Enhancing Circuit Analysis Through the Application of Source Transformation Techniques

The analysis of circuit diagrams constitutes an important part of electrical engineering. This discipline involves network methods which are accurate and easy to implement. One of the methods that can be employed in order to optimize the process is a source transformation technique. This Source transformation technique enables engineers to switch between voltage and current sources while retaining the overall networks function, and as such, makes complex circuits much easier to operate. With the help of source transformation, professionals are able to automate the sifting work, thereby leaving more time for the more sophisticated parts of system design. In this article, the author explains the principles of source transformation, offers a description of real life example applications, and provides instructions for incorporating this method into circuit analysis.

What is Source Transformation in Circuit Analysis?

This is a method that involves changing voltage sources with series resistances into equivalent current sources superimposed by parallel resistances, or into the other way around. The sources could be transformed because our reality is bound by the fact that both of these drawings give the same electrical signals and the result in the circuit remains the same. The transformation of the sources makes it easier to make a circuit design and also simplifies its analysis in theoretical and practical ways.

Understanding Circuit Components

The components of a circuit are its fundamental parts which together make up the entire electrical system, where each component is characteristic and specific in modifying or controlling the flow of electric power. Following are descriptions and other relevant information pertaining to the basic components of circuits:

Function: To restrict or control the passage of electric current.

Key Parameter: Resistance (R) in Ohms (Ω).

Data Example: In a circuit, a resistor having a specified resistance of 1 kΩ or 1000 Ω causes the current to be in accordance with Ohm’s Law (I = V/R).

Function: Store electrical energy temporarily for later use.

Key Parameters: Capacitance (C) in Farads (F) and voltage rating.

Data Example: The total capacitance of 30 µF is achieved by adding a 20 µF capacitor in parallel with a 10 µF (microfarad) capacitor.

Function: Store energy in a magnetic field while opposing current changes.

Key Parameters: Current rating (I) and inductance (L) measured in Henrys (H).

Data Example: In an RL circuit with a 2 H inductor, the time constant (τ = L/R) is determined when R is known.

Function: Block current flowing in one direction while allowing it to pass in the opposite direction.

Key Parameters: Maximum current rating and forward voltage drop (Vf).

Data Example: The forward voltage drop of silicon diode is around 0.7 V.

Function: Amplify or switch electronic signals.

Key Parameters: Maximum current rating, collector-emitter voltage (V_CE), and current gain (h_FE).

Data Example: Common NPN transistor can have 500 mA maximum collector current with 100 current gain.

Fundamentals of Current Source And Voltage Source

In electronics, a current source is a device that maintains a constant current in a circuit, irrespective of the voltage drop at its terminals. Current sources are preferred in applications that require current to remain stable, such as LED drive circuits or current biasing of transistors. In theory, an ideal current source has infinite output impedance which means it can maintain a given current regardless of how much load is placed on it.

Batteries and regulated power supplies are examples of voltage sources that provide a certain voltage, no matter what load is connected to their terminals. Such sources may be termed ideal when their internal impedance is said to be zero which allows the voltage to be maintained, irrespective of the current supplied to the load.

Eager comprehension of the relations between current, voltage, and impedance is essential in developing efficient circuits. These sources are commonly modeled as more simple systems so that real life computations, predictions, and observations may be performed under more simplistic and ideal assumptions.

The Purpose of Source Transformation in Electrical Circuits

In electrical engineering, source transformation is a common technique that is useful for circuit analysis and design. This technique switches between voltage and current sources while maintaining the system’s overall response and features. The advantage of source transformation is that it simplifies calculations and reduces the complexity of circuit networks.

A voltage source with an internal series resistance, Rs, can be converted to an equivalent current source. The value of the current in the equivalent current source, Is, is defined as

Is = Vs/Rs

where \( Vs \) is the voltage of the original voltage source, and Rs is the internal resistance of the voltage source.

The transformed current source will be in parallel with a resistor R s, which is equal to the series resistance of the previous voltage source.

Conversely, a current source with an internal parallel resistance, Rp, can be transformed into an equivalent voltage source. The new voltage source’s voltage, Vs, is defined as

Vs = Is*Rp

where Is is the current of the original current source, and Rp is the internal resistance of the current source.

