1465:
5322:. These are a matrix of four parameters as with the parameters but in the case of the parameters they are a hybrid mixture of impedances, admittances, current gains and voltage gains. In this model the three terminal transistor is considered to be a two port network, one of its terminals being common to both ports. The parameters are quite different depending on which terminal is chosen as the common one. The most important parameter for transistors is usually the forward current gain, h
1100:). For a three terminal network, the three impedances can be expressed as a three node delta (Δ) network or four node star (Y) network. These two networks are equivalent and the transformations between them are given below. A general network with an arbitrary number of nodes cannot be reduced to the minimum number of impedances using only series and parallel combinations. In general, Y-Δ and Δ-Y transformations must also be used. For some networks the extension of Y-Δ to
4616:, then it is not possible to analyse in terms of individual components since they do not exist. The most common approach to this is to model the line as a two-port network and characterise it using two-port parameters (or something equivalent to them). Another example of this technique is modelling the carriers crossing the base region in a high frequency transistor. The base region has to be modelled as distributed resistance and capacitance rather than
4310:
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;
1043:
1119:
180:
218:
190:
170:
175:
185:
165:
505:, 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.
343:
333:
328:
323:
318:
338:
89:
2383:
300:
5338:
78:
73:
1818:
147:
104:
544:
58:
213:
208:
109:
137:
94:
63:
2672:
274:
245:
142:
132:
68:
255:
114:
83:
3532:
3430:
1460:{\displaystyle {\begin{aligned}R_{a}&={\frac {R_{\mathrm {ac} }R_{\mathrm {ab} }}{R_{\mathrm {ac} }+R_{\mathrm {ab} }+R_{\mathrm {bc} }}}\\R_{b}&={\frac {R_{\mathrm {ab} }R_{\mathrm {bc} }}{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{aligned}}}
250:
3497:(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.
99:
557:
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.
223:
295:
290:
269:
264:
4520:
2029:
1478:
5727:
3628:
circuits. Power varies according to the square of total voltage or current and the square of the sum is not generally equal to the sum of the squares. Total power in an element can be found by applying superposition to the voltages and current independently and then calculating power from the total voltage and current.
4228:, 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;
5230:
A switching device is one where the non-linearity is utilised to produce two opposite states. CMOS devices in digital circuits, for instance, have their output connected to either the positive or the negative supply rail and are never found at anything in between except during a transient period when
4598:
These concepts are capable of being extended to networks of more than two ports. However, this is rarely done in reality because, in many practical cases, ports are considered either purely input or purely output. If reverse direction transfer functions are ignored, a multi-port network can always be
4309:
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
3683:
expresses the relationship between an input and an output of a network. For resistive networks, this will always be a simple real number or an expression which boils down to a real number. Resistive networks are represented by a system of simultaneous algebraic equations. However, in the general case
5279:
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). Since resistors are linear components, it is particularly easy to determine the quiescent operating point of the non-linear device from a graph of its transfer function.
5238:
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
Boolean value. For instance, Boolean "1" may be assigned to the state of +5V. The output of the device may be +4.5V but the analyst still considers this
4634:
Transmission lines and certain types of filter design use the image method to determine their transfer parameters. In this method, the behaviour of an infinitely long cascade connected chain of identical networks is considered. The input and output impedances and the forward and reverse transmission
3489:
Nodal analysis uses the concept of a node voltage and considers the node voltages to be the unknown variables. 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
714:
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
556:
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
528:
more) terminal component effectively has two (or more) ports and the transfer function cannot be expressed as a single impedance. The usual approach is to express the transfer function as a matrix of parameters. These parameters can be impedances, but there is a large number of other approaches (see
5314:
For a simple two-terminal device, the small signal equivalent circuit may be no more than two components. A resistance equal to the slope of the v/i curve at the operating point (called the dynamic resistance), and tangent to the curve. A generator, because this tangent will not, in general, pass
5291:
In reality, the designer of the circuit would proceed in the reverse direction to that described. Starting from a plot provided in the manufacturers data sheet for the non-linear device, the designer would choose the desired operating point and then calculate the linear component values required to
5242:
The transients are not entirely uninteresting to the analyst. The maximum rate of switching is determined by the speed of transition from one state to the other. Happily for the analyst, for many devices most of the transition occurs in the linear portion of the devices transfer function and linear
5216:
Another important consideration is the question of stability. A particular solution may exist, but it may not be stable, rapidly departing from that point at the slightest stimulation. It can be shown that a network that is absolutely stable for all conditions must have one, and only one, solution
4654:(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. In special cases, the equations of the dynamic circuit will be in the form of an
5425:
Generalization of circuit theory based on scalar quantities to vectorial currents is a necessity for newly evolving circuits such as spin circuits. Generalized circuit variables consist of four components: scalar current and vector spin current in x, y, and z directions. The voltages and currents
5378:
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
5354:
In this method, the transfer function of the non-linear device is broken up into regions. Each of these regions is approximated by a straight line. Thus, the transfer function will be linear up to a particular point where there will be a discontinuity. Past this point the transfer function will
5358:
A well known application of this method is the approximation of the transfer function of a pn junction diode. The transfer function of an ideal diode has been given at the top of this (non-linear) section. However, this formula is rarely used in network analysis, a piecewise approximation being
4150:
Transfer functions, in general, in control theory are given the symbol H(s). Most commonly in electronics, transfer function is defined as the ratio of output voltage to input voltage and given the symbol A(s), or more commonly (because analysis is invariably done in terms of sine wave response),
5333:
The small signal equivalent circuit in terms of two-port parameters leads to the concept of dependent generators. That is, the value of a voltage or current generator depends linearly on a voltage or current elsewhere in the circuit. For instance the parameter model leads to dependent voltage
5207:
An important consideration in non-linear analysis is the question of uniqueness. For a network composed of linear components there will always be one, and only one, unique solution for a given set of boundary conditions. This is not always the case in non-linear circuits. For instance, a linear
2678:
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
527:
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). A three (or
5034:
There are many other ways that non-linearity can appear in a network. All methods utilising linear superposition will fail when non-linear components are present. There are several options for dealing with non-linearity depending on the type of circuit and the information the analyst wishes to
3623:
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. The total current through or the total voltage
5310:
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
2604:
3627:
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. Superposition of powers cannot be used to find total power consumed by elements even in linear
5258:
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. A transistor amplifier is an example of this kind of network. The essence of this technique is to separate the analysis into two parts. Firstly, the dc
4649:
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
4313:
5295:
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, a diode for instance. In this case however, the plot of the network transfer function onto the device being biased would no longer be a straight line and is
5212:
has up to three solutions for the voltage for a given current. That is, a particular solution for the current through the diode is not unique, there may be others, equally valid. In some cases there may not be a solution at all: the question of existence of solutions must be considered.
