Network Analysis or Circuit Analysis

Network Analysis determines voltages, currents and other responses in an electrical circuit model. A circuit element is connected to other elements in an electrical network, and an electrical circuit is analysed with its stated reference directions and operating conditions. Depending on topology, the circuit elements may be in series, in parallel or in a form that requires simultaneous equations. Common circuit elements include resistors, capacitors, inductors, voltage sources and current sources. Analysis can find Current, voltage, resistance, impedance, reactance, inductance, capacitance, frequency, electric power and electrical energy. The valid method depends on the topology, linearity, time model and DC or AC conditions. In short, network analysis applies element equations and Kirchhoff’s laws to solve the unknown response of an electrical network and its circuit elements.

Graph of an Electrical Network

A circuit graph of the network preserves connectivity: nodes become vertices and element connections become branches or edges. Figure 2 shows the graph corresponding to Figure 1.

Each branch represents a circuit element between two nodes. A reference arrow may be assigned arbitrarily to define positive branch current; a negative solution then means the actual current is opposite that reference. A graph with assigned branch directions is an oriented graph. Figure 3 is an oriented graph of Figure 1.

A topological graph normally retains every element branch, including branches containing voltage and current sources. An oriented topological graph adds branch reference directions while omitting component values. Replacing an independent voltage source by a short circuit and an independent current source by an open circuit is source deactivation for a particular linear-analysis step, not the definition of a topological graph.Figure 4 contains a voltage source and current source. If Figure 5 removes them, it should be read as a source-deactivated network for an equivalent-impedance calculation, not as the only possible oriented topological graph of Figure 4.

Definition of Terms used in Network Analysis

Branch

A branch is an edge of the circuit graph that joins two nodes and represents a circuit element or defined two-terminal subnetwork. Its reference current and voltage directions are assigned consistently.

Node

A node is a set of connected points treated as having the same potential in the ideal-wire model. Branch terminals meet at nodes; a node can join two branches, while an essential node joins three or more.

Subgraph

A subgraph consists of selected vertices and selected edges from the original graph, with every selected edge retaining its endpoints.

Tree

In connected-network analysis, a tree is a spanning, connected and cycle-free subgraph. It contains every node. A tree with n nodes has n − 1 branches, often called twigs. A graph can have more than one valid tree.

Cotree

For a chosen tree, the cotree is the set of graph branches not included in that tree. These remaining branches are often called links or chords. A cotree therefore depends on which spanning tree was selected.

Equivalent Circuit

Two circuits are equivalent at a designated port when they have the same terminal voltage-current relationship over the stated operating range. Equivalence is not a claim that every internal branch voltage and current remains unchanged. Series and parallel combinations, source transformations and Thévenin or Norton models can preserve port behaviour while replacing the internal network. For passive linear networks, the driving-point impedance seen at the selected terminals must match.

Series and Parallel Circuit

Series and parallel reduction is valid only when the topology satisfies the corresponding current or voltage condition.
For n resistances are connected in series, the equivalent resistance is:

For n resistances are connected in parallel, the reciprocal conductances add as shown:

For n uncoupled inductances are connected in series, the ideal equivalent inductance is:

For n uncoupled inductances are connected in parallel, the reciprocal expression is:

Mutual inductance changes these inductor results. For ideal capacitances are connected in series, the reciprocal expression is:

For ideal capacitances are connected in parallel, capacitances add:

For n impedances in series, the equivalent impedance is their sum:

For n nonzero impedances in parallel, the reciprocal admittances combine as shown:

Star Delta Transformation

A three-terminal network may contain a star or delta subnetwork that cannot be reduced by recognising a simple series or parallel pair. A delta connected network can then be converted to a terminal-equivalent star connected network, or vice versa. This transform either delta to star or star to delta preserves the impedance measured between each terminal pair for linear impedances. Let the star arms be Z_a, Z_b, and Z_c, and let the delta arms be Z_ab, Z_bc, and Z_ca. The following relationships make the two three-terminal networks equivalent.

