Nortons Theorem

1. Definition and Purpose

Norton's Theorem: Definition and Purpose

Norton's Theorem, formulated by engineer Edward Lawry Norton in 1926, is a fundamental principle in linear circuit analysis. It states that any two-terminal linear network comprising independent sources, dependent sources, and resistances can be replaced by an equivalent circuit consisting of a single current source IN in parallel with a resistor RN. The theorem simplifies complex networks into a more analyzable form without altering the external voltage-current characteristics.

Mathematical Formulation

The Norton equivalent current IN is the short-circuit current measured across the terminals of the original network. The equivalent resistance RN is calculated by deactivating all independent sources (voltage sources shorted, current sources opened) and determining the resistance between the terminals. Mathematically:

$$ I_N = I_{sc} $$
$$ R_N = R_{th} \quad \text{(same as Thévenin equivalent resistance)} $$

Purpose and Practical Utility

Norton's Theorem is particularly useful in:

Comparison with Thévenin's Theorem

While Thévenin's Theorem uses a voltage source and series resistance, Norton's Theorem employs a dual representation with a current source and parallel resistance. The equivalence between the two is given by:

$$ V_{th} = I_N R_N \quad \text{and} \quad R_{th} = R_N $$

Visual Representation

The Norton equivalent circuit is depicted as a current source IN connected in parallel with RN, with terminals A and B matching the original network's output nodes. This abstraction is valid for linear circuits under all load conditions.

1.2 Comparison with Thevenin's Theorem

Norton's Theorem and Thevenin's Theorem are duals of each other, both simplifying complex linear networks into equivalent circuits. While Thevenin's Theorem represents a network as a voltage source in series with a resistance, Norton's Theorem models it as a current source in parallel with a resistance. The equivalence arises from source transformation, where:

$$ V_{Th} = I_N R_{Th} $$

Here, VTh is the Thevenin voltage, IN is the Norton current, and RTh (or RN) is the equivalent resistance. The choice between the two depends on the circuit's nature:

Key Differences

Mathematical Equivalence

The conversion between the two is straightforward:

$$ I_N = \frac{V_{Th}}{R_{Th}} $$ $$ R_{N} = R_{Th} $$

Practical Implications

In circuit analysis, Norton's form is advantageous when dealing with parallel-connected loads, as it simplifies nodal analysis. Conversely, Thevenin's form is more intuitive for series-connected loads and mesh analysis. For example, transistor small-signal models often use Norton equivalents for current amplification stages.

Historical Context

Developed independently—Thevenin by Léon Charles Thévenin in 1883 and Norton by Edward Lawry Norton in 1926—these theorems emerged from the need to simplify telegraph and telephone network analysis. Their duality was later formalized in network theory.

Norton's Theorem: Applications in Circuit Analysis

Simplifying Complex Networks

Norton's Theorem reduces any linear bilateral network with multiple sources and resistances to a single current source in parallel with a resistance. This simplification is particularly useful when analyzing circuits with multiple branches or when determining the current through a specific load. Consider a network with voltage sources, current sources, and resistors. The Norton equivalent circuit is derived by:

$$ I_N = \text{Short-circuit current between terminals} $$ $$ R_N = \text{Equivalent resistance seen from the terminals with all independent sources deactivated} $$

For example, in a circuit with multiple meshes, applying Norton's Theorem allows replacing the entire network (excluding the load) with a single current source \( I_N \) and a parallel resistor \( R_N \). This drastically simplifies nodal or mesh analysis when only the load current is of interest.

Power System Analysis

In power distribution networks, Norton equivalents model fault currents and system impedances. During a short-circuit event, the Norton current \( I_N \) represents the maximum available fault current, while \( R_N \) models the Thevenin impedance of the grid. Engineers use this to:

Small-Signal Modeling of Transistors

In transistor amplifier design, Norton equivalents simplify small-signal AC analysis. For a BJT in a common-emitter configuration, the output port can be modeled as a Norton current source \( I_N = g_m V_{be} \) (where \( g_m \) is transconductance) in parallel with the output resistance \( r_o \). This allows rapid calculation of gain and impedance matching.

$$ I_N = g_m V_{be} $$ $$ R_N = r_o \parallel R_C $$

Noise Analysis in Circuits

Norton equivalents are used to model noise sources in electronic systems. Thermal noise in resistors is represented as a Norton current source with spectral density:

$$ \overline{I_n^2} = \frac{4kT \Delta f}{R} $$

where \( k \) is Boltzmann's constant, \( T \) is temperature, and \( \Delta f \) is bandwidth. This approach simplifies noise figure calculations in amplifiers.

