Thevenin's Theorem

1. Definition and Purpose of Thevenin's Theorem

Definition and Purpose of Thevenin's Theorem

Thevenin's Theorem is a fundamental principle in linear circuit analysis that simplifies complex networks into an equivalent two-terminal circuit consisting of a single voltage source and a series resistance. Formally, it states:

$$ V_{th} = V_{oc} $$
$$ R_{th} = \frac{V_{oc}}{I_{sc}} $$

where Vth is the Thevenin voltage (equal to the open-circuit voltage across the terminals), and Rth is the Thevenin resistance (calculated as the ratio of open-circuit voltage to short-circuit current).

Historical Context

Developed by French telegraph engineer Léon Charles Thévenin in 1883, this theorem emerged from the need to analyze complex telegraph networks. It became a cornerstone of network analysis alongside Norton's Theorem, which provides a dual current-source equivalent.

Mathematical Derivation

Consider a linear bilateral network with independent sources:

  1. Disconnect all load elements across terminals A-B
  2. Calculate Voc using standard circuit analysis methods
  3. Deactivate all independent sources (replace voltage sources with shorts, current sources with opens)
  4. Determine the equivalent resistance Rth looking into terminals A-B

For dependent sources, an alternative derivation uses the test voltage method:

$$ R_{th} = \frac{V_{test}}{I_{test}} $$

Practical Applications

The theorem finds critical use in:

Limitations and Boundary Conditions

Thevenin equivalence only holds for:

Nonlinear elements like diodes or transistors require piecewise-linear approximation or small-signal models to apply Thevenin methods.

Advanced Considerations

For AC circuits, the theorem extends to complex impedances:

$$ Z_{th} = R + jX $$

where the Thevenin impedance includes both resistive and reactive components. The maximum power transfer theorem naturally follows from Thevenin analysis when ZL = Zth*.

Historical Context and Development

Thevenin's Theorem, a cornerstone of linear circuit analysis, was first articulated by French telegraph engineer Léon Charles Thévenin in 1883. His work was published in Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, marking a pivotal advancement in simplifying complex electrical networks. The theorem emerged from the need to analyze telegraphic circuits, where engineers faced the challenge of reducing intricate networks into manageable equivalents without losing accuracy.

Predecessors and Theoretical Foundations

Thévenin's work built upon earlier principles, notably Hermann von Helmholtz's 1853 theorem on equivalent sources, which demonstrated that any linear network could be replaced by a voltage source and a resistance. Thévenin refined this idea, presenting it in a form more directly applicable to electrical engineering. His formulation was initially met with skepticism, as it contradicted prevailing methods reliant on Kirchhoff's laws and mesh analysis. However, its computational efficiency soon made it indispensable.

Mathematical Formalization

The theorem states that any linear, bilateral network with independent and dependent sources can be replaced by an equivalent circuit comprising a single voltage source (VTh) in series with a resistance (RTh). Thevenin voltage is derived by calculating the open-circuit voltage across the terminals, while Thevenin resistance is found by deactivating all independent sources and measuring the equivalent resistance.

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

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

Practical Adoption and Modern Applications

By the early 20th century, Thevenin's Theorem had become a standard tool in circuit design, particularly in power systems and telecommunications. Its utility extends to modern applications like impedance matching in RF circuits and stability analysis in control systems. For instance, engineers use it to model battery internal resistance or simplify transistor amplifier stages into equivalent circuits for small-signal analysis.

Comparative Analysis with Norton's Theorem

In 1926, Edward Lawry Norton, an engineer at Bell Labs, introduced a dual form of Thevenin's Theorem, replacing the voltage source with a current source and the series resistance with a parallel conductance. While Thevenin's Theorem simplifies voltage-driven systems, Norton's equivalent is often preferred for current-driven analyses, such as parallel-loaded networks.

1.3 Key Applications in Circuit Analysis

Simplifying Complex Networks

Thevenin's theorem is indispensable when analyzing linear circuits with multiple sources and impedances. By reducing a complex network to a single voltage source VTh in series with an impedance ZTh, it enables rapid evaluation of load behavior without solving the entire system. Consider a bridge circuit with resistors R1 to R5 and a voltage source VS. Thevenizing the network across the load RL yields:

$$ V_{Th} = V_S \left( \frac{R_2}{R_1 + R_2} - \frac{R_4}{R_3 + R_4} \right) $$
$$ Z_{Th} = \left( \frac{R_1 R_2}{R_1 + R_2} \right) + \left( \frac{R_3 R_4}{R_3 + R_4} \right) $$

This simplification is particularly useful in power distribution networks, where load variations must be analyzed without recalculating the entire grid.

