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{{short description|Determining all voltages and currents within an electrical network}}
{{redirect|Circuit theory||Circuit (disambiguation)}}
{{Network analysis navigation}}
In [[electrical engineering]] and [[electronics]], a ''[[Electrical network|network]]'' is a collection of interconnected [[Electronic component|components]]. '''Network analysis''' is the process of finding the [[voltage]]s across, and the [[Electric current|current]]s through, all network components. There are many techniques for calculating these values; however, for the most part, the techniques assume [[Linear circuit|linear]] components. Except where stated, the methods described in this article are applicable only to ''linear'' network analysis.
==Definitions==
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|'''Branch'''||The component(s) joining two nodes.
|-valign=top
|'''[[Mesh analysis|Mesh]]'''||A group of branches within a network joined so as to form a complete loop such that there is no other loop inside it
|-valign=top
|'''[[Port (circuit theory)|Port]]'''||Two terminals where the current into one is identical to the current out of the other.
|-valign=top
|'''[[Electrical circuit|Circuit]]'''||A current from one terminal of a [[generator (circuit theory)|generator]], through load component(s) and back into the other terminal. A circuit is, in this sense, a one-port network and is a trivial case to analyse. If there is any connection to any other circuits then a non-trivial network has been formed and at least two ports must exist. Often, "circuit" and "network" are used interchangeably, but many analysts reserve "network" to mean an idealised model consisting of ideal components.<ref>{{cite journal |author=Belevitch V |title=Summary of the history of circuit theory |journal=Proceedings of the IRE |volume=50 |issue=5 |pages=849 |date=May 1962 |doi=10.1109/JRPROC.1962.288301 |s2cid=51666316 |author-link=Vitold Belevitch }} cites {{cite journal |title=IRE Standards on Circuits: Definitions of Terms for Linear Passive Reciprocal Time Invariant Networks, 1960 |journal=Proceedings of the IRE |volume=48 |issue=9 |pages=1609 |date=September 1960 |doi=10.1109/JRPROC.1960.287676 }}to justify this definition.<br />[[Sidney Darlington]] {{cite journal |author=Darlington S |title=A history of network synthesis and filter theory for circuits composed of resistors, inductors, and capacitors |journal= IEEE
|-valign=top
|'''[[Transfer function]]'''||The relationship of the currents and/or voltages between two ports. Most often, an input port and an output port are discussed and the transfer function is described as gain or attenuation.
|-valign=top
|'''Component transfer function'''||For a two-terminal component (i.e. one-port component), the current and voltage are taken as the input and output and the transfer function will have units of impedance or admittance (it is usually a matter of arbitrary convenience whether voltage or current is considered the input).
|}
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[[Image:circuit equivalence.png|200px|right]]
{{main|Equivalent impedance transforms}}
A useful procedure in network analysis is to simplify the network by reducing the number of components. This can be done by replacing physical components with other notional components that have the same effect. A particular technique might directly reduce the number of components, for instance by combining impedances in series. On the other hand, it might merely change the form into one in which the components can be reduced in a later operation. For instance, one might transform a voltage generator into a current generator using Norton's theorem in order to be able to later combine the internal resistance of the generator with a parallel impedance load.
A [[resistive circuit]] is a circuit containing only [[resistors]], ideal [[current source]]s, and ideal [[voltage source]]s. If the sources are constant ([[Direct current|DC]]) sources, the result is a [[direct current circuit|DC circuit]]. Analysis of a circuit consists of solving for the voltages and currents present in the circuit. The solution principles outlined here also apply to [[phasor (electronics)|phasor]] analysis of [[#AC circuits|AC circuits]].
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Two circuits are said to be '''equivalent''' with respect to a pair of terminals if the [[voltage]] across the terminals and [[Current (electricity)|current]] through the terminals for one network have the same relationship as the voltage and current at the terminals of the other network.
If <math>V_2=V_1</math> implies <math>I_2=I_1</math> for all (real) values of
The above is a sufficient definition for a [[one-port]] network. For more than one port, then it must be defined that the currents and voltages between all pairs of corresponding ports must bear the same relationship. For instance, star and delta networks are effectively three port networks and hence require three simultaneous equations to fully specify their equivalence.
===Impedances in series and in parallel===
{{main|Series and parallel circuits}}
Some two terminal network of impedances can eventually be reduced to a single impedance by successive applications of impedances in series or impedances in parallel.
*Impedances in [[Series and parallel circuits#Series circuits|series]]: <math display="block">Z_\mathrm{eq} = Z_1 + Z_2 + \,\cdots\, + Z_n .</math>
*Impedances in [[Series and parallel circuits#Parallel circuits|parallel]]: <math display="block">\frac{1}{Z_\mathrm{eq}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \,\cdots\, + \frac{1}{Z_n} .</math>
===Delta-wye transformation===
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[[Image:Delta-Star Transformation.svg|right|400px]]
A network of impedances with more than two terminals cannot be reduced to a single impedance equivalent circuit. An {{mvar|n}}-terminal network can, at best, be reduced to
For equivalence, the impedances between any pair of terminals must be the same for both networks, resulting in a set of three simultaneous equations. The equations below are expressed as resistances but apply equally to the general case with impedances.