5363:
as the voltage falls. This current, for most purposes, is so small it can be ignored. With increasing voltage, the current increases exponentially. The diode is modelled as an open circuit up to the knee of the exponential curve, then past this point as a resistor equal to the
5371:
The commonly accepted values for the transition point voltage are 0.7V for silicon devices and 0.3V for germanium devices. An even simpler model of the diode, sometimes used in switching applications, is short circuit for forward voltages and open circuit for reverse voltages.
2378:{\displaystyle {\begin{aligned}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{aligned}}}
4635:
functions are then calculated for this infinitely long chain. Although the theoretical values so obtained can never be exactly realised in practice, in many cases they serve as a very good approximation for the behaviour of a finite chain as long as it is not too short.
3636:
Choice of method 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.
5287:
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. The straight line joining these two point is the transfer function of the network.
5311:
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.
4658:(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.
3696:. Working with the equations directly would be described as working in the time (or t) domain because the results would be expressed as time varying quantities. The Laplace transform is the mathematical method of transforming between the s-domain and the t-domain.
1813:{\displaystyle {\begin{aligned}R_{\mathrm {ac} }&={\frac {R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{b}}}\\R_{\mathrm {ab} }&={\frac {R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{c}}}\\R_{\mathrm {bc} }&={\frac {R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{a}}}\end{aligned}}}
5345:
There will always be dependent generators in a two-port parameter equivalent circuit. This applies to the parameters as well as to the and any other kind. These dependencies must be preserved when developing the equations in a larger linear network analysis.
4607:
Where a network is composed of discrete components, analysis using two-port networks is a matter of choice, not essential. The network can always alternatively be analysed in terms of its individual component transfer functions. However, if a network contains
3719:, is the relationship between the current input to the device and the resulting voltage across it. The transfer function, Z(s), will thus have units of impedance, ohms. For the three passive components found in electrical networks, the transfer functions are;
933:
1107:
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.
4732:
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
5280:
The method is as follows: from linear network analysis the output transfer function (that is output voltage against output current) is calculated for the network of resistor(s) and the generator driving them. This will be a straight line (called the
3650:: 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
3666:: 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
3190:
3000:
2401:
3644:: 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.
2011:
3684:
of linear networks, the network is represented by a system of simultaneous linear differential equations. In network analysis, rather than use the differential equations directly, it is usual practice to carry out a
4857:
1024:
4515:{\displaystyle {\begin{bmatrix}V_{1}\\V_{0}\end{bmatrix}}={\begin{bmatrix}z(j\omega )_{11}&z(j\omega )_{12}\\z(j\omega )_{21}&z(j\omega )_{22}\end{bmatrix}}{\begin{bmatrix}I_{1}\\I_{0}\end{bmatrix}}}
3414:
3333:
817:
4290:
2788:
828:
4913:
Most electronic designs are, in reality, non-linear. There are very few that do not include some semiconductor devices. These are invariably non-linear, the transfer function of an ideal semiconductor
2737:
4996:
2034:
1483:
1124:
4216:
4029:
517:
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.
5379:
advantage over small signal however, in that it is equally applicable to signal and dc bias. These can therefore both be analysed in the same operations and will be linearly superimposable.
3970:
4905:, 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.
3500:
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.
5395:. 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
5375:
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.
4716:
3861:
370:
5239:
to be
Boolean "1". Device manufacturers will usually specify a range of values in their data sheets that are to be considered undefined (i.e. the result will be unpredictable).
2661:
3918:
1907:
1097:
5140:
455:
A point at which terminals of more than two components are joined. A conductor with a substantially zero resistance is considered to be a node for the purpose of analysis.
5231:
the device is switching. Here the non-linearity is designed to be extreme, and the analyst can take advantage of that fact. These kinds of networks can be analysed using
4138:
3246:
4665:(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
3808:
688:
648:
3762:
5182:
5093:
4104:
4070:
729:
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.
4594:
4566:
3072:
3045:
2882:
2855:
3511:
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.
3660:
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.
5284:) and can readily be superimposed on the transfer function plot of the non-linear device. The point where the lines cross is the quiescent operating point.
3080:
2890:
1834:
The star-to-delta and series-resistor transformations are special cases of the general resistor network node elimination algorithm. Any node connected by
940:
4761:
4651:
5403:
disclosed a method of analysing such periodic time varying circuits. He developed canonical circuit forms which are analogous to the canonical forms of
2599:{\displaystyle 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}}
584:. Analysis of a circuit consists of solving for the voltages and currents present in the circuit. The solution principles outlined here also apply to
5271:
characteristics of the circuit are analysed using linear network analysis. Examples of methods that can be used for both these stages are given below.
740:
363:
5387:
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,
5318:
A popular form of specifying the small signal equivalent circuit amongst transistor manufacturers is to use the two-port network parameters known as
5098:
This can be thought of as a non-linear resistor. The corresponding constitutive equations for non-linear inductors and capacitors are respectively;
4231:
1923:
5725:, Sidney Darlington, Irwin W. Sandberg, "Synthesis of two-port networks having periodically time-varying elements", issued 1966-08-09
5619:
5250:
that have more than two states. There is not too much use found for these in electronics, although three-state devices are passingly common.