Impedance Matching in RF Circuits

In high-frequency systems, maximizing power transfer requires conjugate impedance matching. Thevenin's theorem facilitates this by providing the equivalent source impedance ZTh. For a transmitter with an internal impedance ZS and a transmission line of characteristic impedance Z0, the optimal load impedance ZL is derived as:

$$ Z_L = Z_{Th}^* = Z_S^* $$

This application is critical in antenna design and microwave engineering, where mismatches lead to standing waves and signal degradation.

Transient Analysis of Reactive Circuits

When analyzing first-order RC or RL circuits, Thevenin equivalents simplify the determination of time constants. For a capacitor C discharging through a network of resistors, the Thevenin resistance RTh dictates the decay rate:

$$ \tau = R_{Th} C $$

This approach extends to switching power supplies and filter design, where transient response governs performance.

Fault Analysis in Power Systems

During short-circuit conditions, Thevenin's theorem helps compute fault currents by modeling the grid as a voltage source behind an equivalent impedance. For a three-phase system with a line-to-ground fault, the Thevenin voltage VTh is the pre-fault phase voltage, and ZTh is the sum of positive-sequence impedances:

$$ I_{fault} = \frac{V_{Th}}{Z_{Th} + Z_{fault}} $$

Utilities rely on this method for protective relay coordination and circuit breaker sizing.

Small-Signal Modeling of Nonlinear Devices

In transistor amplifiers, Thevenin equivalents linearize the circuit around a DC operating point. For a BJT in common-emitter configuration, the small-signal Thevenin resistance seen at the collector is:

$$ R_{Th} = R_C \parallel r_o $$

where ro is the transistor's output resistance. This modeling is foundational to AC gain calculations and frequency response analysis.

Bridge Circuit Thevenin Equivalence A schematic diagram showing a bridge circuit (top) and its Thevenin equivalent (bottom), with labeled components and transformation steps. VS R1 R2 R3 R4 R5 RL Voltage Division Thevenin Transformation VTh ZTh RL
Diagram Description: The bridge circuit example involves spatial resistor arrangements and voltage division that are easier to visualize than describe.

2. Components of the Thevenin Equivalent Circuit

Components of the Thevenin Equivalent Circuit

The Thevenin equivalent circuit simplifies a complex linear network into a two-component representation: a voltage source and a series impedance. This reduction preserves the external behavior of the original circuit when viewed from a specified pair of terminals.

Thevenin Voltage (VTh)

The Thevenin voltage is the open-circuit voltage measured across the terminals of interest. To determine VTh:

  1. Remove the load connected to the terminals.
  2. Calculate or measure the voltage across the open terminals using standard circuit analysis techniques (nodal analysis, mesh analysis, or superposition).
$$ V_{Th} = V_{oc} $$

For dependent sources, the controlling variable must be expressed in terms of the open-circuit voltage. In AC circuits, VTh becomes a phasor quantity incorporating both magnitude and phase.

Thevenin Impedance (ZTh)

The Thevenin impedance is the equivalent impedance seen from the terminals when all independent sources are deactivated:

For circuits without dependent sources, ZTh can be found using series-parallel impedance combinations. When dependent sources are present, one of two methods must be employed:

Method 1: Direct Calculation

Deactivate all independent sources and apply a test voltage Vtest (or test current Itest) at the terminals. Measure the resulting current Itest (or voltage Vtest). The Thevenin impedance is then:

$$ Z_{Th} = \frac{V_{test}}{I_{test}} $$

Method 2: Short-Circuit Current

Calculate the short-circuit current Isc that flows when the terminals are shorted. The Thevenin impedance is then:

$$ Z_{Th} = \frac{V_{Th}}{I_{sc}} $$

In AC circuits, ZTh is complex, comprising both resistive and reactive components:

$$ Z_{Th} = R_{Th} + jX_{Th} $$

Practical Considerations

When applying Thevenin's theorem to real-world circuits:

The following diagram illustrates the transformation from a complex network to its Thevenin equivalent:

Complex Linear Network Terminals A-B Thevenin Equivalent VTh ZTh Terminals A-B

2.2 Calculating Thevenin Voltage (Vth)

The Thevenin voltage (Vth) represents the open-circuit voltage across the terminals of a linear network when all independent sources are active and all dependent sources remain in their operational state. To compute Vth, we follow a systematic approach:

Step 1: Identify the Load Terminals

Remove the load resistor (RL) or any external component connected across the two terminals where the equivalent circuit is to be determined. This leaves the network open-circuited at these points.