====Delta-to-star transformation equations====
:<math>\begin{align}
R_a &= \frac{R_\mathrm{ac}R_\mathrm{ab}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} \\
R_c &= \frac{R_\mathrm{bc}R_\mathrm{ac}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}}
\end{align}</math>
====Star-to-delta transformation equations====
:<math>\begin{align}
R_\mathrm{ac} &= \frac{R_a R_b + R_b R_c + R_c R_a}{R_b} \\
R_\mathrm{bc} &= \frac{R_a R_b + R_b R_c + R_c R_a}{R_a}
\end{align}</math>
===General form of network node elimination===
{{main|Star-mesh transform}}
The star-to-delta and series-resistor transformations are special cases of the general resistor network node elimination algorithm. Any node connected by
:<math>R_\mathrm{xy} =
For a star-to-delta (
:<math>\begin{align}
R_\mathrm{ab} &= R_a R_b \left(\frac 1 R_a+\frac 1 R_b+\frac 1 R_c\right) = \frac{R_a R_b(R_a R_b + R_a R_c + R_b R_c)}{R_a R_b R_c} \\
&= \frac{R_a R_b + R_b R_c + R_c R_a}{R_c}
\end{align}</math>
For a series reduction ({{math|1=''N'' = 2}}) this reduces to:
:<math>R_\mathrm{ab} = R_a R_b \left(\frac 1 R_a+\frac 1 R_b\right) = \frac{R_a R_b(R_a + R_b)}{R_a R_b} = R_a + R_b</math>
For a dangling resistor ({{math|1=''N'' = 1}}) it results in the elimination of the resistor because <math>\tbinom{1}{2} = 0</math>.
===Source transformation===
[[Image:Sourcetransform.svg|thumb]]
A generator with an internal impedance (i.e. non-ideal generator) can be represented as either an ideal voltage generator or an ideal current generator plus the impedance. These two forms are equivalent and the transformations are given below. If the two networks are equivalent with respect to terminals ab, then {{mvar|V}} and {{mvar|I}} must be identical for both networks. Thus,
:<math>V_\mathrm{s} = RI_\mathrm{s}\,\!</math> or <math>I_\mathrm{s} = \frac{V_\mathrm{s}}{R}</math>
* [[Norton's theorem]] states that any two-terminal linear network can be reduced to an ideal current generator and a parallel impedance.
* [[
==Simple networks==
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===Current division of parallel components===
{{main|current division}}
Consider n
:<math>I_i =
for <math>i = 1,2,...,n.</math>
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====Special case: Current division of two parallel components====
:<math>I_1 = \left( \frac{Z_2}{Z_1 + Z_2} \right)I</math>
:<math>I_2 = \left( \frac{Z_1}{Z_1 + Z_2} \right)I</math>
==Nodal analysis==
{{section howto|date=October 2022}}
{{main|nodal analysis}}
Nodal analysis uses the concept of a node voltage and considers the node voltages to be the unknown variables.{{r|"Circuit Analysis, Chen"|p=2-8 - 2-9}} For all nodes, except a chosen reference node, the node voltage is defined as the voltage drop from the node to the reference node. Therefore, there are N-1 node voltages for a circuit with N nodes.{{r|"Circuit Analysis, Chen"|p=2-10}}
In principle, nodal analysis uses [[Kirchhoff's current law]] (KCL) at N-1 nodes to get N-1 independent equations. Since equations generated with KCL are in terms of currents going in and out of nodes, these currents, if their values are not known, need to be represented by the unknown variables (node voltages). For some elements (such as resistors and capacitors) getting the element currents in terms of node voltages is trivial.
For some common elements where this is not possible, specialized methods are developed. For example, a concept called supernode is used for circuits with independent voltage sources.{{r|"Circuit Analysis, Chen"|p=2-12 - 2-13}}
# Label all '''nodes''' in the circuit. Arbitrarily select any node as reference.
# Define a voltage variable from every remaining node to the reference. These voltage variables must be defined as voltage rises with respect to the reference node.
# Write a KCL equation for every node except the reference.
# Solve the resulting system of equations.
==Mesh analysis==
{{section howto|date=October 2022}}
{{main|mesh analysis}}
[[Mesh]] — a loop that does not contain an inner loop.
# Write a [[Kirchhoff's circuit laws|KVL]] equation for every mesh whose current is unknown.
# Solve the resulting equations
==Superposition==
{{Main|Superposition theorem}}
In this method, the effect of each generator in turn is calculated. All the generators other than the one being considered are removed and either short-circuited in the case of voltage generators or open-circuited in the case of current generators.