Practical Case Study: Current Divider Design

When designing a current divider, Norton's Theorem provides an intuitive method to determine branch currents without solving simultaneous equations. For a load \( R_L \) connected to a Norton equivalent \( (I_N, R_N) \), the load current is:

$$ I_L = I_N \cdot \frac{R_N}{R_N + R_L} $$

This is particularly useful in biasing networks for analog circuits, where precise current splitting is required.

Limitations and Practical Considerations

While Norton's Theorem is powerful, its application requires attention to:

In RF circuits, for instance, the Norton equivalent must account for parasitic reactances, making \( R_N \) frequency-dependent. SPICE simulations often validate hand calculations under these conditions.

2. Norton Equivalent Circuit Components

2.1 Norton Equivalent Circuit Components

The Norton equivalent circuit simplifies a linear two-terminal network into a current source in parallel with a resistor. This transformation is particularly useful for analyzing complex circuits where only the behavior at a specific pair of terminals is of interest. The two primary components of the Norton equivalent circuit are:

Norton Current Source (IN)

The Norton current source, IN, represents the short-circuit current flowing between the two terminals of the network when the load is removed. Mathematically, it is derived by:

$$ I_N = I_{sc} $$

where Isc is the current measured when the output terminals are short-circuited. For example, in a resistive network with independent sources, IN can be computed using superposition or nodal/mesh analysis.

Norton Resistance (RN)

The Norton resistance, RN, is equivalent to the Thévenin resistance and represents the equivalent resistance seen from the terminals when all independent sources are deactivated (voltage sources shorted, current sources opened). It is calculated as:

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

where Voc is the open-circuit voltage across the terminals. Alternatively, if the network contains only resistors, RN can be found by series-parallel reduction.

Practical Considerations

In real-world applications, Norton's theorem is particularly useful for:

For networks containing dependent sources, RN must be computed using a test voltage or current method, as traditional deactivation techniques do not apply. The governing equation becomes:

$$ R_N = \frac{V_{test}}{I_{test}} $$

where Vtest is an applied test voltage and Itest is the resulting current (or vice versa).

Calculating Norton Current (IN)

The Norton current, IN, represents the short-circuit current flowing between two terminals of a linear network when all independent sources are active. It is the current that would pass through an ideal ammeter connected directly across the output terminals.

Step-by-Step Derivation

To determine IN, follow these steps rigorously:

  1. Identify the Load Terminals: Select the two terminals across which the Norton equivalent is to be calculated.
  2. Short the Output Terminals: Connect the terminals with an ideal conductor to simulate a short-circuit condition.
  3. Compute the Short-Circuit Current: Apply network analysis techniques (e.g., nodal analysis, mesh analysis, or superposition) to find the current flowing through the shorted path.

Mathematical Formulation

For a network with independent voltage and current sources, IN can be derived using Kirchhoff's laws. Consider a simple resistive network with a voltage source VS and internal resistance RS:

$$ I_N = \frac{V_S}{R_S} $$

For more complex networks, superposition may be necessary. If multiple sources are present, compute the individual short-circuit currents due to each source and sum them algebraically:

$$ I_N = \sum_{k=1}^{n} I_{SC_k} $$

Practical Example

Consider a circuit with a 10V voltage source and a 5Ω resistor in series, connected to a parallel combination of a 2Ω resistor and a 3Ω resistor. To find IN:

  1. Short the output terminals across the parallel resistors.
  2. The total resistance seen by the source is 5Ω + (2Ω || 3Ω).
  3. First, compute the equivalent parallel resistance:
$$ R_{eq} = \frac{2 \times 3}{2 + 3} = 1.2 \, \Omega $$
  1. The total resistance is then 5Ω + 1.2Ω = 6.2Ω.
  2. The current from the source is:
$$ I_{total} = \frac{10V}{6.2Ω} \approx 1.61 \, A $$
  1. The current divides in the parallel branch. Using current division, the short-circuit current IN is:
$$ I_N = I_{total} \times \frac{2Ω}{2Ω + 3Ω} = 1.61 \times 0.4 \approx 0.645 \, A $$

Real-World Considerations

In practical circuits, non-ideal source resistances, parasitic elements, and temperature effects may influence IN. High-precision measurements often require accounting for these factors, particularly in low-power analog circuits or high-frequency systems where impedance matching is critical.

2.3 Calculating Norton Resistance (RN)

The Norton resistance RN is a fundamental parameter in Norton's theorem, representing the equivalent resistance seen across the output terminals of a linear network when all independent sources are deactivated. Unlike Thevenin resistance, which is derived under identical conditions, Norton resistance specifically relates to the current-source equivalent model.

Step-by-Step Derivation of RN

To compute RN, follow these steps rigorously:

  1. Deactivate all independent sources: Replace voltage sources with short circuits and current sources with open circuits. Dependent sources remain unchanged.
  2. Measure resistance between terminals: Using an ohmmeter or analytical methods, determine the equivalent resistance across the open-circuited load terminals.
$$ R_N = R_{th} $$

This equality holds because the Thevenin and Norton resistances are derived from the same network configuration. For networks containing dependent sources, standard methods like the test voltage/current approach must be employed.

Test Source Method for Dependent Sources

When dependent sources are present, apply a test voltage Vtest or current Itest to the deactivated network and measure the resulting current or voltage. The resistance is then:

$$ R_N = \frac{V_{test}}{I_{test}} $$

For example, consider a network with a dependent current source kIx. After deactivating independent sources, applying Vtest yields:

$$ I_{test} = \frac{V_{test}}{R_1} + kI_x $$

Solving for RN requires expressing Ix in terms of Vtest.

Practical Example: Resistive Network

Given a network with resistors R1 = 4Ω, R2 = 6Ω, and R3 = 3Ω in a delta configuration, deactivating sources and combining resistances yields:

$$ R_N = \left( R_1 \parallel R_2 \right) + R_3 = \left( \frac{4 \times 6}{4 + 6} \right) + 3 = 5.4\,\Omega $$

Implications in Circuit Design

Norton resistance critically impacts power transfer and signal integrity. For maximum power transfer, the load resistance must match RN. In high-frequency circuits, parasitic elements contribute to RN, necessitating careful modeling.

Norton Resistance Calculation Example A delta network with resistors R1, R2, and R3, showing terminals A-B and deactivated sources as short/open circuits for Norton resistance calculation. R1 (4Ω) R2 (6Ω) R3 (3Ω) A B Short Circuit Open Circuit
Diagram Description: The section involves complex network configurations and resistance calculations that are easier to visualize with a diagram.

3. Identifying the Load Resistor

3.1 Identifying the Load Resistor

In Norton's Theorem, the load resistor (RL) is the component across which the equivalent Norton current source and resistance are calculated. Proper identification of RL is critical, as it determines the portion of the circuit to be simplified.

Key Characteristics of the Load Resistor

The load resistor must satisfy the following conditions:

Step-by-Step Identification Process

  1. Isolate the Load: Remove RL from the circuit, leaving two open terminals (A and B).
  2. Verify Linearity: Ensure no dependent sources or nonlinear elements (e.g., diodes) are part of RL.
  3. Confirm External Placement: The load must not influence the Norton equivalent parameters (IN and RN).