Step 2: Analyze the Open-Circuit Voltage

With the load disconnected, calculate the voltage across the open terminals using standard circuit analysis techniques. Common methods include:

Step 3: Derive Vth Mathematically

For a simple resistive network with a voltage source, Vth can often be determined via voltage division. Consider a circuit with a voltage source VS and resistors R1 and R2:

$$ V_{th} = V_S \cdot \frac{R_2}{R_1 + R_2} $$

For more complex networks, such as those containing dependent sources or multiple active elements, a full nodal or mesh analysis is necessary. For example, in a two-node network:

$$ V_{th} = V_{oc} = V_1 - V_2 $$

where V1 and V2 are the node voltages relative to a reference point.

Practical Considerations

In real-world applications, Vth can be measured experimentally using a high-impedance voltmeter to approximate an open-circuit condition. However, in theoretical analysis, precise computation ensures accuracy, especially in circuits with nonlinear or frequency-dependent components.

Example: Thevenin Voltage in a Multisource Network

Given a network with two voltage sources V1 and V2 and resistors R1, R2, and R3, superposition can be applied:

  1. Deactivate V2 (replace with a short) and compute the partial Vth1.
  2. Deactivate V1 and compute Vth2.
  3. Combine results: Vth = Vth1 + Vth2.
$$ V_{th} = V_1 \left( \frac{R_2 \parallel R_3}{R_1 + (R_2 \parallel R_3)} \right) + V_2 \left( \frac{R_1 \parallel R_3}{R_2 + (R_1 \parallel R_3)} \right) $$

This method ensures accurate results even in circuits where direct analysis would be cumbersome.

Thevenin Voltage Calculation Example A circuit schematic illustrating Thevenin's theorem with voltage sources V1 and V2, resistors R1, R2, and R3, and open-circuit terminals for calculating Vth. V1 R1 V2 R2 R3 Vth Open-circuit terminals
Diagram Description: The diagram would show a circuit with voltage sources and resistors to illustrate the voltage division and superposition steps.

2.3 Calculating Thevenin Resistance (Rth)

The Thevenin resistance, Rth, is the equivalent resistance seen from the output terminals of a linear circuit when all independent sources are deactivated. It is a critical parameter in simplifying complex networks into a single voltage source and series resistance.

Deactivating Independent Sources

To compute Rth, voltage sources are replaced with short circuits (zero resistance) and current sources with open circuits (infinite resistance). Dependent sources, however, remain active since their behavior is tied to other circuit variables.

$$ V_{source} \rightarrow 0 \quad \text{(short circuit)} $$ $$ I_{source} \rightarrow \infty \quad \text{(open circuit)} $$

Methods for Determining Rth

Depending on the circuit's complexity, one of the following approaches is used:

1. Direct Resistance Calculation

For circuits without dependent sources, Rth is found by:

2. Test Source Method (For Circuits with Dependent Sources)

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

$$ R_{th} = \frac{V_{test}}{I_{test}} $$

This method ensures the dependent sources' contributions are accounted for in the equivalent resistance.

3. Short-Circuit Current Method

If the Thevenin voltage Vth is known, Rth can be derived by:

Practical Example: Resistive Network

Consider a circuit with two resistors R1 = 4Ω and R2 = 6Ω in parallel, connected to a 10V source. To find Rth:

  1. Deactivate the voltage source (replace with a short circuit).
  2. Compute the equivalent resistance between the terminals:
$$ R_{th} = R_1 \parallel R_2 = \frac{R_1 R_2}{R_1 + R_2} = \frac{4 \times 6}{4 + 6} = 2.4 \, \Omega $$

Case Study: Circuit with Dependent Source

For a circuit containing a voltage-controlled current source (VCCS), the test source method is necessary. Applying Vtest = 1V and solving the modified circuit yields Itest, from which Rth is derived.

$$ R_{th} = \frac{1V}{I_{test}} $$

This approach is indispensable in amplifier and transistor modeling, where dependent sources dominate circuit behavior.

3. Identifying the Load Resistor

3.1 Identifying the Load Resistor

In applying Thevenin's theorem, the load resistor RL represents the component or network segment whose behavior we wish to analyze when disconnected from the original circuit. Proper identification of RL is critical because it determines the boundary between the "source network" (to be Thevenized) and the external load.

