{Th}$.
When you connect any load resistor $R_L$ between A and B, the load current and voltage will be the same as if the entire original network were still there.
Mathematically, with a load $R_L$ connected, the current through the load is
$$I_L = \frac{V_\text{Th}}{R_\text{Th} + R_L}$$
and the voltage across the load is
$$V_L = I_L R_L = \frac{V_\text{Th} R_L}{R_\text{Th} + R_L}.$$
Finding Thevenin Voltage and Resistance 🧲
There are two quantities that define the Thevenin equivalent: the Thevenin voltage $V_\text{Th}$ and the Thevenin resistance $R_\text{Th}$. The general procedure always focuses on the two terminals of interest.
Thevenin Voltage: Open Circuit Voltage 🔓
$V_\text{Th}$ is the voltage that appears across the two terminals when nothing is connected there, so when the terminals are open.
To find $V_\text{Th}$:
- Remove the load resistor (if there is one) from the terminals.
- Leave the terminals open, which means no current flows out of the network through them.
- Use your circuit analysis tools to compute the voltage between the two terminals. That voltage is $V_\text{Th}$.
This may involve using Ohm’s law, Kirchhoff’s laws, or nodal or mesh analysis. The key is that the terminals are open, so any current that would have gone into a load is zero.
Definition of Thevenin voltage
$V_\text{Th}$ is exactly the open circuit voltage at the chosen pair of terminals.
Thevenin Resistance: Seen from the Terminals 🔍
$R_\text{Th}$ represents how the network behaves in terms of resistance when you look into the terminals with all independent sources handled in a standard way.
For circuits that have only independent sources (no controlled sources), a widely used method is:
- Turn off all independent sources:
- Replace each independent voltage source with a short circuit.
- Replace each independent current source with an open circuit.
- Remove the load so the terminals are free.
- Compute the equivalent resistance between the two terminals. That resistance is $R_\text{Th}$.
This essentially treats the original network as a purely resistive network whose equivalent resistance is what the outside world sees.
Source suppression for Thevenin resistance
To find $R_\text{Th}$ for a network with only independent sources:
- Set all independent voltage sources to $0$ V by short circuiting them.
- Set all independent current sources to $0$ A by open circuiting them.
- Then find the equivalent resistance between the two terminals.
For networks that include dependent (controlled) sources, this simple suppression is not enough. In such cases, a common approach is to find $R_\text{Th}$ by applying a test source at the terminals:
- Turn off all independent sources using the rules above, but keep dependent sources active.
- Connect a test voltage source $V_\text{test}$ across the terminals and compute the resulting current $I_\text{test}$.
- Then
$$R_\text{Th} = \frac{V_\text{test}}{I_\text{test}}.$$
Alternatively, you can use a test current and measure the resulting voltage. The ratio is still the resistance.
Norton Equivalent: Concept and Structure 🔁
The Norton equivalent is the current source version of the same idea. Instead of a series voltage source, you use a parallel current source.
The Norton equivalent at two terminals consists of:
- an ideal current source of value $I_\text{N}$,
- in parallel with a resistor $R_\text{N}$.
Again, for any load connected across the terminals, this simple network will produce the same voltage and current at the load as the original complex network.
With a load resistor $R_L$ connected across the terminals, the source current will split between $R_\text{N}$ and $R_L$ according to current division. The resulting load current and voltage can be analyzed using parallel circuit rules.
Finding Norton Current and Resistance 💧
Just as Thevenin equivalents are defined by $V_\text{Th}$ and $R_\text{Th}$, Norton equivalents are defined by $I_\text{N}$ and $R_\text{N}$.
Norton Resistance
The resistance in the Norton model is always the same as in the Thevenin model:
$$R_\text{N} = R_\text{Th}.$$
So any method that gives you $R_\text{Th}$ automatically gives you $R_\text{N}$.
Norton Current: Short Circuit Current 🔒
$I_\text{N}$ is the current that would flow if the two terminals were shorted together.
To find $I_\text{N}$:
- Remove any load resistor from the terminals.
- Replace the terminals by a short circuit, meaning connect them with a wire.
- Use circuit analysis to find the current that flows through this short between the terminals. That current is $I_\text{N}$.
Definition of Norton current
$I_\text{N}$ is exactly the short circuit current between the two terminals.
In practice, you often prefer to find the Thevenin equivalent first, then convert to Norton, rather than directly calculating the short circuit current.
Relationship Between Thevenin and Norton Equivalents 🔄
Thevenin and Norton equivalents are just two different ways of describing the same behavior at the terminals. You can convert freely between them.
They share the same equivalent resistance:
$$R_\text{Th} = R_\text{N}.$$
The source values are related by Ohm’s law:
$$V_\text{Th} = I_\text{N} R_\text{Th}$$
and
$$I_\text{N} = \frac{V_\text{Th}}{R_\text{Th}}.$$
The following table summarizes the relationships:
| Quantity | Thevenin form | Norton form | Relationship |
|---|---|---|---|
| Source type | Voltage source in series | Current source in parallel | Equivalent at terminals |
| Source value | $V_\text{Th}$ | $I_\text{N}$ | $V_\text{Th} = I_\text{N} R_\text{Th}$ |
| Resistance value | $R_\text{Th}$ | $R_\text{N}$ | $R_\text{Th} = R_\text{N}$ |
Source conversion rule between Thevenin and Norton
A Thevenin equivalent with $V_\text{Th}$ in series with $R_\text{Th}$ is interchangeable with a Norton equivalent with:
$$I_\text{N} = \frac{V_\text{Th}}{R_\text{Th}}, \quad R_\text{N} = R_\text{Th}.$$
Typical Procedure to Find Equivalents in Practice 🧰
In practical problems you usually deal with a network that drives one particular load. You want to replace that network by a Thevenin or Norton equivalent at the load terminals, in order to analyze the effect of the load more easily.
A common step by step approach is:
- Identify the two terminals where the load connects. Label them clearly.
- Remove the load resistor so the network is defined only by its internal elements and the two open terminals.
- Find $V_\text{Th}$, the open circuit voltage at the terminals.
- Find $R_\text{Th}$, the equivalent resistance seen at the terminals with sources handled properly.
- Assemble the Thevenin equivalent using $V_\text{Th}$ and $R_\text{Th}$, or convert to a Norton equivalent if preferred.
Once you have the equivalent, you can reconnect the load as a simple series or parallel combination and compute currents, voltages, and power with basic formulas.
This technique combines well with other analysis methods. For example, nodal analysis might be used to find the open circuit voltage, and then series parallel simplification might be used to find the resistance. The power of Thevenin and Norton equivalents comes from how they allow you to isolate and simplify one part of a circuit while focusing on the behavior at a particular pair of terminals.
Choosing Between Thevenin and Norton 🔁⚡
Both equivalents describe the same behavior, so in theory you can use either. In practice, one form may be more convenient depending on the situation.
If the load is easier to analyze in series with a source, using a Thevenin equivalent usually simplifies the equations. When the load is in parallel with the source, or when you deal with current splitting among branches, a Norton equivalent can make calculations more direct.
The important idea is that both Thevenin and Norton equivalents let you think about a complicated circuit as a simple source and resistor, without losing accuracy at the point where your load connects.