413:
through, all network components. There are many techniques for calculating these values; however, for the most part, the techniques assume
3542:
3440:
356:
5235:
by assigning the two states ("on"/"off", "positive"/"negative" or whatever states are being used) to the
Boolean constants "0" and "1".
4162:
5544:
Darlington S (1984). "A history of network synthesis and filter theory for circuits composed of resistors, inductors, and capacitors".
3339:
3258:
5710:
5686:
5460:
2742:
607:
through the terminals for one network have the same relationship as the voltage and current at the terminals of the other network.
5208:
resistor with a fixed current applied to it has only one solution for the voltage across it. On the other hand, the non-linear
217:
5758:
5435:
3663:
551:
198:
5589:
4655:
2693:
4923:
4644:
5388:
3981:
3667:
822:
734:
724:
155:
4873:
If all circuit components were linear or the circuit was linearized beforehand, the equation system at this point is a
5635:
Kumar, Ankush; Vidhyadhiraja, N. S.; Kulkarni, G. U . (2017). "Current distribution in conducting nanowire networks".
5247:
477:
A group of branches within a network joined so as to form a complete loop such that there is no other loop inside it.
5440:
3602:
1049:
A network of impedances with more than two terminals cannot be reduced to a single impedance equivalent circuit. An
928:{\displaystyle {\frac {1}{Z_{\mathrm {eq} }}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}+\,\cdots \,+{\frac {1}{Z_{n}}}.}
5763:
5021:
is an arbitrary parameter called the reverse leakage current whose value depends on the construction of the device.
4874:
3494:
502:
5030:
is a parameter proportional to temperature called the thermal voltage and equal to about 25mV at room temperature.
4036:
Finally, for a network to which only steady dc is applied, s is replaced with zero and dc network theory applies.
2797:
states that any two-terminal linear network can be reduced to an ideal current generator and a parallel impedance.
5470:
4609:
4224:
standing for attenuation, or amplification, depending on context. In general, this will be a complex function of
2803:
states that any two-terminal linear network can be reduced to an ideal voltage generator plus a series impedance.
268:
263:
5520:"IRE Standards on Circuits: Definitions of Terms for Linear Passive Reciprocal Time Invariant Networks, 1960".
4878:
3929:
31:
2800:
244:
5450:
5281:
5232:
4886:
4755:
4675:
3819:
581:
440:
384:
179:
5260:
4860:
4662:
3716:
3657:
3618:
2619:
604:
483:
299:
189:
169:
5465:
3883:
273:
5722:
1871:
1061:
5644:
5445:
5365:
5104:
1058:
585:
433:
398:
254:
174:
122:
2812:
Some very simple networks can be analysed without the need to apply the more systematic approaches.
332:
4115:
3688:
on them first and then express the result in terms of the
Laplace parameter s, which in general is
2794:
1829:
249:
184:
164:
3715:
For two terminal components the transfer function, or more generally for non-linear elements, the
3624:
across a particular branch is then calculated by summing all the individual currents or voltages.
3198:
5509:
5305:
5048:
4734:
3773:
653:
613:
495:
393:
234:
48:
3730:
342:
322:
88:
5706:
5682:
5615:
5585:
5540:
5400:
5392:
5146:
5057:
4898:
4617:
4613:
3704:
3685:
3680:
561:
511:
417:
components. Except where stated, the methods described in this article are applicable only to
5315:
through the origin. With more terminals, more complicated equivalent circuits are required.
4661:
Simulation-based methods for time-based network analysis solve a circuit that is posed as an
4081:
4047:
3598:
Count the number of “window panes” in the circuit. Assign a mesh current to each window pane.
317:
57:
5652:
5553:
5529:
5501:
5489:
5404:
5319:
4882:
4300:
3011:
2821:
1037:
529:
410:
308:
212:
207:
5426:
each become vector quantities with conductance described as a 4x4 spin conductance matrix.
4571:
4529:
3050:
3023:
2860:
2833:
1042:
4629:
449:
337:
77:
72:
327:
5648:
3700:
3689:
3641:
3484:
577:
573:
569:
489:
Two terminals where the current into one is identical to the current out of the other.
414:
289:
222:
113:
82:
5337:
3553:
3451:
5752:
5408:
3647:
3586:
3185:{\displaystyle I_{i}=Y_{i}V=\left({\frac {Y_{i}}{Y_{1}+Y_{2}+\cdots +Y_{n}}}\right)I}
2995:{\displaystyle V_{i}=Z_{i}I=\left({\frac {Z_{i}}{Z_{1}+Z_{2}+\cdots +Z_{n}}}\right)V}
471:
294:
5513:
5268:
5209:
4914:
4881:
methods. Otherwise, it is a nonlinear algebraic equation system and is solved with
3651:
543:
5455:
146:
5743:
5609:
5563:
follows
Belevitch but notes there are now also many colloquial uses of "network".
5420:
388:
103:
62:
17:
5533:
5505:
4718:. Since finding numerical results for the infinite number of time points from
3868:
For a network to which only steady ac signals are applied, s is replaced with
2671:
108:
67:
5557:
5359:
used instead. It can be seen that the diode current rapidly diminishes to -I
2006:{\displaystyle R_{\mathrm {xy} }=R_{x}R_{y}\sum _{i=1}^{N}{\frac {1}{R_{i}}}}
4306:
565:
136:
93:
98:
4305:
The concept of a two-port network can be useful in network analysis as a
3693:
711:
141:
131:
5264:
600:
406:
5656:
1019:{\displaystyle Z_{\mathrm {eq} }={\frac {Z_{1}Z_{2}}{Z_{1}+Z_{2}}}.}
4852:{\displaystyle x'(t_{n+1})\approx {\frac {x_{n+1}-x_{n}}{h_{n+1}}}}
5336:
5243:
analysis can be applied to obtain at least an approximate answer.