Key Characteristics of the Load Resistor

Practical Identification Steps

  1. Determine the specific component or subcircuit whose voltage/current response needs evaluation.
  2. Disconnect RL from the network, leaving open terminals at the separation points.
  3. Verify that no dependent sources or control variables are affected by the removal.
$$ V_{Th} = V_{oc} \quad \text{(Open-circuit voltage at terminals } a-b) $$
$$ R_{Th} = \frac{V_{oc}}{I_{sc}} \quad \text{(Ratio of open-circuit voltage to short-circuit current)} $$

Common Pitfalls

Source Network RL Terminal a Terminal b

Advanced Considerations

For networks with dependent sources, the load must not interfere with control variables. In such cases, verify that:

$$ \frac{\partial V_{Th}}{\partial I_{sc}} = 0 $$

This ensures the source network's behavior remains invariant during Thevenin resistance calculation.

Removing the Load and Calculating Vth

To determine the Thevenin voltage (Vth), the first step involves removing the load resistor (RL) from the circuit. This isolates the network, allowing us to compute the open-circuit voltage across the terminals where the load was previously connected. The voltage measured at these terminals under no-load conditions is Vth.

Step-by-Step Derivation

Consider a linear circuit with independent and dependent sources. The procedure for calculating Vth is as follows:

  1. Disconnect the load resistor (RL) from the circuit.
  2. Analyze the remaining network to find the open-circuit voltage (Voc) across the load terminals. This voltage is equivalent to Vth.
  3. Apply standard circuit analysis techniques (e.g., nodal analysis, mesh analysis, or superposition) to solve for Voc.

Example Calculation

Assume a simple DC circuit with a voltage source (VS = 10 V) and two resistors (R1 = 2 Ω, R2 = 3 Ω) in series. The load resistor (RL = 5 Ω) is connected across R2.

  1. Remove RL to create an open circuit.
  2. The open-circuit voltage across R2 is found using voltage division:
$$ V_{th} = V_{oc} = V_S \cdot \frac{R_2}{R_1 + R_2} = 10 \cdot \frac{3}{2 + 3} = 6 \text{ V} $$

Handling Dependent Sources

If the circuit contains dependent sources, their influence must be accounted for when computing Vth. The analysis proceeds similarly, but the controlling variables must be expressed in terms of the open-circuit conditions.

Case Study: Circuit with a Current-Controlled Voltage Source (CCVS)

Given a network with a CCVS (V = kIx), where Ix is the controlling current:

  1. Disconnect RL.
  2. Solve for Ix under open-circuit conditions.
  3. Compute Vth by evaluating the voltage across the terminals, including the contribution from the dependent source.
$$ V_{th} = V_{oc} = V_{\text{independent}} + kI_x $$

Practical Considerations

In real-world applications, measuring Vth experimentally involves using a high-impedance voltmeter to approximate open-circuit conditions. The internal resistance of the meter must be significantly larger than the Thevenin resistance (Rth) to avoid loading effects.

For AC circuits, the same principle applies, but phasor analysis must be used to account for impedance and phase differences. The Thevenin voltage becomes a complex quantity:

$$ \tilde{V}_{th} = |\tilde{V}_{th}| \angle \theta $$

where θ is the phase angle relative to a reference signal.

3.3 Deactivating Sources to Find Rth

To determine the Thevenin equivalent resistance (Rth), all independent sources in the original circuit must be deactivated. This involves:

Dependent sources, however, remain active since their behavior is tied to other circuit variables. The resulting network, devoid of independent sources, allows Rth to be calculated using standard resistance combination techniques (series, parallel, or delta-wye transformations).

Mathematical Derivation

Consider a linear circuit with independent voltage and current sources. After deactivating all independent sources, the circuit reduces to a purely resistive network. The Thevenin resistance is then:

$$ R_{th} = \frac{V_{oc}}{I_{sc}} $$

where:

Alternatively, if the network contains no dependent sources, Rth can be found directly by computing the equivalent resistance between the two terminals.

Practical Example

Given a circuit with a 10V voltage source and a 2Ω resistor in series with a 3Ω resistor:

  1. Deactivate the voltage source (replace it with a short).
  2. The remaining resistors (2Ω and 3Ω) are in series.
  3. The Thevenin resistance is Rth = 2Ω + 3Ω = 5Ω.

Handling Dependent Sources

If dependent sources are present, deactivate only the independent sources and apply a test voltage Vtest or test current Itest at the terminals. Measure the resulting current or voltage and compute:

$$ R_{th} = \frac{V_{test}}{I_{response}} $$

This method ensures the dependent sources' contributions are accounted for in the equivalent resistance.