Electrical Source Transformation

A source transformation replaces one linear two-terminal source model with another that has the same port voltage-current relation in the connected electrical network. A practical voltage source model with an ideal voltage source V in series with impedance Z is equivalent to an ideal current source I = V/Z in parallel with the same Z. The transformation requires a finite, defined impedance; an ideal voltage source alone cannot be converted to a finite current source. The Norton current equals the short-circuit current of the Thévenin form, and the Thévenin voltage equals the open-circuit voltage of the Norton form. Independent sources may be deactivated when finding an equivalent impedance, but dependent sources remain in the model and may require a test source.

Voltage Current Division Rule

The voltage and current division rule applies directly to series impedances or parallel admittances. The Voltage division rule gives the voltage across one impedance in a series chain when the same current passes through every element. Suppose impedances Z₁, Z₂, Z₃ …..Z_n are in series across V_s. The voltage across Z₁ is:
The illustration uses resistors, but the same phasor-domain relation applies to impedances at a common frequency.

The voltage across Z_i is:

For a current source I_s feeding parallel admittance branches Y₁, Y₂, Y₃ …Y_n:

The current through admittance Y₁ is:

The current through admittance Y_i is:

Nodal Analysis and Mesh Analysis

nodal analysis solves node voltages, while mesh analysis solves assigned mesh currents. Nodal analysis selects a reference node and applies Kirchhoff’s Current Law. With consistent current signs, the algebraic sum of currents at a node is zero. If a node has branch currents I₁, I₂, I₃ ….. I_n, then:

A mesh is a loop containing no other loop within it in a planar circuit. mesh analysis applies Kirchhoff’s voltage law around each mesh. Shared-branch current is the algebraic difference or sum of adjacent mesh currents, according to their assigned directions. Mesh analysis is limited to planar networks, while nodal analysis is more general. The algebraic sum of branch voltages around each closed path is zero by Kirchhoff’s Voltage Law. For branch voltages V₁, V₂, V₃, and V_n, this gives:

Superposition Theorem

The superposition theorem applies to linear circuits. To find a voltage or current response, retain one independent source at a time and set the other independent sources to zero: replace an ideal independent voltage source by a short circuit and an ideal independent current source by an open circuit. Keep all dependent sources active because their values remain tied to circuit variables. Calculate each signed voltage or current contribution, then add the contributions algebraically. Power is nonlinear in voltage or current, so powers found in the separate source cases must not be added as though they were superposed.

Norton Theorem

Norton Theorem replaces a linear two-terminal electrical network by an ideal current source in parallel with an equivalent impedance. The Norton current is the short-circuit current at the chosen port. The Norton impedance is the driving-point impedance with independent sources set to zero, or the port voltage-current ratio obtained with a test source when dependent sources are present. The model preserves the port behaviour for any connected load within the linear operating model. The equivalent named for Edward Lawry Norton is described in more detail under Norton Theorem.

Thevenin Theorem

Thevenin Theorem is the voltage-source form dual to Norton Theorem. It replaces a linear two-terminal network with an ideal voltage source in series with an equivalent impedance. The source voltage is the open-circuit port voltage, and the series impedance is the same driving-point impedance used in the Norton model. The equivalent associated with Leon Charles Thevenin simplifies the response seen by a load to a single voltage source and impedance; it does not reproduce internal branch variables.

Maximum Power Transfer Theorem

For a linear DC source network represented by a positive Thévenin resistance, the load receives maximum power when its resistance equals the Thévenin resistance. For sinusoidal steady-state AC with an adjustable complex load, the Maximum Power Transfer Theorem requires the load impedance to equal the complex conjugate of the source’s Thévenin impedance. This condition maximises load power, not efficiency: in the simple resistive match, equal power is dissipated in the source resistance and load, so efficiency is 50%. Power-delivery systems often choose a different load when efficiency or voltage regulation matters more than maximum load power.