5044:
2670:
27:
Determining all voltages and currents within an electrical network
5341:
parameter equivalent circuit showing dependent voltage generators
5263:
are analysed using some non-linear method. This establishes the
3591:
3409:{\displaystyle I_{2}=\left({\frac {Z_{1}}{Z_{1}+Z_{2}}}\right)I}
3328:{\displaystyle I_{1}=\left({\frac {Z_{2}}{Z_{1}+Z_{2}}}\right)I}
812:{\displaystyle Z_{\mathrm {eq} }=Z_{1}+Z_{2}+\,\cdots \,+Z_{n}.}
4285:{\displaystyle A(\omega )=\left|{\frac {V_{o}}{V_{i}}}\right|}
3525:
3423:
2783:{\displaystyle I_{\mathrm {s} }={\frac {V_{\mathrm {s} }}{R}}}
4758:
is used to replace the derivatives with differences, such as
1101:
5326:, in the common emitter configuration. This is designated h
3872:
and the more familiar values from ac network theory result.
1041:
542:
5744:
The
Feynman Lectures on Physics Vol. II Ch. 22: AC Circuits
4524:
The matrix may be abbreviated to a representative element;
3565:
3463:
5672:
5670:
5668:
5666:
3707:
of a system, for instance, in an amplifier with feedback.
3557:
3455:
937:
The above simplified for only two impedances in parallel:
3514:
Write a KCL equation for every node except the reference.
3508:
in the circuit. Arbitrarily select any node as reference.
3251:
Special case: Current division of two parallel components
5575:
5573:
5571:
5569:
5492:(May 1962). "Summary of the history of circuit theory".
3561:
3459:
2616:) it results in the elimination of the resistor because
3549:
3447:
4477:
4365:
4322:
2732:{\displaystyle V_{\mathrm {s} }=RI_{\mathrm {s} }\,\!}
2624:
1876:
1066:
5149:
5107:
5060:
4991:{\displaystyle i=I_{o}\left(e^{{v}/{V_{T}}}-1\right)}
4926:
4764:
4678:
4574:
4532:
4316:
4234:
4165:
4118:
4084:
4050:
3984:
3932:
3886:
3822:
3776:
3733:
3342:
3261:
3201:
3083:
3053:
3026:
2893:
2863:
2836:
2745:
2696:
2622:
2404:
2032:
1926:
1874:
1481:
1122:
1064:
943:
831:
743:
656:
616:
30:"Circuit theory" redirects here. For other uses, see
5705:(Ed: Wai-Kai Chen), pp. 79–81, Academic Press, 2005
3594: — a loop that does not contain an inner loop.
5603:
5601:
5176:
5134:
5087:
4990:
4851:
4710:
4588:
4560:
4514:
4284:
4211:{\displaystyle A(j\omega )={\frac {V_{o}}{V_{i}}}}
4210:
4132:
4098:
4064:
4024:{\displaystyle Z(j\omega )={\frac {1}{j\omega C}}}
4023:
3964:
3912:
3855:
3802:
3756:
3408:
3327:
3240:
3184:
3066:
3039:
2994:
2876:
2849:
2782:
2731:
2655:
2598:
2377:
2005:
1901:
1812:
1459:
1091:
1018:
927:
811:
682:
642:
4897:Simulation methods are much more applicable than
4129:
4095:
4061:
3961:
3909:
3799:
3753:
3605:equation for every mesh whose current is unknown.
2728:
3670:measures and geometrical properties of networks.
4599:decomposed into a number of two-port networks.
443:into which, or out of which, current may flow.
5582:Circuit Analysis and Feedback Amplifier Theory
5267:operating point of the circuit. Secondly, the
4917:is given by the very non-linear relationship;
3654:network, that is, with no crossing components.
1053:-terminal network can, at best, be reduced to
3016:Consider n admittances that are connected in
2640:
2627:
1913:nodes. The resistance between any two nodes
1892:
1879:
1082:
1069:
364:
8:
5608:Nilsson, James W.; Riedel, Susan A. (2007).
5355:again be linear but with a different slope.
4740:In an IVP, when finding a solution for time
2826:Consider n impedances that are connected in
4639:Time-based network analysis with simulation
2687:must be identical for both networks. Thus,
710:The above is a sufficient definition for a
599:with respect to a pair of terminals if the
5701:Ljiljana Trajković, "Nonlinear circuits",
5012:are the instantaneous current and voltage.
4652:differential-algebraic system of equations
707:, circuit 1 and circuit 2 are equivalent.
