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# The Modes (and Nodes) of Circuit Analysis Methods

### Key Takeaways

• Circuit analysis begins with Ohm’s Law and Kirchoff’s Laws.

• Using Thevenin and Norton equivalent circuits and source transforms to measure response.

• The subject of superposition and linearity. Circuit analysis methods descend from nodal and mesh techniques

At the outset, looking at a schematic can be daunting: circuit diagrams contain an incredible amount of information in a few symbols and connections. While familiarity with various circuit topologies helps readers discern function, there’s always a time when designers will want to study a circuit’s operation more granularly. There are numerous tools available to perform these scans. Though simulation software will be able to analyze more rapidly and accurately than pen-and-paper calculations, understanding how these sophisticated models operate will impart a greater appreciation as well as a boost to creativity. Circuit analysis methods require some level of practice to gain familiarity, but they are an invaluable tool at any level of circuit design for a quick ballpark measurement check.

## Foundational Circuit Analysis Methods

Circuit analysis begins with a legendary equation that’s synonymous with the electromagnetism field itself: Ohm’s Law. Ohm’s Law relates the three fundamental parameters of passive and linear system analysis that govern basic electronic operations. While resistance is crucial, we’ll let its superficial role as a scaling factor give way to the proportional relationship between current and voltage. Current and voltage are intrinsically linked; one of the defining aspects of electromagnetism is its duality, and these two, respectively, represent the “individual” magnetic and electric contributions.

Kirchhoff’s laws follow Ohm’s Law as the bedrock of network analysis techniques. Unsurprisingly, Kirchhoff’s laws use either current or voltage as a definition and the other to manipulate in equations:

• Kirchhoff’s Junction Rule, also known as Kirchhoff’s First Law or Kirchhoff’s Current Law (KCL), states that in any node (a connection point between two or more components), all of the current flowing into the node must be equal to all of the current flowing out of the node. That is, all current branches flowing into and out of the node must sum to zero. A node connecting only two components is the trivial solution for KCL, as current in and current out are identical. Therefore, nodes with three or more components are known as essential nodes and the subject of nodal analysis.
• Kirchhoff’s Loop Law, also known as Kirchhoff’s Second Law or Kirchhoff’s Voltage Law (KVL), evaluates the sum of the voltages of a closed loop at zero. More plainly stated, the change in potential proceeding either clockwise or counterclockwise around a closed circuit and arriving at the same position is zero. Similar to the node/essential node dichotomy, KVL builds mesh currents from branches and essential branches of a circuit: the former connects any two or more consecutive nodes, while the latter starts and ends on an essential node without any interceding essential nodes.

These rules are widely used and highly useful, but it is valuable to understand their drawbacks. Both rules operate with the lumped-circuit element model, which ignores parasitics in favor of idealized components. For DC, this assumption is negligible, but AC frequencies upend the model due to flux leakage and evolving charge densities.

How can an engineer determine whether to apply KVL or KCL? Assuming equations are representative of the circuit and the algebra is sound, both formats will arrive at complementary answers (KCL yields voltage at a node, while KVL provides current through a branch). The motivating factor should be the fewest number of equations to solve the circuit unknowns. Not only is this more expedient than a method that requires additional equations to arrive at the same answer, but more equations also compound the probability of error when describing equations or transcribing them to a matrix solver. Determining the best course of action will come with experience, but identifying supernodes (an exclusive voltage source between adjacent nodes) and supermeshes (a current source between two adjacent nodes) is a great first step toward reducing a network’s complexity before deciding upon nodal or mesh analysis.

## Transforms Enable Ease of Analysis for Certain Circuit Classes

Although KCL and KVL are indispensable, circuits containing many discrete elements can quickly overwhelm the formulas – applying only Kirchoff’s laws on even a modest modern circuit would result in an innumerable amount of terms to keep track of without the assistance of computer simulation. For human use, a more elegant model is needed that can encapsulate the critical attributes of the circuit: enter equivalent circuits, which can simplify networks down to resistance and voltage or current, depending on the algorithm:

• Thévenin - A Thévenin equivalent circuit is defined by a voltage source, series resistance, and open circuit voltage.
• Mayer-Norton - A Mayer-Norton equivalent circuit is defined by a current source, parallel resistance, and open circuit voltage.

Similar to nodal/mesh analysis, Thévenin and Mayer-Norton equivalent circuits are complementary and formed based on source transformation; a voltage source with a series resistance can be substituted for a current source with a parallel resistance (and vice versa). Note that because of source transformation, the Thévenin resistance and Mayer-Norton resistance are identical. The ability to convert between sources can greatly aid in the ability to rapidly trivialize  circuits into two idealized values that capture the entirety of the circuit response as seen by the load.

Source transformation and Thévenin/Mayer-Norton equivalents are not deployable in all situations. Dependent voltage and current sources – which rely on voltage or current elsewhere in the circuit as an input – may preclude the use of one or more of the use cases outlined below:

1. Thévenin equivalent by way of open circuit voltage (typically using KCL) and short circuit current, where current is calculated when the output terminals are shorted. The Thévenin resistance can be calculated from the open circuit voltage and short circuit current. Cannot be performed when a circuit contains only dependent sources.

2. Source transformation to yield a Thévenin or Mayer-Norton equivalent circuit. Can only be performed with independent sources.

3. Replace all voltage sources with a short and all current sources with an open, then determine the Thévenin resistance by looking into the circuit from the terminals and combining series and parallel resistors. An equivalent circuit can be built by solving for the open circuit voltage or short circuit current. Can only be performed with independent sources.

4. Thévenin resistance can also be determined by applying the appropriate short/open to independent sources as outlined in 3 and plugging in a test voltage at the open terminals for circuits containing dependent sources. From there, the open circuit voltage or short circuit current can be obtained as outlined in 1.

## A Circuit’s Elements Constrain Analysis Methods

One of the keys to understanding circuit analysis methods is a bit more abstract, but an application has already been demonstrated in the prior section. The superposition principle, which arises due to linearity, states that independent sources can be replaced with an open (current) or short (voltage) to isolate the response of a single source. This can be done in sequence for all the remaining sources, and the individual contributions can be summed to find the total response of the circuit with all sources active simultaneously.

Importantly, the superposition principle only holds in linear circuits or circuits that exhibit additivity and homogeneity. Although these concepts serve to prove linearity, it is more straightforward to identify a circuit as linear or nonlinear by studying its components: resistors, capacitors, inductors, and op-amps form the bulk of the basis of linear circuits. To be clear, superposition only applies to idealized forms of the aforementioned elements, but approximations by hand need not be as exacting as more intensive computations for the sake of time and ease.

The benefits of linearity do not end there, however. Translations into the frequency domain, specifically Laplace and Fourier transforms, can replace differential equations and other relatively complex expressions with rudimentary algebraic operators. Nonlinear circuits, on the other hand, require mathematical models to act as approximations; although highly accurate, they are more demanding from a resource standpoint.

Support for High-Level Simulation and Analysis With Cadence

Aerospace analysis methods are vast: engineers should have an idea of which techniques are viable depending on the parameters of the circuit in question. Introductory topics like KCL/KVL and source transforms provide a basis for circuit analysis but may falter when dealing with more expansive designs. For the circuits better suited for exhaustive approaches, Cadence offers industry-leading PCB Design and Analysis Software that can easily integrate and progress from schematic to simulation. From there, the OrCAD PCB Designer offers an easy-to-use interface that expedites the layout process while keeping track of the board’s manufacturing intricacies.

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