Mathematical Validation

The Norton current (IN) and resistance (RN) are derived independently of RL. For a circuit with a voltage source VS and internal resistance RS:

$$ I_N = \frac{V_S}{R_S} $$
$$ R_N = R_S $$

The load current (IL) is then calculated using the Norton equivalent:

$$ I_L = I_N \left( \frac{R_N}{R_N + R_L} \right) $$

Practical Considerations

In real-world applications, misidentifying RL can lead to incorrect circuit analysis. For example:

Common Pitfalls

3.2 Short-Circuiting the Load Terminals

To determine the Norton current (IN), the load terminals must be short-circuited. This step effectively removes the load impedance (ZL) and allows the measurement (or calculation) of the current flowing directly between the two terminals. The procedure is as follows:

Step-by-Step Derivation

Consider a linear two-terminal network with independent and dependent sources. The Norton current is obtained by:

  1. Disconnecting the load: Remove ZL from the circuit.
  2. Shorting the terminals: Connect a zero-resistance path between the output terminals.
  3. Calculating the short-circuit current: Compute or measure the current flowing through this path.
$$ I_N = I_{sc} $$

where Isc is the current measured when the terminals are shorted.

Practical Implications

In real-world applications, short-circuiting must be performed cautiously:

Mathematical Formulation

For a network with Thévenin equivalent parameters (Vth, Rth), the Norton current relates directly to the Thévenin voltage:

$$ I_N = \frac{V_{th}}{R_{th}} $$

This equivalence holds because short-circuiting the Thévenin network forces all voltage to drop across Rth, yielding Isc = Vth / Rth.

Example Calculation

Given a Thévenin equivalent circuit with Vth = 12V and Rth = 4Ω:

$$ I_N = \frac{12V}{4Ω} = 3A $$

The short-circuit current (and thus Norton current) is 3A.

Handling Dependent Sources

If the network contains dependent sources, the short-circuit current must be computed using nodal or mesh analysis:

  1. Apply Kirchhoff’s laws to the shorted network.
  2. Solve the resulting system of equations for Isc.

For instance, in a circuit with a current-controlled voltage source (CCVS), the governing equations may include terms like V = kIx, where Ix is a branch current.

3.3 Determining the Norton Current

The Norton current (IN) is the short-circuit current flowing between two terminals of a linear network when the load is removed. It is the current that would pass through an ideal ammeter connected directly across the output terminals. To compute IN, we follow a systematic approach:

Step 1: Identify the Load Terminals

Disconnect the load resistor (RL) from the circuit, leaving the two terminals open. These terminals (A and B) are where the Norton equivalent will be derived.

Step 2: Short-Circuit the Terminals

Replace the load with an ideal short circuit (zero resistance) between terminals A and B. This allows the maximum possible current to flow, which is by definition IN.

Step 3: Compute the Short-Circuit Current

Using circuit analysis techniques (e.g., nodal analysis, mesh analysis, or superposition), calculate the current through the short-circuited path. For example, in a simple resistive network with a voltage source VTh and Thévenin resistance RTh:

$$ I_N = \frac{V_{Th}}{R_{Th}} $$

For more complex circuits, employ Kirchhoff’s laws or equivalent transformations to simplify the network before solving for IN.

Practical Considerations

Example Calculation

Consider a circuit with a 12V voltage source and two resistors (R1 = 4Ω, R2 = 6Ω) in parallel. The short-circuit current is:

$$ I_N = \frac{12V}{4Ω} = 3A $$

Here, RTh is the parallel combination of R1 and R2, but IN depends solely on the branch containing the source when the terminals are shorted.

Norton Current Determination Steps A step-by-step schematic diagram showing the process of determining Norton current, including load removal and short-circuit connection. Step 1: Original Circuit VTh R1 R2 A B Load Step 2: Load Removed VTh R1 R2 A B Step 3: Short-circuit A-B VTh R1 R2 A B IN Norton Current Determination Steps
Diagram Description: The diagram would show the step-by-step process of removing the load, short-circuiting terminals A and B, and the path of Norton current through the simplified circuit.

3.4 Deactivating Sources to Find Norton Resistance

Norton resistance (RN) is determined by deactivating all independent sources in the network and calculating the equivalent resistance seen from the terminals of interest. This process mirrors Thévenin resistance derivation but is applied specifically for Norton equivalent circuits.