371:
357:
5546:IEEE Transactions on Circuits and Systems
5148:
5106:
5059:
4968:
4963:
4958:
4953:
4952:
4937:
4925:
4835:
4824:
4805:
4798:
4780:
4763:
4702:
4683:
4677:
4573:
4531:
4498:
4484:
4472:
4458:
4434:
4408:
4384:
4360:
4343:
4329:
4317:
4315:
4270:
4260:
4254:
4233:
4200:
4190:
4184:
4164:
4128:
4117:
4094:
4083:
4060:
4049:
4003:
3983:
3965:{\displaystyle Z(j\omega )=j\omega L\,\!}
3960:
3931:
3908:
3885:
3838:
3821:
3798:
3775:
3752:
3732:
3711:Two terminal component transfer functions
3390:
3377:
3366:
3360:
3347:
3341:
3309:
3296:
3285:
3279:
3266:
3260:
3200:
3166:
3147:
3134:
3123:
3117:
3101:
3088:
3082:
3058:
3052:
3031:
3025:
2976:
2957:
2944:
2933:
2927:
2911:
2898:
2892:
2868:
2862:
2841:
2835:
2768:
2767:
2761:
2751:
2750:
2744:
2727:
2720:
2719:
2702:
2701:
2695:
2639:
2626:
2623:
2621:
2590:
2577:
2561:
2551:
2536:
2523:
2510:
2500:
2493:
2479:
2469:
2459:
2449:
2437:
2427:
2410:
2409:
2403:
2363:
2352:
2342:
2329:
2319:
2306:
2296:
2289:
2270:
2260:
2250:
2235:
2225:
2212:
2202:
2189:
2179:
2166:
2156:
2149:
2135:
2125:
2115:
2105:
2095:
2085:
2073:
2063:
2042:
2041:
2033:
2031:
1995:
1986:
1980:
1969:
1959:
1949:
1932:
1931:
1925:
1891:
1878:
1875:
1873:
1798:
1787:
1777:
1764:
1754:
1741:
1731:
1724:
1707:
1706:
1690:
1679:
1669:
1656:
1646:
1633:
1623:
1616:
1599:
1598:
1582:
1571:
1561:
1548:
1538:
1525:
1515:
1508:
1491:
1490:
1482:
1480:
1440:
1439:
1422:
1421:
1404:
1403:
1387:
1386:
1372:
1371:
1364:
1351:
1330:
1329:
1312:
1311:
1294:
1293:
1277:
1276:
1262:
1261:
1254:
1241:
1220:
1219:
1202:
1201:
1184:
1183:
1167:
1166:
1152:
1151:
1144:
1131:
1123:
1121:
1081:
1068:
1065:
1063:
1004:
991:
979:
969:
962:
949:
948:
942:
914:
905:
901:
897:
886:
877:
866:
857:
842:
841:
832:
830:
800:
792:
788:
779:
766:
749:
748:
742:
674:
661:
655:
634:
621:
615:
5246:It is mathematically possible to derive
4672:, and the analysis is done for the time
3517:Solve the resulting system of equations.
1909:resistors interconnecting the remaining
1824:General form of network node elimination
5614:(8th ed.). Pearson Prentice Hall.
5481:
3543:instructions, advice, or how-to content
3441:instructions, advice, or how-to content
3006:Current division of parallel components
5254:Separation of bias and signal analyses
5226:Boolean analysis of switching networks
4902:
3692:. This is described as working in the
1471:Star-to-delta transformation equations
1112:Delta-to-star transformation equations
1104:transformations may also be required.
5334:generators as shown in this diagram;
4711:{\displaystyle t_{0}\leq t\leq t_{f}}
2816:Voltage division of series components
7:
5411:used for analysing linear circuits.
3856:{\displaystyle Z(s)={\frac {1}{sC}}}
719:Impedances in series and in parallel
465:The component(s) joining two nodes.
5703:The Electrical Engineering Handbook
5203:Existence, uniqueness and stability
5047:equation above is an example of an
3552:so that it is more encyclopedic or
3450:so that it is more encyclopedic or
4146:Two port network transfer function
4125:
3493:In principle, nodal analysis uses
2769:
2752:
2721:
2703:
2656:{\displaystyle {\tbinom {1}{2}}=0}
2631:
2414:
2411:
2046:
2043:
1936:
1933:
1883:
1711:
1708:
1603:
1600:
1495:
1492:
1444:
1441:
1426:
1423:
1408:
1405:
1391:
1388:
1376:
1373:
1334:
1331:
1316:
1313:
1298:
1295:
1281:
1278:
1266:
1263:
1224:
1221:
1206:
1203:
1188:
1185:
1171:
1168:
1156:
1153:
1073:
953:
950:
846:
843:
753:
750:
397:is a collection of interconnected
25:
5461:Reciprocity (electrical networks)
5296:consequently more tedious to do.
3913:{\displaystyle Z(j\omega )=R\,\!}
699:, then with respect to terminals
501:A current from one terminal of a
5368:of the semiconducting material.
5195:is the stored magnetic flux and
3530:
3428:
1902:{\displaystyle {\tbinom {N}{2}}}
1092:{\displaystyle {\tbinom {n}{2}}}
589:
341:
336:
331:
326:
321:
316:
298:
293:
288:
272:
267:
262:
253:
248:
243:
221:
216:
211:
206:
188:
183:
178:
173:
168:
163:
145:
140:
135:
130:
112:
107:
102:
97:
92:
87:
81:
76:
71:
66:
61:
56:
36:
5300:Small signal equivalent circuit
5275:Graphical method of dc analysis
5135:{\displaystyle f(v,\varphi )=0}
4656:ordinary differential equations
3664:Effective medium approximations
576:. If the sources are constant (
552:Equivalent impedance transforms
5165:
5153:
5123:
5111:
5076:
5064:
4792:
4773:
4550:
4541:
4455:
4445:
4431:
4421:
4405:
4395:
4381:
4371:
4244:
4238:
4178:
4169:
3997:
3988:
3945:
3936:
3899:
3890:
3832:
3826:
3786:
3780:
3743:
3737:
3703:and is useful for determining
2542:
2516:
2241:
2172:
405:is the process of finding the
1:
5389:voltage controlled amplifiers
5049:element constitutive equation
4645:Electronic circuit simulation
4133:{\displaystyle Z=\infty \,\!}
3699:This approach is standard in
3608:Solve the resulting equations
564:is a circuit containing only
5436:Bartlett's bisection theorem
5217:for each set of conditions.
3241:{\displaystyle i=1,2,...,n.}
725:Series and parallel circuits
595:Two circuits are said to be
156:Series and parallel circuits
5538:to justify this definition.
5528:(9): 1609. September 1960.
5191:is any arbitrary function,
4893:Comparison to other methods
4883:nonlinear numerical methods
4612:, such as in the case of a
3803:{\displaystyle Z(s)=sL\,\!}
683:{\displaystyle I_{2}=I_{1}}
643:{\displaystyle V_{2}=V_{1}}
580:) sources, the result is a
523:Component transfer function
5780:
5637:Journal of Applied Physics
5534:10.1109/JRPROC.1960.287676
5506:10.1109/JRPROC.1962.288301
5418:
5303:
4875:system of linear equations
4642:
4627:
4298:
3757:{\displaystyle Z(s)=R\,\!}
3616:
3584:
3482:
3009:
2819:
1827:
1035:
722:
549:
520:
508:
492:
480:
468:
458:
446:
439:A device with two or more
430:
29:
5681:. John Wiley & Sons.
