Norton Equivalent CircuitEdit
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The Norton equivalent circuit is a standard representation in linear circuit analysis that replaces any two-terminal network by a current source in parallel with a resistor. This form is convenient because it directly expresses the short-circuit current the network can deliver to a load connected across the terminals and the impedance that appears in parallel with that load. The Norton form is the dual of the Thevenin form: any linear two-terminal network can be represented either as a current source in parallel with a resistor (the Norton model) or as a voltage source in series with a resistor (the Thevenin model). The two representations are related by I_N = V_th / R_th and R_N = R_th, where V_th is the open-circuit voltage across the terminals and R_th is the equivalent resistance seen from the terminals with all independent sources turned off. For networks that include dependent (or controlled) sources, the Norton form remains valid, but the method for determining R_N requires care (the dependent sources cannot be simply turned off; a test source is typically used).
Definition and theory
The Norton equivalent of a linear, bilateral network N as seen from two terminals a and b consists of: - a current source I_N connected between a and b, and - a resistor R_N connected in parallel with that current source, also between a and b.
The pair (I_N, R_N) is selected so that the external behavior across terminals a–b matches that of the original network for all possible load connections. This representation is valid for networks containing independent sources (voltage or current), passive elements, and, in many cases, dependent sources.
- Short-circuit current: I_N is the current that flows when terminals a and b are shorted together. This short-circuit current is one way to determine I_N directly.
- Parallel impedance: R_N is the equivalent impedance seen by any load connected across a–b when all independent sources are deactivated (replaced by their internal impedances). For networks with only independent sources, deactivating the sources typically means replacing voltage sources with shorts and current sources with opens.
- Thevenin relationship: The Norton and Thevenin equivalents describe the same external behavior. If you know V_th (the open-circuit voltage across a–b) and R_th (the resistance seen into a–b with sources deactivated), then I_N and R_N follow from I_N = V_th / R_th and R_N = R_th.
If the network contains dependent sources, deactivating independent sources is not sufficient to determine R_N. In such cases, a test source (voltage or current) is applied at terminals a–b, and the resulting voltage or current is used to compute R_N as R_N = V_test / I_test (or equivalently R_N = I_test / V_test, depending on which quantity is imposed).
Computation steps
A practical procedure to obtain the Norton equivalent is: - Step 1: Find the short-circuit current I_N. Short the output terminals a and b and compute the current that flows through the short. - Step 2: Find the open-circuit voltage V_th. Remove the load from the terminals and determine the voltage across a–b with no load connected. - Step 3: Determine the equivalent resistance R_N (or R_th). If the network contains only independent sources, deactivate them (voltage sources become shorts, current sources become opens) and compute the resistance seen across a–b. If dependent sources are present, apply a test source across a–b and compute the resulting current or voltage to obtain R_N. - Step 4: Use the relationship between the two representations. In particular, R_N = R_th, and I_N = V_th / R_th.
An alternative route uses the Thevenin form directly. If V_th and R_th are known, you can immediately write the Norton form with I_N = V_th / R_th and R_N = R_th. This is often convenient when you start from the open-circuit voltage and the response to a short circuit, respectively.
Example
Consider a simple network left of terminals a–b consisting of a 12 V voltage source in series with a 4 Ω resistor, connected to the terminals a–b where a is the top node and b is the bottom node.
- Open-circuit voltage V_th: With no load connected, the full 12 V appears across a–b, so V_th = 12 V.
- Short-circuit current I_N: If a and b are shorted, the configuration reduces to the 12 V source across a 4 Ω shorted path, so I_N = 12 V / 4 Ω = 3 A.
- Equivalent resistance R_th (and hence R_N): Deactivating the independent source would normally leave a 4 Ω path seen from a–b, so R_th = 4 Ω and R_N = 4 Ω.
- Norton representation: A 3 A current source in parallel with a 4 Ω resistor across terminals a–b produces the same external behavior as the original network for any load connected across a–b.
With the Norton form, the current through a load R_L connected across a–b is I_load = I_N − (V across the parallel network)/R_N, or more directly, current division in the parallel combination can be used to determine load currents.
Applications and limitations
- When to use Norton form: The Norton representation is particularly convenient when multiple loads are connected in parallel across the same terminals, or when the primary interest is in currents delivered to a load or in short-circuit behavior.
- Relationship to Thevenin form: Since every Norton equivalent has a corresponding Thevenin equivalent, choosing between them is a matter of convenience for the analysis at hand.
- Limitations: The Norton equivalent applies to linear, bilateral networks with time-invariant behavior. If the circuit contains nonlinear elements (for example, diode junctions or transistor regions that vary with operating point) in the portion being replaced, the equivalence only holds for the specific operating point or for small-signal linearizations about a bias point.
- Dependent sources: When dependent sources are present, care must be taken to determine R_N using a test source, because simply deactivating all sources does not generally yield the correct equivalent resistance.
- Frequency response: For linear time-invariant networks, the Norton model preserves the frequency response characteristics seen at the terminals, enabling straightforward analysis of filters, amplifiers, and power transfer scenarios.