3.4 Constructing the Thevenin Equivalent Circuit

Once the Thevenin voltage (VTh) and Thevenin resistance (RTh) have been determined, the equivalent circuit can be constructed. The Thevenin equivalent simplifies a complex linear network into a single voltage source in series with a resistor, enabling efficient analysis of load behavior without solving the entire original circuit.

Step-by-Step Construction

1. Thevenin Voltage Source: The open-circuit voltage (VTh) becomes the independent voltage source in the equivalent circuit. If the original circuit contains dependent sources, ensure VTh accounts for their influence. For example, in a voltage divider:

$$ V_{Th} = V_{oc} = V_{source} \cdot \frac{R_2}{R_1 + R_2} $$

2. Thevenin Resistance: Place RTh in series with VTh. If the original network includes dependent sources, use the test voltage/current method to derive RTh:

$$ R_{Th} = \frac{V_{test}}{I_{test}} \quad \text{(with independent sources deactivated)} $$

Practical Considerations

Nonlinear Components: Thevenin’s theorem applies only to linear networks. Nonlinear elements (e.g., diodes, transistors) require piecewise-linear approximation or small-signal analysis.

Frequency-Dependent Networks: For AC circuits, replace resistances with impedances (ZTh) and use phasor representation for VTh.

Validation

Verify the equivalent circuit by comparing the load voltage/current with the original network. For a load resistor RL:

$$ I_L = \frac{V_{Th}}{R_{Th} + R_L} \quad \text{and} \quad V_L = I_L R_L $$

should match the values derived from the original circuit.

Example: Bridge Circuit

For a Wheatstone bridge with resistors R1–R4 and a voltage source VS:

  1. Calculate VTh as the voltage across the open load terminals.
  2. Deactivate VS (short-circuit) to find RTh as the parallel/series combination of resistors.
  3. Construct the equivalent circuit: VTh in series with RTh.
VTh RTh

4. Example 1: Simple Resistive Network

Thevenin's Theorem: Example 1 - Simple Resistive Network

Problem Statement

Consider a linear DC network consisting of two resistors, R1 = 4 Ω and R2 = 6 Ω, connected in series with a voltage source VS = 10 V. A load resistor RL = 5 Ω is connected across the terminals A and B. Determine the Thevenin equivalent circuit with respect to terminals A and B.

Step 1: Identify the Open-Circuit Voltage (VTH)

The Thevenin voltage VTH is the open-circuit voltage across terminals A and B. Since RL is removed, the circuit reduces to a voltage divider:

$$ V_{TH} = V_S \times \frac{R_2}{R_1 + R_2} $$

Substituting the given values:

$$ V_{TH} = 10 \times \frac{6}{4 + 6} = 6 \text{ V} $$

Step 2: Determine the Thevenin Resistance (RTH)

The Thevenin resistance is found by deactivating all independent sources (replacing the voltage source with a short circuit) and calculating the equivalent resistance seen from terminals A and B:

$$ R_{TH} = R_1 \parallel R_2 = \frac{R_1 \times R_2}{R_1 + R_2} $$

Substituting the values:

$$ R_{TH} = \frac{4 \times 6}{4 + 6} = 2.4 \text{ Ω} $$

Step 3: Construct the Thevenin Equivalent Circuit

The equivalent circuit now consists of VTH = 6 V in series with RTH = 2.4 Ω. The load RL = 5 Ω can be reconnected to terminals A and B to analyze the simplified network.

Step 4: Verify the Solution

To confirm correctness, compare the load current IL in both the original and Thevenin equivalent circuits.

Original Circuit: Total resistance Rtotal = R1 + R2 \parallel RL:

$$ R_{total} = 4 + \frac{6 \times 5}{6 + 5} = 4 + 2.727 = 6.727 \text{ Ω} $$

Total current from the source:

$$ I_{total} = \frac{10}{6.727} \approx 1.487 \text{ A} $$

Load current via current divider:

$$ I_L = I_{total} \times \frac{R_2}{R_2 + R_L} = 1.487 \times \frac{6}{11} \approx 0.811 \text{ A} $$

Thevenin Equivalent Circuit: Load current:

$$ I_L = \frac{V_{TH}}{R_{TH} + R_L} = \frac{6}{2.4 + 5} \approx 0.811 \text{ A} $$

The results match, validating the Thevenin equivalent.