5471:Symbolic circuit analysis
2609:For a dangling resistor (
690:for all (real) values of
603:across the terminals and
314:
307:
286:
282:Network analysis methods
281:
204:
197:
161:
154:
128:
121:
54:
47:
39:
5558:10.1109/TCS.1984.1085415
5441:Kirchhoff's circuit laws
5177:{\displaystyle f(v,q)=0}
5088:{\displaystyle f(v,i)=0}
4879:numerical linear algebra
4754:is already known. Then,
4747:, the solution for time
2388:For a series reduction (
1032:Delta-wye transformation
32:Circuit (disambiguation)
5677:Najm, Farid N. (2010).
5451:Modified nodal analysis
5383:Time-varying components
5350:Piecewise linear method
4901:based methods, such as
4887:Root-finding algorithms
4756:temporal discretization
4099:{\displaystyle Z=0\,\!}
4065:{\displaystyle Z=R\,\!}
3495:Kirchhoff's current law
3047:through any admittance
5759:Electrical engineering
5580:Chen, Wai-Kai (2005).
5522:Proceedings of the IRE
5494:Proceedings of the IRE
5342:
5199:is the stored charge.
5178:
5136:
5089:
5039:Constitutive equations
4992:
4853:
4712:
4610:distributed components
4603:Distributed components
4590:
4562:
4516:
4286:
4212:
4134:
4100:
4066:
4025:
3966:
3914:
3857:
3804:
3758:
3410:
3329:
3242:
3186:
3068:
3041:
2996:
2878:
2851:
2784:
2733:
2675:
2657:
2600:
2379:
2007:
1985:
1903:
1814:
1461:
1093:
1046:
1020:
929:
813:
684:
644:
547:
385:electrical engineering
5723:US patent 3265973
5415:Vector circuit theory
5340:
5179:
5137:
5090:
5051:of the general form,
4993:
4861:backward Euler method
4854:
4713:
4663:initial value problem
4591:
4589:{\displaystyle \left}
4563:
4561:{\displaystyle \left}
4517:
4287:
4213:
4135:
4101:
4067:
4026:
3967:
3915:
3858:
3805:
3759:
3717:constitutive equation
3619:Superposition theorem
3411:
3330:
3243:
3187:
3069:
3067:{\displaystyle Y_{i}}
3042:
3040:{\displaystyle I_{i}}
2997:
2879:
2877:{\displaystyle Z_{i}}
2857:across any impedance
2852:
2850:{\displaystyle V_{i}}
2785:
2734:
2674:
2667:Source transformation
2658:
2601:
2380:
2016:For a star-to-delta (
2008:
1965:
1904:
1815:
1462:
1094:
1057:impedances (at worst
1045:
1021:
930:
814:
685:
645:
546:
5147:
5105:
5058:
4924:
4762:
4676:
4572:
4530:
4314:
4232:
4163:
4116:
4082:
4048:
3982:
3930:
3884:
3820:
3774:
3731:
3340:
3259:
3199:
3081:
3051:
3024:
2891:
2861:
2834:
2743:
2694:
2620:
2402:
2030:
1924:
1872:
1479:
1120:
1062:
941:
829:
741:
654:
614:
199:Impedance transforms
5649:2017JAP...122d5101K
4909:Non-linear networks
4877:and is solved with
4870:is the time step.
4295:Two port parameters
3550:rewrite the content
3448:rewrite the content
2679:terminals ab, then
2395:) this reduces to:
2023:) this reduces to:
1868:can be replaced by
1830:Star-mesh transform
715:their equivalence.
539:Equivalent circuits
309:Two-port parameters
231:Generator theorems
5679:Circuit Simulation
5466:Tellegen's theorem
5343:
5306:Small-signal model
5174:
5132:
5085:
4988:
4903:transfer functions
4849:
4708:
4586:
4558:
4512:
4506:
4466:
4351:
4282:
4208:
4130:
4096:
4062:
4021:
3962:
3910:
3853:
3800:
3754:
3406:
3325:
3238:
3182:
3064:
3037:
2992:
2874:
2847:
2801:Thévenin's theorem
2780:
2729:
2676:
2653:
2645:
2596:
2375:
2373:
2003:
1899:
1897:
1810:
1808:
1457:
1455:
1089:
1087:
1047:
1016:
925:
809:
680:
640:
548:
421:network analysis.
5764:Electronic design
5657:10.1063/1.4985792
5621:978-0-13-198925-2
5611:Electric Circuits
5541:Sidney Darlington
5446:Millman's theorem
5401:Sidney Darlington
4899:Laplace transform
4847:
4618:lumped components
4614:transmission line
4276:
4206:
4143:
4142:
4034:
4033:
4019:
3866:
3865:
3851:
3686:Laplace transform
3681:transfer function
3675:Transfer function
3583:
3582:
3481:
3480:
3397:
3316:
3173:
2983:
2778:
2638:
2568:
2477:
2457:
2369:
2277:
2133:
2113:
2093:
2001:
1890:
1804:
1696:
1588:
1451:
1341:
1231:
1080:
1011:
920:
892:
872:
852:
562:resistive circuit
536:
535:
512:Transfer function
381:
380:
16:(Redirected from
5771:
5732:
5731:
5730:
5726:
5719:
5713:
5699:
5693:
5692:
5674:
5661:
5660:
5632:
5626:
5625:
5605:
5596:
5595:
5577:
5564:
5561:
5537:
5517:
5486:
5405:Ronald M. Foster
5330:on data sheets.