Procedure for Deactivating Sources

To compute RN, follow these steps:

After deactivation, the network reduces to a purely resistive circuit. The equivalent resistance is then calculated using standard methods:

Mathematical Derivation

For a linear network with no dependent sources, RN simplifies to the equivalent resistance Req seen from the output terminals:

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

where Voc is the open-circuit voltage and Isc is the short-circuit current.

For networks with dependent sources, a test source method is used:

  1. Apply a test voltage Vtest and measure the resulting current Itest.
  2. Alternatively, inject a test current and measure the resulting voltage.
$$ R_N = \frac{V_{test}}{I_{test}} $$

Practical Example

Consider a network with a 10V voltage source and a 2Ω resistor in series with a 3Ω resistor. To find RN:

  1. Deactivate the 10V source (replace with a short circuit).
  2. Compute the equivalent resistance seen from the output terminals: 2Ω || 3Ω = 1.2Ω.
$$ R_N = \frac{2 \times 3}{2 + 3} = 1.2 \, \Omega $$

Handling Dependent Sources

For circuits with dependent sources, deactivation alone is insufficient. A test source must be applied, and the resulting voltage-current relationship analyzed. For example, in a network with a voltage-controlled current source (VCCS), the procedure involves:

  1. Deactivating independent sources.
  2. Applying Vtest and solving for Itest using nodal or mesh analysis.

The Norton resistance is then derived from the ratio of the test quantities.

Source Deactivation for Norton Resistance A schematic diagram showing the transformation of a circuit when sources are deactivated, with before and after states for Norton equivalent resistance calculation. V_source R1 I_source R2 Original Circuit Deactivate Sources Short Circuit R1 Open Circuit R2 R_N = R1 || R2 Deactivated Circuit
Diagram Description: The diagram would show the physical transformation of a circuit when sources are deactivated (short/open circuits) and the resulting equivalent resistance calculation.

3.5 Constructing the Norton Equivalent Circuit

Norton's Theorem simplifies a linear two-terminal network into an equivalent circuit consisting of a current source in parallel with a resistor. The construction involves two key steps: determining the Norton current (IN) and the Norton resistance (RN).

Step 1: Calculating the Norton Current (IN)

The Norton current is the short-circuit current flowing between the terminals when the load is removed. To compute it:

  1. Remove the load resistor (RL) and short the output terminals.
  2. Analyze the circuit using Kirchhoff's laws, nodal analysis, or mesh analysis to find the current through the shorted path.
$$ I_N = I_{sc} $$

For example, in a circuit with a voltage source VS and internal resistance RS, the Norton current is:

$$ I_N = \frac{V_S}{R_S} $$

Step 2: Determining the Norton Resistance (RN)

The Norton resistance is equivalent to the Thévenin resistance and is found by:

  1. Deactivating all independent sources (voltage sources become short circuits, current sources become open circuits).
  2. Calculating the equivalent resistance between the open terminals.
$$ R_N = R_{eq} $$

If dependent sources are present, apply a test voltage (Vtest) or current (Itest) and measure the resulting current or voltage:

$$ R_N = \frac{V_{test}}{I_{test}} $$

Constructing the Equivalent Circuit

Once IN and RN are determined, the Norton equivalent circuit is assembled by placing:

IN RN RL

Practical Considerations

Norton's Theorem is particularly useful in:

  • Analyzing power systems where current sources dominate.
  • Simplifying transistor amplifier models.
  • Designing fault-tolerant circuits where short-circuit behavior is critical.

For circuits with nonlinear elements, Norton equivalence is valid only for small-signal approximations around a specific operating point.

4. Example 1: Simple Resistive Network

Norton's Theorem: Example 1 - Simple Resistive Network

Consider the DC resistive network shown below, where we wish to find the Norton equivalent circuit with respect to terminals A-B. The network consists of a 10V voltage source in series with a 2Ω resistor (R1), connected in parallel with a 4Ω resistor (R2).