The matching value of parallel resistance to the primary current source will now be designated as the series resistance \( R_p \) for the equivalent voltage source.

Let us assume the voltage source to be \( V_s = 12\ \text{V} \) with a series resistance of \( R_s = 4\ \Omega \). According to the transformation formula,

I_s = \frac{V_s}{R_s} = \frac{12}{4} = 3\ \text{A}.

An equivalent current source can thus be defined with a current \( I_s=3 \text{A}\) and a parallel resistance \( R_p = 4 \:\Omega\) .

The following can also be transformed: a current source of \( I_s = 5\text{A}\) with a parallel resistance of \( R_p = 8\ \Omega\).

V_s = I_s \cdot R_p = 5 \cdot 8 = 40\ \text{V}.

The voltage of the equivalent voltage source is \(40\ \text{V}\) while its series resistance becomes \( R_s = 8 \:\Omega \) .

How to Perform a Voltage Source Transformation?

Process for Changing a Voltage Source into a Current Source

Recognize the Voltage Source: Start with a voltage source that has a voltage, \( V_s \), and a series resistance, \( R_s \).

Find the Current: Calculate the current from the output of the equivalent current source with the formula:

I_s = \frac{V_s}{R_s}

Employ the Parallel Resistance: The series resistance \( R_s \) is converted to parallel resistance \( R_p \) of the equivalent current source.

Construct the Current Source: Describe the transformed current source in terms of \( I_s \) and \( R_p \) where \( R_p = R_s \). Thus the circuit will be similar from a load standpoint.

The as-a-planes methodology can be particularly beneficial for the simplification of circuits and for the analysis of networks. This method allows any voltage source to be transformed into a current source without changing the load at pole being attached to the pole.

Example of Voltage Source Transformation

Think of a voltage source with \(V_s =10 V\) and a series resistance of \(R_s= 5\Omega\). To effect a transformation into an equivalent current source:

Calculate the Current Source (\( I_s \)):

The current source is computed by means of the formula: a voltage of the source divided by the series resistance, therefore,

\(I_s = \frac{V_s}{R_s} = \frac{10 V}{5 \Omega}=2 A\)

Determine the Parallel Resistance (\( R_p \)):

The parallel resistance is defined as the series resistance, therefore, \(R_p=5 \Omega \).

The equivalent current source is thus represented as \(I_s=2 A\) in parallel with \(R_p= 5\Omega\).

Connecting a load \(R_L = 10 \Omega \) to the circuit yields:

For the voltage source,

\begin{equation}

I_{load} = \frac{V_s}{R_s + R_L} = \frac{10 V}{5 \Omega + 10 \Omega} = 0.667 A

\end{equation}

And using the current division rule,

\begin{equation}

I_{load}=I_s \cdot \frac{R_p}{R_p + R_L} = 2 A \cdot \frac{5 \Omega}{5 \Omega + 10 \Omega} = 0.667 \, A

\end{equation}

This verifies that the transformed current source behaves the same way as the original voltage source with respect to the load, so they are equivalent.

Common Mistakes in Voltage Source Transformation

Overlooking the correct proportionality of a voltage source with series resistance to an equivalent current source with parallel resistance is one of the mistakes done mathematically transforming voltage sources. It is important to remember that Ohm’s law and principles of circuit transformation need to be rigorously followed for the ‘equivalence’ to be obtained. Another common mistake is the misestimation of the load voltage or current due to incorrect combinations of resistances used in the analysis. Moreover, not paying attention to mismatches of parallel resistance when applying the current division rule can cause errors in the way current is allocated among the branches of the circuit. Careful calculation and checking with the original circuit behavior helps in avoiding these errors.

How to Execute a Current Source Transformation?

Transforming a Current Source into a Voltage Source

For a current source to be transformed into an equivalent voltage source, consider performing the following steps accurately.

Identify The Source Values:

Identify the current source \( I_s \) and the associated parallel resistance \( R_p \) of the current source.

Apply The Transformation:

Transforms the current source into a voltage source, where \(I_s\) is changed to \(V_s\) using the formula below.

V_s = I_s \times R_p

Whatever the determined value of \( V_s \), that will serve as voltage of an equivalent voltage source. The resistance \( R_p \) will be remaining constant and will be placed in series with the voltage source.