Practical Implications

Thevenin's theorem simplifies complex networks into a single voltage source and series resistance, enabling rapid analysis of load variations. This is particularly useful in power systems, amplifier design, and sensor interfacing, where load conditions frequently change.

VS = 10V R1 = 4Ω R2 = 6Ω A B RL = 5Ω VTH = 6V RTH = 2.4Ω A B RL = 5Ω
Original Circuit and Thevenin Equivalent The diagram shows the original circuit with resistors and voltage source, and the Thevenin equivalent circuit with the simplified voltage source and resistance. Original Circuit VS 10V R1 4Ω R2 6Ω RL 5Ω A B Thevenin Equivalent VTH 6V RTH 2.4Ω A B
Diagram Description: The diagram shows the original circuit with resistors and voltage source, and the Thevenin equivalent circuit with the simplified voltage source and resistance.

Thevenin's Theorem – Example 2: Circuit with Dependent Sources

When analyzing circuits containing dependent sources, Thevenin's theorem remains applicable, but the methodology requires careful treatment of the dependent relationships. Unlike independent sources, dependent sources introduce additional constraints that must be incorporated into the analysis.

Problem Statement

Consider a linear circuit with a voltage-controlled current source (VCCS) as shown below:

VCCS: gVâ‚“ Vâ‚“

Step 1: Find Thevenin Equivalent Voltage (Vth)

To determine Vth, we compute the open-circuit voltage across terminals A and B. Since the dependent source remains active, we must express its controlling variable (Vâ‚“) in terms of the open-circuit conditions.

$$ V_{th} = V_{AB} \big|_{I_{load}=0} $$

For the given circuit, applying KVL and incorporating the VCCS relationship yields:

$$ V_{th} = V_s - I R_1 $$

where the current I is influenced by the dependent source gVâ‚“. Substituting the constraint Vâ‚“ = V_s - I R_1 leads to a solvable equation for Vth.

Step 2: Find Thevenin Equivalent Resistance (Rth)

For circuits with dependent sources, Rth cannot be found simply by deactivating independent sources. Instead, we apply a test voltage (or current) at the terminals and compute the resulting current (or voltage):

$$ R_{th} = \frac{V_{test}}{I_{test}} $$

Deactivate all independent sources, introduce Vtest, and analyze the circuit while preserving the dependent source relationships. The dependent source's contribution modifies the equivalent resistance, often resulting in non-intuitive values.

Step 3: Solve the Example Circuit

Assume the following parameters:

Applying nodal analysis with the dependent source constraint:

$$ I = gV_x = g(V_s - I R_1) $$

Solving for I:

$$ I = \frac{g V_s}{1 + g R_1} = \frac{0.005 \times 10}{1 + 0.005 \times 2000} \approx 0.00455 \text{ A} $$

The open-circuit voltage (Vth) is then:

$$ V_{th} = V_s - I R_1 = 10 - (0.00455 \times 2000) = 0.9 \text{ V} $$

For Rth, deactivate Vₛ and apply Vtest = 1V. The dependent source generates a current gVₓ, where Vₓ now equals -Itest R₁. The equivalent resistance becomes:

$$ R_{th} = \frac{V_{test}}{I_{test}} = \frac{R_1}{1 + g R_1} \approx 181.8 \text{ Ω} $$

Practical Implications

Dependent sources model active components like transistors and amplifiers. Thevenin equivalents of such circuits enable simplified analysis of loaded conditions, stability criteria, and frequency response without solving the full network repeatedly.

Circuit with Voltage-Controlled Current Source A schematic diagram illustrating a circuit with a voltage source, resistor, voltage-controlled current source (VCCS), and output terminals A and B. Vₛ R₁ gVₓ VCCS Vₓ A B
Diagram Description: The section involves a circuit with dependent sources and requires visualization of the VCCS and its relationship with other components.

4.3 Example 3: Complex Network Analysis

Consider a multi-loop resistive network with dependent and independent sources, as shown below:

Given the network parameters: V1 = 12V, V2 = 9V, R1 = 4Ω, R2 = 6Ω, R3 = 3Ω, and a current-controlled voltage source (CCVS) with gain coefficient k = 2Ω, we will determine the Thevenin equivalent between terminals A and B.

Step 1: Identify the Open-Circuit Voltage (Voc)

With terminals A-B open, we analyze the network using nodal analysis. The CCVS introduces the constraint Vx = 2I1, where I1 is the current through R1.

$$ \text{Node 1: } \frac{V_1 - V_a}{R_1} + \frac{V_2 - V_a}{R_2} = 0 $$
$$ \text{Constraint: } V_a - V_b = 2\left(\frac{V_1 - V_a}{R_1}\right) $$

Solving the system yields Va = 8.4V and Vb = 6V, giving Voc = Va - Vb = 2.4V.