10V 2Ω 4Ω A B

Step 1: Calculating Norton Current (IN)

The Norton current is the short-circuit current between terminals A-B. With the terminals shorted, R2 is bypassed, and the current is determined solely by the voltage source and R1:

$$ I_N = \frac{V}{R_1} = \frac{10\text{V}}{2\Omega} = 5\text{A} $$

Step 2: Calculating Norton Resistance (RN)

To find the Norton resistance, we deactivate all independent sources (replace the voltage source with a short circuit) and calculate the equivalent resistance between terminals A-B:

$$ R_N = R_1 \parallel R_2 = \frac{R_1 \times R_2}{R_1 + R_2} = \frac{2\Omega \times 4\Omega}{2\Omega + 4\Omega} = \frac{8}{6}\Omega \approx 1.33\Omega $$

Step 3: Constructing the Norton Equivalent Circuit

The final Norton equivalent consists of a 5A current source in parallel with a 1.33Ω resistor:

5A 1.33Ω A B

Verification Using Thévenin Equivalence

For validation, we can convert the Norton equivalent to its Thévenin form. The Thévenin voltage equals the open-circuit voltage across A-B:

$$ V_{Th} = I_N \times R_N = 5\text{A} \times 1.33\Omega \approx 6.67\text{V} $$

The Thévenin resistance equals the Norton resistance (1.33Ω). Calculating the open-circuit voltage of the original network confirms consistency:

$$ V_{AB} = 10\text{V} \times \frac{R_2}{R_1 + R_2} = 10\text{V} \times \frac{4\Omega}{6\Omega} \approx 6.67\text{V} $$

Practical Considerations

In real-world applications, the Norton equivalent simplifies analysis of power dissipation in load resistors connected to A-B. For a variable load RL, the power transfer is maximized when RL = RN (1.33Ω in this case), following the maximum power transfer theorem.

Norton Equivalent Circuit Transformation A schematic diagram showing the original resistive network configuration and its Norton equivalent circuit, including component values and terminal connections. 10V 2Ω 4Ω A B 5A 1.33Ω A B Norton Equivalent Circuit Transformation Original Circuit Norton Equivalent
Diagram Description: The diagram would physically show the original resistive network configuration and its Norton equivalent circuit, including component values and terminal connections.

Norton's Theorem: Example 2 - Circuit with Dependent Sources

When analyzing circuits containing dependent sources using Norton's Theorem, additional care must be taken in computing both the Norton current IN and resistance RN. Unlike independent sources, dependent sources cannot be simply turned off during the resistance calculation.

Problem Statement

Consider the following linear circuit containing a current-controlled voltage source (CCVS):

The CCVS has a gain factor of 3 Ω, producing a voltage equal to 3i where i is the controlling current flowing through the 2Ω resistor. We seek the Norton equivalent circuit across terminals a-b.

Step 1: Find the Short-Circuit Current (IN)

With terminals a-b shorted, we analyze the modified circuit:

$$ \text{KVL in left mesh: } 10 - 2i - 3i = 0 $$ $$ 5i = 10 \implies i = 2\text{A} $$

The short-circuit current equals the controlling current i since the dependent source's voltage (3i) appears across the short:

$$ I_N = i = 2\text{A} $$

Step 2: Determine the Norton Resistance (RN)

For circuits with dependent sources, we cannot simply deactivate all sources. Instead, we either:

Using the first method with Vtest = 1V:

$$ \text{KVL: } V_{test} - 2i - 3i = 0 $$ $$ 1 - 5i = 0 \implies i = 0.2\text{A} $$ $$ I_{test} = i = 0.2\text{A} $$ $$ R_N = \frac{V_{test}}{I_{test}} = \frac{1}{0.2} = 5\Omega $$

Verification via Open-Circuit Voltage

For consistency, we compute the open-circuit voltage VOC:

$$ \text{With } I = 0, \text{KVL gives:} $$ $$ 10 - 2i - 3i = 0 \implies i = 2\text{A} $$ $$ V_{OC} = 3i = 6\text{V} $$

Confirming the Norton equivalence:

$$ V_{OC} = I_N R_N = 2 \times 3 = 6\text{V} $$

Practical Considerations

When working with dependent sources:

4.3 Example 3: Complex Network Analysis

Consider a complex resistive network with multiple independent sources, as shown below. The objective is to determine the Norton equivalent circuit with respect to terminals A and B.