Validate The Transformation:

To confirm the transformation was done accurately, it is required to check whether the voltage source and series resistance produces the same current, voltage values across the load as the initial current source with parallel resistance.

As an example, let us consider a parallel resistance of \( R_p = 10 \Omega \) for the current source \( I_s = 2 A \):

The value for the ideal voltage source becomes:

\[ V_s = 2 A * 10 \Omega = 20 V. \]

The circuit diagram is reconfigured into a voltage source of twenty volts in series with a resistor of 10 ohms.

This shows that the simplification was done without loosing the essential features of the circuit, making it possible for the latter to be analyzed or simplified more readily. In any case, always check with real circuit operation for correct performance.

Example for Current Source Transformation

Let us take a practical case for applying the current source transformation to make the circuit easier to analyze. Assume that we have a current source \( I_s = 3 \, A\) which is in parallel with a resistor \( R_p = 15 \, \Omega \). We want to transform this into an equivalent voltage source.

First, let’s calculate the equivalent voltage \( V_s \).

V_s= I_s \times R_p = 3 \, A \times 15 \, \Omega = 45 \, V

Now the circuit consists of:

A voltage source \( 45 \, V\) in series with a resistor \( 15 \, \Omega \).

This change enables us to process the circuit with simplifed components without changing the electrical parameters of the circuit. As an example, for an external load of \(R_L = 30 \,\Omega\), the total resistance of the circuit can be calculated as follows:

R_{total} = R_p + R_L = 15 + 30 = 45 \, \Omega

The total current in the circuit can then be calculated using Ohm’s Law:

I_{total} = \frac{V_s}{R_{total}} = \frac{45 V}{45 \Omega} = 1 A

This simple example illustrates how current source transformation simplifies the analysis of electrical circuits while ensuring the accuracy of how the circuit behaves electrically.

Problems Related to Current Source Transformation

One of the most severe problems in current source transformation is taking into consideration all the non-ideal effects from the real components in the circuit. For instance, voltage sources in the real world are often accompanied by a source resistance (R_s). On the other hand, current sources may have a limited compliance voltage range. If these non-idealities are not properly set, the accuracy of the transformation is concerned.

To understand, let us analyze a practical current source with a internal resistance of \(R_i=5 \, \Omega \) and compliance voltage of \(\pm50 \, V \). After transforming the source into an equivalent voltage source, the voltage across its terminals becomes: \[V_s=I_s\cdot R_i\]. \\ For a values of source current \(I_s=2 \, A\) the resulting voltage source parameters are: \\ \[V_s = 2 \, A \cdot 5 \, \Omega = 10 \, V \] and \[R_s=5 \, \Omega.\] \\ However, when the equivalent circuit is applied to a connected load of high resistance say \( R_L=100 \, \Omega \), then the output voltage comes close to the compliance limit. \\ \[V_{out}=I_s\cdot R_L=2 \, A \cdot 100 \, \Omega = 200 \, V \] \\ This situation shows how real limits of a component need to be taken into consideration – in most cases, they require some adjustments or even a redesign in the analysis of circuits. Including this additional information on limits into the analysis will allow engineers to create systems within the expected limits.

How to Analyze Circuits Using Source Transformation?

Methods of Easing the Analysis Of Complicated Circuits

The transformation of sources greatly aids the analysis of a circuit as it simplifies it to a level where the required values of voltage, current, or resistance can be defined easily. This technique includes the transformation of a voltage source in series with a resistor to an equivalent current source in parallel with the same resistor and vice versa, or the other way around. This process is determined by Ohm’s law, which states:

From Voltage Source to Current Source: \( I = \frac{V}{R} \)

From Current Source to Voltage Source: \( V = I \cdot R \)

This conversion allows engineers to represent various intricate components of a circuit in a single simplified form. For instance, many engineers would be familiar with systems having a great number of sources and resistors over such systems, they can be simplified step by step. Such systems may be further interlinked and to achieve a straight forward independent solution, they need to be transformed further. But the transformed circuit should perform the same function as the original circuit in terms of input and output for the results to be accurate.