Step 2: Determine the Thevenin Resistance (Rth)

To find Rth, we deactivate all independent sources (replace voltage sources with shorts) and apply a test voltage Vtest = 1V at terminals A-B:

$$ \text{Mesh 1: } (R_1 + R_2)I_1 - R_2I_2 = 0 $$
$$ \text{Mesh 2: } -R_2I_1 + (R_2 + R_3)I_2 = V_{test} - 2I_1 $$

The dependent source modifies the second mesh equation. Solving gives Itest = 0.3A, thus:

$$ R_{th} = \frac{V_{test}}{I_{test}} = 3.\overline{3}\Omega $$

Step 3: Construct the Thevenin Equivalent Circuit

The final equivalent circuit consists of Vth = 2.4V in series with Rth = 10/3Ω. This simplified representation accurately models the original network's behavior at terminals A-B.

Practical Considerations

When analyzing networks with dependent sources:

This approach extends to nonlinear networks when small-signal analysis is appropriate, with Vth representing the operating point and Rth the incremental resistance.

Complex Network with CCVS for Thevenin Analysis A detailed circuit schematic illustrating a complex multi-loop network with dependent sources (CCVS) and labeled components for Thevenin analysis. V1=12V V2=9V R1=4Ω R2=6Ω R3=3Ω 2I1 CCVS I1 I2 A B
Diagram Description: The section describes a complex multi-loop network with dependent sources and nodal/mesh analysis, where spatial relationships between components are critical.

5. Non-linear and Time-varying Circuits

Non-linear and Time-varying Circuits

Thevenin's Theorem, while fundamentally derived for linear time-invariant (LTI) circuits, encounters significant limitations when applied to non-linear or time-varying networks. In such systems, the superposition principle no longer holds, and the concept of a fixed Thevenin equivalent becomes ambiguous.

Non-linear Circuits

For non-linear circuits, the Thevenin equivalent resistance RTh and voltage VTh are not constant but depend on the operating point. Consider a diode-resistor circuit:

$$ V_{Th} = V_S - I_D R_S $$

where ID is the diode current, governed by the Shockley diode equation:

$$ I_D = I_S \left( e^{\frac{V_D}{n V_T}} - 1 \right) $$

Here, VTh becomes a function of VD, making the Thevenin equivalent valid only for small-signal perturbations around a bias point. The small-signal Thevenin equivalent can be derived by linearizing the circuit at the operating point, yielding:

$$ r_{Th} = \left. \frac{\partial V}{\partial I} \right|_{Q} $$

where Q is the quiescent point.

Time-varying Circuits

For time-varying components (e.g., circuits with switches or time-modulated elements), Thevenin's Theorem must be generalized using dynamic impedance concepts. The equivalent voltage VTh(t) and impedance ZTh(t) become functions of time. For a switched capacitor network, the Thevenin equivalent can be approximated in the Laplace domain:

$$ Z_{Th}(s) = \frac{1}{s C_{eq}} $$

where Ceq is the equivalent capacitance derived from charge conservation principles.

Practical Implications

In power electronics, non-linear loads (e.g., rectifiers) require piecewise Thevenin equivalents for different conduction intervals. Similarly, in RF circuits with time-varying components (e.g., mixers), the equivalent impedance must account for harmonic balance conditions. These extensions are critical for accurate modeling in:

Advanced techniques like harmonic balance or Volterra series expansions are often employed to approximate Thevenin equivalents in such scenarios.

Thevenin Equivalents for Non-linear and Time-varying Circuits A schematic diagram illustrating Thevenin equivalents for non-linear diode-resistor circuits and time-varying switched capacitor networks, including I-V curves and Laplace domain representations. V_Th R_Th Diode I_D V_D Q-point Shockley equation C R V_Th Switch Z_Th(s) C_eq = C/T T = switching period Thevenin Equivalents for Non-linear and Time-varying Circuits
Diagram Description: The section discusses non-linear diode-resistor circuits and time-varying switched capacitor networks, which require visual representation of component relationships and dynamic behavior.

5.2 Limitations with Dependent Sources

Thevenin's Theorem provides a powerful simplification for linear circuits with independent sources, but its application becomes nuanced when dependent (controlled) sources are present. Unlike independent sources, dependent sources introduce constraints that complicate the determination of Thevenin equivalent parameters.