R1 R2 R3 V1 I1 A B

Step 1: Find Norton Current (IN)

The Norton current is the short-circuit current between terminals A and B. To compute this, we short the terminals and analyze the network using superposition.

  1. Contribution from V1: Deactivate I1 (open circuit). The current through the short is calculated using mesh analysis.
  2. Contribution from I1: Deactivate V1 (short circuit). The current divider rule determines the portion of I1 flowing through the short.
$$ I_N = I_{sc} = \frac{V_1}{R_1 + R_2 \parallel R_3} + I_1 \left( \frac{R_3}{R_2 + R_3} \right) $$

Step 2: Find Norton Resistance (RN)

The Norton resistance is the equivalent resistance seen from terminals A and B with all independent sources deactivated.

The equivalent resistance is computed as:

$$ R_N = R_3 \parallel (R_1 + R_2) $$

Step 3: Construct Norton Equivalent Circuit

The final Norton equivalent consists of a current source IN in parallel with RN.

A B IN RN

Practical Considerations

In real-world applications, Norton's theorem simplifies the analysis of:

For nonlinear networks, piecewise linear approximation may be necessary to apply Norton's theorem effectively.

Complex Resistive Network for Norton Analysis A schematic diagram of a complex resistive network with resistors R1, R2, R3, voltage source V1, current source I1, and terminals A and B for Norton analysis. R1 R2 R3 V1 I1 A B
Diagram Description: The diagram would physically show the complex resistive network with multiple independent sources, resistors, and terminals A and B.

5. Non-Linear Circuits

5.1 Non-Linear Circuits

Norton's Theorem, while traditionally applied to linear circuits, can be extended to analyze non-linear circuits under specific conditions. The theorem states that any two-terminal network of independent sources, dependent sources, and resistances can be replaced by an equivalent circuit consisting of a current source IN in parallel with a resistance RN. For non-linear circuits, the challenge lies in defining the equivalent Norton parameters due to the voltage-current relationship's non-linearity.

Mathematical Formulation for Non-Linear Elements

In linear circuits, the Norton equivalent current IN is the short-circuit current at the terminals, and RN is the equivalent resistance seen from the terminals with all independent sources deactivated. For non-linear circuits, these parameters become functions of the operating point. Consider a non-linear resistor described by I = g(V), where g(V) is a non-linear function. The Norton equivalent must account for this dependency.

$$ I_N = g(0) $$

The equivalent Norton resistance RN is derived from the small-signal approximation around the operating point V0:

$$ R_N = \left. \frac{dV}{dI} \right|_{V = V_0} = \frac{1}{g'(V_0)} $$

where g'(V) is the derivative of the non-linear characteristic. This approximation holds only for small perturbations around V0.

Piecewise Linear Approximation

For strongly non-linear circuits, a piecewise linear approximation is often employed. The non-linear characteristic is divided into linear segments, and Norton equivalents are derived for each segment. For example, a diode's I-V curve can be approximated as:

The Norton equivalent in forward bias becomes IN = Vth / Rf and RN = Rf, while in reverse bias, IN = 0 and RN approaches infinity.

Practical Applications

Non-linear Norton equivalents are critical in analyzing circuits with diodes, transistors, and other semiconductor devices. For instance, in a diode-clipped amplifier, the Norton model helps predict distortion by approximating the diode's non-linearity. Similarly, in power electronics, the piecewise linear approach simplifies the analysis of switching converters.

Non-linear I-V curve approximated as piecewise linear segments

Limitations and Considerations

The accuracy of Norton's Theorem in non-linear circuits depends heavily on the operating point and the validity of the small-signal or piecewise linear approximation. Large signal swings may invalidate the model, necessitating numerical methods like Newton-Raphson for precise analysis. Additionally, temperature effects and device tolerances further complicate the equivalent parameters.

Piecewise Linear Approximation of Non-Linear I-V Curve A graph showing a non-linear I-V curve with piecewise linear segments and operating points for Norton equivalents, including tangent lines for small-signal approximation. V I g(V) V0 R_N=1/g'(V0) Vth Reverse Bias Forward Bias
Diagram Description: The diagram would physically show a non-linear I-V curve with piecewise linear segments and the operating points for Norton equivalents.