Role of Equivalent Circuit in Analysis

An electrical network can be simplified to its most basic elements while still embodying its key electrical features, and this is referred to as equivalent circuit. It is concerned with general behavioral pattern of the circuit rather than its specific details. There are several approaches used for bypassing the complexities presented by real electrical circuits, with the most known ones being Norton’s and Thevenin’s theorems, which are used widely in electrical engineering.

Thevenin’s theorem states that any circuit consisting of independent and dependent sources, as well as ohmic resistors, can be simplified to a single voltage source (Vₜ), in series with an equivalent resistance (Rₜ). The steps for applying Thevenin’s theorem are as follows:

Define the Boundary of the Segment of Interest – Determine the section of the circuit you wish to derive an equivalent for.

Determining Open-Circuit Voltage (Vₒ) – The value of the voltage at the terminals is the value at open circuit where load is not connected is established.

Determine and Calculate the Thevenin Resistance (Rₜ) – All independent voltage sources are replaced with short circuits and all current sources are open circuited, the remaining resistance at the terminals is Rₜ and it is called the Thevenin Resistance.

Construct the circuit using given components – Build the constructed Thevenin equivalent circuit from the Vₜand Rₜ .

Analyze the following circuit parameters and values:

Source Voltage(Vs) = 12V

Resistor: R1 = 4Ω R2 = 6Ω R3 = 2Ω

Load Resistance (Rl) = 8Ω

Procedure for deriving the thevenin equivalent:

Employing the voltage division on the remaining resistors steps,

\( Vₒ = \frac{R₃}{R₁ + R₂ + R₃} \cdot V_s = \frac{2}{4 + 6 + 2} \cdot 12 = 2V \).

With Vs deactivated (short cicuit), thus, Rt = R1 || (R2 + R3).

Compute parallel resistance,

\( R_T = R_1 || (R_2 + R_3) = 4 || (6 + 2) = \frac{1}{\frac{1}{4} + \frac{1}{8}} = 2.67\Omega \).

The Thevenin equivalent circuit is therefore a 2V voltage source in series with a 2.67 resistor.

This method provides simpler ways of analyzing circuits especially in examining the effect of different load resistances. Precision in the application of such theorems guarantees reliability in the design and diagnosis in electrical systems.

Effect of Source Transformation on Circuit Analysis

By using source transformation, a voltage source in series with a resistor can be analyzed as an equivalent current source in parallel to the same resistor, or the other way around. This simplification makes analysis easier and ensures the circuit’s performance from the perspective of external components remains unchanged. This method is of great help when attempting to compute complex networks or while trying to determine equivalent circuits in a orderly fashion.

What are the Advantages of Source Transformation in Electrical Circuits?

Gains from Altering Circuit Reasoning

The conversion of sources has some measurable gains in the designing and analysis of electrical circuits:

Reduced Effort: The transformation of sources in their equivalent form gives rise to a simpler circuit. For example, a complex network having numerous voltage and current sources is often reduced to an equivalent simple circuit with the help of this technique.

Simplified Parameter Determination: Using source transformations, finding pegged parameters like load current or voltage is much easier. For instance, a 10V voltage source in series with a 5 ohm resistor can be transformed into a 2 A current source in parallel with the same resistor, without changing the outcome.

Greater Efficiency in Large Circuits: When analyzing large circuits, this technique remarkably decreases the time spent computing. For example, it can be very helpful in circuits with many meshes or nodes, source transformation can reduce the number of equations that need to be solved.

Continued accuracy: Accuracy is guaranteed as long as equivalence is maintained, no matter what form of the circuit is used. Transformation provides assurance that the external behavior of the circuit will not be altered, therefore precise calculations can be made.

These transformations have been passed on from one engineer to another, so reliability in system performance and design efficiency can be guaranteed.

Efficiency in Solving Complex Circuit Problems

The analysis of electrical circuits is made easier through source transformation which involves changing the voltage sources into current sources and the other way around. This strategy is effective in lowering the difficulty of circuits and lets engineers devise solutions to circuit behavior with ease. This method guarantees that the performance metrics of the circuit, like the voltage and current at different nodes, are precise and reliable, making the approach useful in theoretical and practical cases. This method is well-known for its capability of reducing calculations in theory and practical situations.