Mathematical Constraints in Thevenin Equivalent Calculation

For a circuit containing dependent sources, the Thevenin voltage (VTh) and resistance (RTh) cannot be determined purely through source suppression. The presence of a dependent source means that RTh must be computed using either:

$$ R_{Th} = \frac{V_{test}}{I_{test}} $$

This differs from independent source cases, where RTh is found simply by deactivating all sources and computing the equivalent resistance.

Practical Challenges

Three key complications arise when applying Thevenin's Theorem to dependent-source circuits:

Example: Voltage-Controlled Voltage Source (VCVS)

Consider a circuit with a VCVS (μVx), where Vx is a voltage across an internal resistor. The Thevenin voltage includes contributions from both independent sources and the dependent term:

$$ V_{Th} = V_{OC} = V_{ind} + \mu V_x $$

Meanwhile, RTh must be computed by applying a test source while keeping the dependent source active. Attempting to suppress the dependent source leads to incorrect results.

Special Cases Where Thevenin's Theorem Fails

Two scenarios render Thevenin's Theorem inapplicable:

Workarounds and Alternatives

When Thevenin's Theorem is limited by dependent sources, engineers often resort to:

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Dependent Source in Thevenin Equivalent Circuit A schematic diagram showing a Thevenin equivalent circuit with a dependent voltage source (VCVS), resistors, and open-circuit terminals where V_Th is measured. V_ind R_Th μV_x V_OC Test voltage source V_x
Diagram Description: A diagram would show the arrangement of dependent sources in a circuit and how they interact with independent sources and resistors, which is complex to visualize from text alone.

5.3 Practical vs. Theoretical Constraints

Thevenin's Theorem provides a powerful theoretical framework for simplifying complex linear circuits into an equivalent voltage source and series resistance. However, real-world applications introduce constraints that deviate from idealized assumptions. Understanding these limitations is critical for accurate circuit analysis and design.

Nonlinear Components and Thevenin Equivalence

Thevenin's Theorem strictly applies to linear, time-invariant (LTI) networks. Practical circuits often include nonlinear elements (diodes, transistors, magnetic cores), where the theorem's assumptions break down. For weakly nonlinear systems, a small-signal approximation can extend Thevenin's validity around a DC operating point:

$$ R_{th} = \left. \frac{\partial V}{\partial I} \right|_{Q} $$

where \( Q \) is the quiescent point. Strong nonlinearities require piecewise-linear modeling or numerical simulation.

Frequency-Dependent Behavior

The classical Thevenin model assumes resistive networks, but real sources exhibit frequency-dependent impedance. For AC systems, the equivalent impedance \( Z_{th}(j\omega) \) must account for inductive/capacitive effects:

$$ Z_{th}(j\omega) = R_{th} + jX_{th}(\omega) $$

This complicates wideband analysis, as the Thevenin equivalent becomes valid only at a single frequency unless \( Z_{th} \) is purely resistive.

Measurement Uncertainties

Experimental determination of \( V_{th} \) and \( R_{th} \) faces practical hurdles:

A robust approach uses weighted least-squares fitting to multiple load conditions:

$$ \min_{V_{th},R_{th}} \sum_{i=1}^N w_i \left( V_{meas,i} - V_{th} \frac{R_L}{R_{th}+R_L} \right)^2 $$

Parasitic Elements

Stray capacitance (<5 pF) and lead inductance (<10 nH) become significant at high frequencies, introducing reactive components not captured in the DC Thevenin model. For a 50 Hz power system, these are negligible, but at 2.4 GHz (Wi-Fi), they dominate:

\( L_{lead} \) \( C_{stray} \)

Thermal and Aging Effects

Thevenin parameters drift with temperature and operational history. A lithium-ion battery's \( R_{th} \) increases by 30-50% over discharge cycles due to electrolyte degradation. Empirical models incorporate Arrhenius aging:

$$ R_{th}(t) = R_{0} \exp\left( \frac{E_a}{kT} \right) t^n $$

where \( E_a \) is activation energy and \( n \approx 0.5 \) for diffusion-limited processes.

Numerical Conditioning

Matrix methods for large networks (e.g., nodal analysis) suffer from ill-conditioning when \( R_{th} \) values span orders of magnitude (e.g., 1 mΩ power bus vs. 1 MΩ sensor). Double-precision arithmetic and preconditioning become essential.

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6. Recommended Textbooks

6.1 Recommended Textbooks

6.2 Online Resources and Tutorials

6.3 Research Papers and Advanced Readings