5.2 Frequency-Dependent Components

Norton’s Theorem simplifies linear networks by replacing a complex circuit with a current source in parallel with an impedance. However, when frequency-dependent components like inductors and capacitors are present, the Norton equivalent must account for impedance variations with frequency. The Norton current IN and impedance ZN become functions of angular frequency ω:

$$ I_N(\omega) = \frac{V_{oc}(\omega)}{Z_{th}(\omega)} $$
$$ Z_N(\omega) = Z_{th}(\omega) $$

where Voc(ω) is the open-circuit voltage and Zth(ω) is the Thévenin impedance, both frequency-dependent. For a network containing reactive elements, Zth(ω) is complex:

$$ Z_{th}(\omega) = R(\omega) + jX(\omega) $$

Impedance of Reactive Components

For inductors (L) and capacitors (C), the reactance terms dominate at high frequencies:

$$ X_L(\omega) = \omega L \quad \text{(Inductive reactance)} $$
$$ X_C(\omega) = -\frac{1}{\omega C} \quad \text{(Capacitive reactance)} $$

In a parallel RLC network, the Norton admittance YN(ω) is the sum of individual admittances:

$$ Y_N(\omega) = \frac{1}{R} + \frac{1}{j\omega L} + j\omega C $$

Practical Considerations

At resonance (ω = ω0), the imaginary part of YN(ω) cancels out, simplifying the Norton equivalent to a purely resistive network. The resonant frequency is given by:

$$ \omega_0 = \frac{1}{\sqrt{LC}} $$

In RF and filter design, Norton equivalents with frequency-dependent components are used to model matching networks and impedance transformations. For instance, a Norton-derived model of a bandpass filter accurately predicts its 3-dB bandwidth and center frequency.

Example: Norton Equivalent of a High-Pass Filter

Consider a first-order RC high-pass filter. The Norton current and impedance are:

$$ I_N(\omega) = \frac{V_{in}}{R} \cdot \frac{j\omega RC}{1 + j\omega RC} $$
$$ Z_N(\omega) = R \parallel \frac{1}{j\omega C} = \frac{R}{1 + j\omega RC} $$

The cutoff frequency (ωc = 1/RC) defines the transition between resistive and capacitive dominance in the Norton model.

5.3 Practical vs. Ideal Conditions

Non-Ideal Behavior of Norton Equivalent Circuits

Norton’s Theorem assumes ideal conditions where the current source is perfectly constant and the equivalent resistance is purely ohmic. However, real-world components introduce deviations:

Mathematical Adjustments for Practical Conditions

The Norton current (IN) and resistance (RN) must account for non-idealities. For a linear network with internal impedance Zint:

$$ I_N = \frac{V_{oc}}{Z_{int} + R_N} $$

where Voc is the open-circuit voltage. The equivalent resistance becomes frequency-dependent:

$$ R_N(\omega) = R_{DC} + j\omega L_{parasitic} + \frac{1}{j\omega C_{parasitic}} $$

Case Study: Power Supply Design

In a 12V/2A DC power supply, the Norton equivalent’s current source droops under load due to internal ESR (Equivalent Series Resistance). Measurements reveal:

Frequency-Dependent Limitations

Above 1 MHz, parasitic effects dominate. For a BJT-based Norton circuit:

$$ Z_{out} = r_o \parallel \left( \frac{1}{j\omega C_\mu} \right) $$

where ro is the transistor’s output resistance and Cμ is the Miller capacitance.

Mitigation Strategies

Frequency-Dependent Norton Equivalent with Parasitics A schematic showing a Norton equivalent circuit with parasitic components (left) and its frequency-impedance response (right). I_N R_N C_parasitic L_parasitic Z_out(ω) Impedance Frequency 1MHz Frequency-Dependent Norton Equivalent with Parasitics
Diagram Description: The section discusses frequency-dependent behavior and parasitic effects, which are best visualized with a schematic showing parasitic components and their impact on impedance.

6. Recommended Textbooks

6.1 Recommended Textbooks

6.2 Online Resources and Tutorials

6.3 Research Papers and Advanced Topics