Network analysis (electrical circuits): Difference between revisions

Revision as of 12:56, 16 August 2022

Revision as of 17:57, 12 October 2022

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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>

Impedances**The inabove [[Seriessimplified andfor parallelonly circuits#Paralleltwo impedances in circuits|parallel]]: <math display="block">\frac{1}{Z_\mathrm{eq}} = \frac{1Z_1Z_2}{Z_1} + \frac{1}{Z_2} + \,\cdots\, + \frac{1}{Z_n} .</math>

:The above simplified for only two impedances in parallel: <math>Z_\mathrm{eq} = \frac{Z_1Z_2}{Z_1 + Z_2} .</math>

===Delta-wye transformation===

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====Delta-to-star transformation equations====

:<math>R_a = \frac{R_\mathrm{ac}R_\mathrm{ab}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} </math>

:<math>R_b = \frac{R_\mathrm{ab}R_\mathrm{bc}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} </math>

:<math>R_c = \frac{R_\mathrm{bc}R_\mathrm{ac}}{R_\mathrm{ac} + R_\mathrm{ab} + R_\mathrm{bc}} </math>

====Star-to-delta transformation equations====

:<math>R_\mathrm{ac} = \frac{R_aR_bR_a R_b + R_bR_cR_b R_c + R_cR_aR_c R_a}{R_b}</math>

:<math>R_\mathrm{bcab} = \frac{R_aR_bR_a R_b + R_bR_cR_b R_c + R_cR_aR_c R_a}{R_aR_c}</math>

:<math>R_\mathrm{abbc} = \frac{R_aR_bR_a R_b + R_bR_cR_b R_c + R_cR_aR_c R_a}{R_cR_a}</math>

:<math>R_\mathrm{bc} = \frac{R_aR_b + R_bR_c + R_cR_a}{R_a}</math>

===General form of network node elimination===

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The star-to-delta and series-resistor transformations are special cases of the general resistor network node elimination algorithm. Any node connected by <math>N</math> resistors (<math>R_1</math> .. <math>R_N</math>) to nodes '''1''' .. '''N''' can be replaced by <math>{N \choose 2}</math> resistors interconnecting the remaining <math>N</math> nodes. The resistance between any two nodes <math>x</math> and <math>y</math> is given by:

:<math>R_\mathrm{xy} = R_xR_yR_x R_y\sum_{i=1}^N \frac{1}{R_i}</math>

For a star-to-delta (<math>N=3</math>) this reduces to:

:<math>R_\mathrm{ab} = R_aR_bR_a R_b \left(\frac 1 R_a+\frac 1 R_b+\frac 1 R_c\right) = \frac{R_aR_bR_a R_b(R_aR_bR_a R_b +R_aR_c R_a R_c +R_bR_c R_b R_c)}{R_aR_bR_cR_a R_b R_c}=\frac{R_aR_bR_a R_b + R_bR_cR_b R_c + R_cR_aR_c R_a}{R_c}</math>

For a series reduction (<math>N=2</math>) this reduces to:

:<math>R_\mathrm{ab} = R_aR_bR_a R_b \left(\frac 1 R_a+\frac 1 R_b\right) = \frac{R_aR_bR_a R_b(R_a + R_b)}{R_aR_bR_a R_b} = R_a + R_b</math>

For a dangling resistor (<math>N=1</math>) it results in the elimination of the resistor because <math>{1 \choose 2} = 0</math>.

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* [[Norton's theorem]] states that any two-terminal linear network can be reduced to an ideal current generator and a parallel impedance.

* [[Thévenin's theorem]] states that any two-terminal linear network can be reduced to an ideal voltage generator plus a series impedance.

*

==Simple networks==

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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==

{{main|nodal analysis}}

1.# Label all '''nodes''' in the circuit. Arbitrarily select any node as reference.

2.# 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.

3.# Write a [[Kirchhoff's circuit laws|KCL]] equation for every node except the reference.

2. 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.

4.# Solve the resulting system of equations.

3. Write a [[Kirchhoff's circuit laws|KCL]] equation for every node except the reference.

4. Solve the resulting system of equations.

==Mesh analysis==

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[[Mesh]]&nbsp;— a loop that does not contain an inner loop.

1.# Count the number of “window panes” in the circuit. Assign a mesh current to each window pane.

2.# Write a [[Kirchhoff's circuit laws|KVL]] equation for every mesh whose current is unknown.

3.# Solve the resulting equations

2. Write a [[Kirchhoff's circuit laws|KVL]] equation for every mesh whose current is unknown.

3. Solve the resulting equations

==Superposition==

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==Choice of method==

Choice of method<ref name="ElectricCircuits">{{cite book |authorlast1 = Nilsson, J| first1 = James W,. | last2 = Riedel, S| first2 = Susan A. |title=Electric Circuits |publisher=Pearson Prentice Hall |year=2007 |isbn=978-0-13-198925-2 |edition=8th |pages=112–113 |url=https://books.google.com/books?id=sxmM8RFL99wC&q=112&pg=PA112 }}</ref>{{rp|pp=112–113}} is to some extent a matter of taste. If the network is particularly simple or only a specific current or voltage is required then ad-hoc application of some simple equivalent circuits may yield the answer without recourse to the more systematic methods.

* [[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.

* [[Mesh analysis]]: The number of current variables, and hence simultaneous equations to solve, equals the number of meshes. Every current source in a mesh reduces the number of unknowns by one. Mesh analysis can only be used with networks which can be drawn as a [[Planar graph|planar]] network, that is, with no crossing components.<ref name="ElectricCircuits"/>{{cite book rp|author=Nilsson, J W, Riedel, S A |title=Electric Circuits |publisher=Pearson Prentice Hall |year=2007 |isbn=978-0-13-198925-2 |edition=8th |pagep=94 |url=https://books.google.com/books?id=sxmM8RFL99wC&pg=PA94 }}</ref>

* [[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==

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{{main|Two-port network}}

The concept of a two-port network can be useful in network analysis as a [[black box]] approach to analysis. The behaviour of the two-port network in a larger network can be entirely characterised without necessarily stating anything about the internal structure. However, to do this it is necessary to have more information than just the A(jω) described above. It can be shown that four such parameters are required to fully characterise the two-port network. These could be the forward transfer function, the input impedance, the reverse transfer function (i.e., the voltage appearing at the input when a voltage is applied to the output) and the output impedance. There are many others (see the main article for a full listing), one of these expresses all four parameters as impedances. It is usual to express the four parameters as a matrix;

<math display="block">

<math>

\begin{bmatrix}

V_1 \\

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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 \,</math>

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>

:<math>f(v, q) = 0 \,</math>

where ''f'' is any arbitrary function, ''φ'' is the stored magnetic flux and ''q'' is the stored charge.

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===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. A non-linear component excited with a periodic signal, for instance, can be represented as a periodically varying ''linear'' component. [[Sidney Darlington]] disclosed a method of analysing such periodic time varying circuits. He developed canonical circuit forms which are analogous to the canonical forms of [[Ronald M. Foster]] and [[Wilhelm Cauer]] used for analysing linear circuits.<ref>{{Ref patent |country=US |number=3265973 |status=patent |title=Synthesis of two-port networks having periodically time-varying elements |gdate=1966-08-09 |fdate=1962-05-16 |inventor=Sidney Darlington, Irwin W. Sandberg }}</ref>

|country=US

|number=3265973

|status=patent

|title=Synthesis of two-port networks having periodically time-varying elements

|gdate=1966-08-09

|fdate=1962-05-16

|inventor=Sidney Darlington, Irwin W. Sandberg

}}

</ref>

===Vector circuit theory===