There is an underlying assumption to this method that the total current or voltage is a linear superposition of its parts. Therefore, the method cannot be used if non-linear components are present.
==Choice of method==
Choice of method<ref name="ElectricCircuits">{{cite book |
* [[Nodal analysis]]: The number of voltage variables, and hence simultaneous equations to solve, equals the number of nodes minus one. Every voltage source connected to the reference node reduces the number of unknowns and equations by one.
* [[
* [[Superposition theorem|Superposition]] is possibly the most conceptually simple method but rapidly leads to a large number of equations and messy impedance combinations as the network becomes larger.
* [[Effective medium approximations]]: For a network consisting of a high density of random resistors, an exact solution for each individual element may be impractical or impossible. Instead, the effective resistance and current distribution properties can be modelled in terms of [[Graph (discrete mathematics)|graph]] measures and geometrical properties of networks.<ref>{{Cite journal|last1=Kumar|first1=Ankush|last2=Vidhyadhiraja|first2=N. S.| last3=Kulkarni|first3=G. U .|year=2017|title=Current distribution in conducting nanowire networks|journal=Journal of Applied Physics| volume=122|issue=4|pages=045101|doi=10.1063/1.4985792|bibcode=2017JAP...122d5101K}}</ref>
==Transfer function==
A [[transfer function]] expresses the relationship between an input and an output of a network.
This approach is standard in [[control theory]] and is useful for determining [[Stable polynomial|stability]] of a system, for instance, in an amplifier with feedback.
===Two terminal component transfer functions===
For two terminal components the transfer function, or more generally for non-linear elements, the [[constitutive equation]], is the relationship between the current input to the device and the resulting voltage across it.
{|
|-
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===Two port network transfer function===
Transfer functions, in general, in control theory are given the symbol H(s).
<math display="block">A(j\omega)=\frac{V_o}{V_i}</math>
The ''A'' standing for attenuation, or amplification, depending on context. In general, this will be a complex function of ''jω'', which can be derived from an analysis of the impedances in the network and their individual transfer functions. Sometimes the analyst is only interested in the magnitude of the gain and not the phase angle. In this case the complex numbers can be eliminated from the transfer function and it might then be written as;
<math display="block">A(\omega)=\left|{\frac{V_o}{V_i}}\right|</math>
====Two port parameters====
{{main|Two-port network}}
The concept of a two-port network can be useful in network analysis as a [[black box]] approach to analysis.
<math display="block">
\begin{bmatrix}
V_1 \\
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<math> \left [z(j\omega) \right] </math> or just <math> \left [z \right] </math>
These concepts are capable of being extended to networks of more than two ports.
====Distributed components====
Where a network is composed of discrete components, analysis using two-port networks is a matter of choice, not essential.
====Image analysis====
{{Main|Image impedance}}
Transmission lines and certain types of filter design use the image method to determine their transfer parameters.
== Time-based network analysis with simulation ==
{{see also|Electronic circuit simulation}}
Most analysis methods calculate the voltage and current values for static networks, which are circuits consisting of memoryless components only but have difficulties with complex dynamic networks. In general, the equations that describe the behaviour of a dynamic circuit are in the form of a [[differential-algebraic system of equations]] (DAEs). DAEs are challenging to solve and the methods for doing so are not yet fully understood and developed (as of 2010). Also, there is no general theorem that guarantees solutions to DAEs will exist and be unique. <ref name="Circuit Simulation, Najm">{{cite book |last=Najm |first=Farid N. |date=2010 |title=Circuit Simulation |publisher=John Wiley & Sons |isbn=9780470538715}}</ref>{{rp|pages=204-205}} In special cases, the equations of the dynamic circuit will be in the form of an [[ordinary differential equation|ordinary differential equations]] (ODE), which are easier to solve, since numerical methods for solving ODEs have a rich history, dating back to the late 1800s. One strategy for adapting ODE solution methods to DAEs is called direct discretization and is the method of choice in circuit simulation. {{r|"Circuit Simulation, Najm"|p=204-205}}
Simulation-based methods for time-based network analysis solve a circuit that is posed as an [[initial value problem]] (IVP). That is, the values of the components with memories (for example, the voltages on capacitors and currents through inductors) are given at an initial point of time {{math|t<sub>0</sub>}}, and the analysis is done for the time <math>t_0\leq t\leq t_f</math>. {{r|"Circuit Simulation, Najm"|p=206-207}} Since finding numerical results for the infinite number of time points from {{math|t<sub>0</sub>}} to {{math|t<sub>f</sub>}} is not possible, this time period is discretized into discrete time instances, and the numerical solution is found for every instance. The time between the time instances is called the time step and can be fixed throughout the whole simulation or may be [[adaptive step size|adaptive]].