Comparison with Other Methods of Circuit Analysis

In specific cases, source transformation stands out from other circuit analysis techniques like nodal analysis, mesh analysis, and Thevenin/Norton equivalent circuits. Unlike nodal and mesh analyses that are systematic approaches needing the physician to solve sets of simultaneous equations of the whole network, source transformation deals with the simplification of a localized portions of a circuit. Source transformation works faster than Thevenin or Norton equivalents in converting and not deriving fully the equivalent source impedance. Source transformation becomes the best approach in circuits that are repeatedly configured, thereby saving time and improving accuracy during the design and testing phases.

Can Source Transformation Handle Parallel and Series Configurations?

Transforming Sources in Parallel Circuits

Source transformation can be employed in parallel configurations of circuits by transforming voltage sources in parallel with resistances into equivalent current sources or the other way round. As an illustration, let us for the time being ignore a parallel resistance R with voltage source Vs. We can later transform this combination into a current source Is that is parallel with R and where, IL = P / R.

Example Calculation:

Voltage source (\( V_s \)): 12V

Parallel resistor (\( R \)): 4Ω

Transforming this configuration:

\[ I_s = \frac{V_s}{R} = \frac{12}{4} = 3A \]

The derived circuit contains a 3A current source in parallel with 4Ω resistor, thus transforming the sources. This is particularly useful for analysis when attempting to solve larger systems that contain multiple parallel configurations, since it simplifies network analysis.

Other Advantages:

Parallel Network Simplification: Mitigates complex connections to provide simple analysis.

Efficient Computed Transformed: Eliminates solving numerous equations, since transformations are based on equivalent relationships.

Stepwise Facilitation of Circuit Analysis: Single aids facilitates reduction of mixed configurations into simpler forms, one step at a time.

Transforming sources in parallel circuits is a good demonstration of practicality when one seeks to simplify the design into more manageable equivalent circuits without altering the functional behavior of the network.

Source Transformation with Voltage Source in Series

As a voltage source is said to be in series with a resistor, it can be transformed into a current source in parallel with a resistor. The current source can also be considered as a voltage source in series with a resistor. The governing relationships are as follows:

Voltage to Current Conversion:

The current of the equivalent current source is calculated using:

\[ I_s = \frac{V_s}{R} \]

where \( V_s \) is the voltage of the source and R is the resistance.

Retaining Functional Equivalence:

The parallel resistor in the transformed circuit retains the same resistance value \( R \) as the series combination.

For series circuits that have an embedded parallel connection, the analysis can be made easier using source or circuit transformation. In the example above, a 24V voltage source is in series with an 8\(\Omega\) resistor. So using the above formula:

I_s = \frac{V_s}{R} = \frac{24}{8} = 3A

So now the circuit can be drawn as a parallel configuration with a 3A current source in parallel with the 8 ohm resistor. This is also termed as functional equivalence.

By applying such transformations it is possible to approach the problem of solving circuits of mixed configurations where series and parallel arrangements are combined and which are otherwise complex in nature and this simplification is really very helpful for performing the analysis and design in electrical engineering.

Handling Complicated Circuit Arrangements

Source transformations is a technique for the simplification of circuits with a complicated configuration. In this method, the sources in question: voltage or current get transformed into the other type. This type of transforming helps in cases which require studying circuits with complex interrelations. With the use of ‘equivalent’ representations of the circuit elements noted above, one can derive: \( V_s = I_s \cdot R \) or \( I_s = \frac{V_s}{R} \). These transformations enable the analyst to estimate the total current more easily, as well as calculating drops in voltage and power dissipation in different parts of the circuit.

Frequently Asked Questions (FAQs)

Q: What is source transformation in the context of circuit analysis?

A: Source transformation pertains to the process of simplifying circuits by rearranging elements within a circuit. For example, the method involves the conversion of a voltage with an accompanying series resistance into a current that has a parallel resistance, and the opposite is true. The combination of sources can thus be substituted with a simplified source and the circuit’s behavior will not be altered.

Q: How do you convert a voltage source to a current source?

A: In order to convert a voltage source to a current source, take the voltage from the source and divide it by the resistor in series with the source. The value obtained is the value of the current transformer. A resistor is then placed in parallel with the current source.

Q: Can a current source be transformed into a voltage source?

A: Indeed, a current source is transformed into a voltage source when the current is multiplied by the value of the resistor which is in parallel with the current source. This gives a source of voltage in series with the resistance.