In an IVP, when finding a solution for time {{math|t<sub>n+1</sub>}}, the solution for time {{math|t<sub>n</sub>}} is already known. Then, [[temporal discretization]] is used to replace the derivatives with differences, such as <math display="block">x'(t_{n+1}) \approx \frac{x_{n+1}-x_n}{h_{n+1}}</math> for the [[backward Euler method]], where {{math|h<sub>n+1</sub>}} is the time step. {{r|"Circuit Simulation, Najm"|p=266}}
If all circuit components were linear or the circuit was linearized beforehand, the equation system at this point is a [[system of linear equations]] and is solved with [[numerical linear algebra]] methods. Otherwise, it is a nonlinear algebraic equation system and is solved with [[Equation solving#Numerical methods|nonlinear numerical methods]] such as [[Root-finding algorithms]].
=== Comparison to other methods ===
Simulation methods are much more applicable than [[Laplace transform]] based methods, such as [[Network analysis (electrical circuits)#Transfer function|transfer functions]], which only work for simple dynamic networks with capacitors and inductors. Also, the input signals to the network cannot be arbitrarily defined for Laplace transform based methods.
==Non-linear networks==
Most electronic designs are, in reality, non-linear.
:<math>i = I_o \left(e^{
where;
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* ''V<sub>T</sub>'' is a parameter proportional to temperature called the thermal voltage and equal to about 25mV at room temperature.
There are many other ways that non-linearity can appear in a network.
===Constitutive equations===
The [[diode]] equation above is an example of an [[Electrical element#Non-linear elements|element constitutive equation]] of the general form,
:<math>f(v,i) = 0
This can be thought of as a non-linear resistor. The corresponding constitutive equations for non-linear inductors and capacitors are respectively;
:<math>f(v, \varphi) = 0
:<math>f(v, q) = 0
where ''f'' is any arbitrary function, ''φ'' is the stored magnetic flux and ''q'' is the stored charge.
===Existence, uniqueness and stability===
An important consideration in non-linear analysis is the question of uniqueness.
Another important consideration is the question of stability.
===Methods===
====Boolean analysis of switching networks====
A switching device is one where the non-linearity is utilised to produce two opposite states.
The transients are ignored in this analysis, along with any slight discrepancy between the state of the device and the nominal state assigned to a
The transients are not entirely uninteresting to the analyst.
It is mathematically possible to derive [[Boolean algebra (structure)|
====Separation of bias and signal analyses====
This technique is used where the operation of the circuit is to be essentially linear, but the devices used to implement it are non-linear.
====Graphical method of dc analysis====
In a great many circuit designs, the dc bias is fed to a non-linear component via a resistor (or possibly a network of resistors).
Perhaps the easiest practical method is to calculate the (linear) network open circuit voltage and short circuit current and plot these on the transfer function of the non-linear device.
In reality, the designer of the circuit would proceed in the reverse direction to that described.
It is still possible to use this method if the device being biased has its bias fed through another device which is itself non-linear
====Small signal equivalent circuit====
{{main|Small
This method can be used where the deviation of the input and output signals in a network stay within a substantially linear portion of the non-linear devices transfer function, or else are so small that the curve of the transfer function can be considered linear. Under a set of these specific conditions, the non-linear device can be represented by an equivalent linear network. It must be remembered that this equivalent circuit is entirely notional and only valid for the small signal deviations. It is entirely inapplicable to the dc biasing of the device.
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The model of a forward biased pn junction having an approximately constant 0.7V is also a much used approximation for transistor base-emitter junction voltage in amplifier design.
The piecewise method is similar to the small signal method in that linear network analysis techniques can only be applied if the signal stays within certain bounds. If the signal crosses a discontinuity point then the model is no longer valid for linear analysis purposes. The model does have the advantage over small signal however, in that it is equally applicable to signal and dc bias.
===Time-varying components===
In linear analysis, the components of the network are assumed to be unchanging, but in some circuits this does not apply, such as sweep oscillators, [[voltage controlled amplifier]]s, and variable [[Electronic filter|equalisers]]. In many circumstances the change in component value is periodic.
===Vector circuit theory===
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==See also==
* [[Bartlett's bisection theorem]]
* [[Kirchhoff's circuit laws]]
* [[Millman's theorem]]
* [[Modified nodal analysis]]
* [[Ohm's law]]
* [[Reciprocity (electrical networks)]]
* [[Tellegen's theorem]]
* [[Symbolic circuit analysis]]
==References==
{{reflist|25em}}
==External links==
* [https://feynmanlectures.caltech.edu/II_22.html The Feynman Lectures on Physics Vol. II Ch. 22: AC Circuits]
{{DEFAULTSORT:Network Analysis (Electrical Circuits)}}
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