Q: Why is source transformation useful in circuit analysis?

A: Source transformation is beneficial because it assists in simplifying complicated circuits by enabling the user to toggle between voltage and current sources. This further enhances analysis regarding how current and voltage behave in different areas of the circuit.

Q: What are the limitations of using source transformation in circuits?

A: Circuits which contain dependent sources or non-linear elements are not suitable for source transformation. This process is most often used with linear, resistive circuits which have ideal voltage and current sources.

Q: In what ways does source transformation impact the analysis of parallel circuits that have a current source?

A: By using source transformation, the analysis of parallel circuits with a current source can be simplified by replacing the current source with a voltage source while keeping the resistor in series which makes it possible to use mesh or nodal analysis techniques.

Q: Are there any possibilities for employing source transformation in circuits with dependent sources?

A: Source transformations are typically applied to independent sources only. While one can apply it to dependent sources in theory, it is far more complicated and is usually best left out when attempting to analyze the circuit.

Q: In what way do ideal and real current sources differ regarding source transformation integration?

A: An ideal current source will always have a constant current flowing through it regardless of the voltage it is connected to, while real current sources will have an element of internal resistance which will affect the output current. Source transformation is made under ideal conditions which assists in the conversion.

Q: Is source transformation applicable to circuits with several sources?

A: Certainly, source transformation is applicable to circuitry with several voltage or current sources, particularly when combined with other methods such as series-parallel reduction to eliminate some sources and lessens the complexity in the circuit.

Reference Sources

1. Title: A 580 nW Dual-Input Energy Harvester IC Using Multi-Task MPPT and a Current Boost Converter for Heterogeneous Source Combining
   • Authors: Seneke Chamith Chandrarathna, Jong‐Wook Lee
   • Journal: IEEE Transactions on Circuits and Systems Part 1: Regular Papers
   • Publication Date: December 1, 2020
   • Citation Token: (Chandrarathna & Lee, 2020, pp. 5650–5663)
   • Summary:
     • This paper presents a multi-energy harvesting interface integrated circuit (IC) designed to efficiently combine heterogeneous power sources, specifically an electromagnetic vibration energy generator (EVG) and a thermoelectric energy generator (TEG).
     • The authors propose a current boost converter (CBC) that facilitates resistance transformation and source type conversion, addressing the challenges of combining sources with significantly different internal resistances.
     • A multi-task maximum power point tracking (MPPT) controller is introduced, which is shared between the two sources to optimize energy harvesting under varying conditions.
     • The IC was fabricated using a 0.18 μm process, achieving an output power of up to 119 μW with a peak efficiency of 82% while consuming only 582 nW.
2. Title: Norton-Transformed Wideband High-Order Analog All-Pass Delays
   • Authors: Z. Kabirkhoo, M. Radpour, Leonid Belostotski
   • Journal: IEEE Transactions on Circuits and Systems – II – Express Briefs
   • Publication Date: August 1, 2024
   • Citation Token: (Kabirkhoo et al., 2024, pp. 3705–3709)
   • Summary:
     • This study introduces a novel compact wideband high-order all-pass filter (APF) topology for use as an analog delay circuit, which avoids conventional cascading that typically degrades bandwidth and increases power consumption.
     • The proposed topology consists of a common-source transistor and a passive impedance network, with the order of the filter increased through passive-impedance-network extension.
     • Experimental results demonstrate a single-stage 5th-order APF with a delay of 75 ps and a 10% bandwidth from 0 to 16.4 GHz, showcasing the effectiveness of the design in achieving high performance with reduced circuit area.
3. Title: Source Transformation Theorem Revisited
   • Authors: T. Rathore
   • Journal: IETE Journal of Education
   • Publication Date: January 1, 2008
   • Citation Token: (Rathore, 2008, pp. 13–17)
   • Summary:
     • This paper revisits the source transformation theorem, which states that a parallel combination of a current source and a resistance can be replaced by a series combination of a voltage source and a resistance, and vice versa.
     • The author discusses the existence of numerous transformations beyond the basic current-to-voltage and voltage-to-current conversions, suggesting a broader application of the theorem in circuit analysis.

Series and parallel circuits

Current source