5248:Boolean algebras
5183:
5181:
5180:
5175:
5141:
5139:
5138:
5133:
5094:
5092:
5091:
5086:
4997:
4995:
4994:
4989:
4987:
4983:
4976:
4975:
4974:
4973:
4972:
4962:
4957:
4942:
4941:
4869:
4858:
4856:
4855:
4850:
4848:
4846:
4845:
4830:
4829:
4828:
4816:
4815:
4799:
4791:
4790:
4772:
4753:
4746:
4731:
4724:
4717:
4715:
4714:
4709:
4707:
4706:
4688:
4687:
4671:
4595:
4593:
4592:
4587:
4585:
4567:
4565:
4564:
4559:
4557:
4553:
4521:
4519:
4518:
4513:
4511:
4510:
4503:
4502:
4489:
4488:
4471:
4470:
4463:
4462:
4439:
4438:
4413:
4412:
4389:
4388:
4356:
4355:
4348:
4347:
4334:
4333:
4301:Two-port network
4291:
4289:
4288:
4283:
4281:
4277:
4275:
4274:
4265:
4264:
4255:
4217:
4215:
4214:
4209:
4207:
4205:
4204:
4195:
4194:
4185:
4139:
4137:
4136:
4131:
4105:
4103:
4102:
4097:
4071:
4069:
4068:
4063:
4039:
4038:
4030:
4028:
4027:
4022:
4020:
4018:
4004:
3971:
3969:
3968:
3963:
3919:
3917:
3916:
3911:
3875:
3874:
3862:
3860:
3859:
3854:
3852:
3850:
3839:
3809:
3807:
3806:
3801:
3763:
3761:
3760:
3755:
3722:
3721:
3632:Choice of method
3578:
3575:
3569:
3534:
3533:
3526:
3476:
3473:
3467:
3432:
3431:
3424:
3415:
3413:
3412:
3407:
3402:
3398:
3396:
3395:
3394:
3382:
3381:
3371:
3370:
3361:
3352:
3351:
3334:
3332:
3331:
3326:
3321:
3317:
3315:
3314:
3313:
3301:
3300:
3290:
3289:
3280:
3271:
3270:
3247:
3245:
3244:
3239:
3191:
3189:
3188:
3183:
3178:
3174:
3172:
3171:
3170:
3152:
3151:
3139:
3138:
3128:
3127:
3118:
3106:
3105:
3093:
3092:
3073:
3071:
3070:
3065:
3063:
3062:
3046:
3044:
3043:
3038:
3036:
3035:
3012:current division
3001:
2999:
2998:
2993:
2988:
2984:
2982:
2981:
2980:
2962:
2961:
2949:
2948:
2938:
2937:
2928:
2916:
2915:
2903:
2902:
2883:
2881:
2880:
2875:
2873:
2872:
2856:
2854:
2853:
2848:
2846:
2845:
2822:voltage division
2795:Norton's theorem
2789:
2787:
2786:
2781:
2779:
2774:
2773:
2772:
2762:
2757:
2756:
2755:
2738:
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2726:
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2708:
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2686:
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2643:
2630:
2615:
2605:
2603:
2602:
2597:
2595:
2594:
2582:
2581:
2569:
2567:
2566:
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2555:
2545:
2541:
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2528:
2527:
2515:
2514:
2505:
2504:
2494:
2489:
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2484:
2483:
2478:
2470:
2464:
2463:
2458:
2450:
2442:
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2432:
2431:
2419:
2418:
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2394:
2384:
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2370:
2368:
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2358:
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2347:
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2334:
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2324:
2323:
2311:
2310:
2301:
2300:
2290:
2282:
2278:
2276:
2275:
2274:
2265:
2264:
2255:
2254:
2244:
2240:
2239:
2230:
2229:
2217:
2216:
2207:
2206:
2194:
2193:
2184:
2183:
2171:
2170:
2161:
2160:
2150:
2145:
2141:
2140:
2139:
2134:
2126:
2120:
2119:
2114:
2106:
2100:
2099:
2094:
2086:
2078:
2077:
2068:
2067:
2051:
2050:
2049:
2022:
2012:
2010:
2009:
2004:
2002:
2000:
1999:
1987:
1984:
1979:
1964:
1963:
1954:
1953:
1941:
1940:
1939:
1916:
1912:
1908:
1906:
1905:
1900:
1898:
1896:
1895:
1882:
1867:
1855:
1837:
1819:
1817:
1816:
1811:
1809:
1805:
1803:
1802:
1793:
1792:
1791:
1782:
1781:
1769:
1768:
1759:
1758:
1746:
1745:
1736:
1735:
1725:
1716:
1715:
1714:
1697:
1695:
1694:
1685:
1684:
1683:
1674:
1673:
1661:
1660:
1651:
1650:
1638:
1637:
1628:
1627:
1617:
1608:
1607:
1606:
1589:
1587:
1586:
1577:
1576:
1575:
1566:
1565:
1553:
1552:
1543:
1542:
1530:
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1520:
1519:
1509:
1500:
1499:
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1466:
1464:
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1458:
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1452:
1450:
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1413:
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1411:
1397:
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1394:
1381:
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1379:
1365:
1356:
1355:
1342:
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1337:
1321:
1320:
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1303:
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1301:
1287:
1286:
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1284:
1271:
1270:
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1255:
1246:
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1232:
1230:
1229:
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1227:
1211:
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1193:
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1177:
1176:
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1135:
1098:
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1090:
1088:
1086:
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1072:
1056:
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1025:
1023:
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1017:
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1010:
1009:
1008:
996:
995:
985:
984:
983:
974:
973:
963:
958:
957:
956:
934:
932:
931:
926:
921:
919:
918:
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893:
891:
890:
878:
873:
871:
870:
858:
853:
851:
850:
849:
833:
818:
816:
815:
810:
805:
804:
784:
783:
771:
770:
758:
757:
756:
706:
702:
698:
689:
687:
686:
681:
679:
678:
666:
665:
649:
647:
646:
641:
639:
638:
626:
625:
530:two-port network
429:
428:
409:across, and the
403:Network analysis
373:
366:
359:
345:
340:
335:
330:
325:
320:
302:
297:
292:
276:
271:
266:
257:
252:
247:
225:
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215:
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192:
187:
182:
177:
172:
167:
149:
144:
139:
134:
116:
111:
106:
101:
96:
91:
85:
80:
75:
70:
65:
60:
42:network analysis
37:
21:
18:Circuit analysis
5779:
5778:
5774:
5773:
5772:
5770:
5769:
5768:
5749:
5748:
5740:
5735:
5728:
5721:
5720:
5716:
5700:
5696:
5689:
5676:
5675:
5664:
5634:
5633:
5629:
5622:
5607:
5606:
5599:
5592:
5579:
5578:
5567:
5562:
5543:
5539:
5519:
5488:
5487:
5483:
5479:
5432:
5423:
5417:
5391:, and variable
5385:
5366:bulk resistance
5362:
5352:
5329:
5325:
5308:
5302:
5277:
5256:
5233:Boolean algebra
5228:
5223:
5205:
5145:
5144:
5103:
5102:
5056:
5055:
5041:
5028:
5019:
4964:
4948:
4947:
4943:
4933:
4922:
4921:
4911:
4895:
4868:
4864:
4831:
4820:
4801:
4800:
4776:
4765:
4760:
4759:
4752:
4748:
4745:
4741:
4730:
4726:
4723:
4719:
4698:
4679:
4674:
4673:
4670:
4666:
4647:
4641:
4632:
4630:Image impedance
4626:
4605:
4575:
4570:
4569:
4537:
4533:
4528:
4527:
4505:
4504:
4494:
4491:
4490:
4480:
4473:
4465:
4464:
4454:
4440:
4430:
4415:
4414:
4404:
4390:
4380:
4361:
4350:
4349:
4339:
4336:
4335:
4325:
4318:
4312:
4311:
4303:
4297:
4266:
4256:
4250:
4230:
4229:
4196:
4186:
4161:
4160:
4148:
4114:
4113:
4080:
4079:
4046:
4045:
4008:
3980:
3979:
3928:
3927:
3882:
3881:
3843:
3818:
3817:
3772:
3771:
3729:
3728:
3713:
3677:
3634:
3621:
3615:
3589:
3579:
3573:
3570:
3547:
3535:
3531:
3524:
3487:
3477:
3471:
3468:
3445:
3433:
3429:
3422:
3386:
3373:
3372:
3362:
3356:
3343:
3338:
3337:
3305:
3292:
3291:
3281:
3275:
3262:
3257:
3256:
3253:
3197:
3196:
3162:
3143:
3130:
3129:
3119:
3113:
3097:
3084:
3079:
3078:
3054:
3049:
3048:
3027:
3022:
3021:
3014:
3008:
2972:
2953:
2940:
2939:
2929:
2923:
2907:
2894:
2889:
2888:
2864:
2859:
2858:
2837:
2832:
2831:
2824:
2818:
2810:
2808:Simple networks
2763:
2746:
2741:
2740:
2715:
2697:
2692:
2691:
2684:
2680:
2669:
2625:
2618:
2617:
2610:
2586:
2573:
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2292:
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2280:
2279:
2266:
2256:
2246:
2245:
2231:
2221:
2208:
2198:
2185:
2175:
2162:
2152:
2151:
2124:
2104:
2084:
2083:
2079:
2069:
2059:
2052:
2037:
2028:
2027:
2017:
1991:
1955:
1945:
1927:
1922:
1921:
1914:
1910:
1877:
1870:
1869:
1857:
1852:
1846:
1839:
1835:
1832:
1826:
1807:
1806:
1794:
1783:
1773:
1760:
1750:
1737:
1727:
1726:
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1699:
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1686:
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1652:
1642:
1629:
1619:
1618:
1609:
1594:
1591:
1590:
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1567:
1557:
1544:
1534:
1521:
1511:
1510:
1501:
1486:
1477:
1476:
1473:
1454:
1453:
1435:
1417:
1399:
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1367:
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1347:
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1343:
1325:
1307:
1289:
1288:
1272:
1257:
1256:
1247:
1237:
1234:
1233:
1215:
1197:
1179:
1178:
1162:
1147:
1146:
1137:
1127:
1118:
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1114:
1067:
1060:
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1054:
1050:
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986:
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910:
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837:
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738:
727:
721:
704:
700:
697:
691:
670:
657:
652:
651:
630:
617:
612:
611:
574:voltage sources
570:current sources
554:
541:
427:
377:
86:
35:
28:
23:
22:
15:
12:
11:
5:
5777:
5775:
5767:
5766:
5761:
5751:
5750:
5747:
5746:
5739:
5738:External links
5736:
5734:
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5714:
5694:
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5304:Main article:
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4721:
4705:
4701:
4697:
4694:
4691:
4686:
4682:
4668:
4640:
4637:
4628:Main article:
4625:
4624:Image analysis
4622:
4604:
4601:
4584:
4581:
4578:
4556:
4552:
4549:
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4540:
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4359:
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4338:
4337:
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4299:Main article:
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3825:
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3785:
3782:
3779:
3769:
3765:
3764:
3751:
3748:
3745:
3742:
3739:
3736:
3726:
3712:
3709:
3701:control theory
3676:
3673:
3672:
3671:
3661:
3655:
3645:
3642:Nodal analysis
3633:
3630:
3617:Main article:
3614:
3611:
3610:
3609:
3606:
3599:
3585:Main article:
3581:
3580:
3538:
3536:
3529:
3523:
3520:
3519:
3518:
3515:
3512:
3509:
3485:nodal analysis
3483:Main article:
3479:
3478:
3436:
3434:
3427:
3421:
3420:Nodal analysis
3418:
3417:
3416:
3405:
3401:
3393:
3389:
3385:
3380:
3376:
3369:
3365:
3359:
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3350:
3346:
3335:
3324:
3320:
3312:
3308:
3304:
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3278:
3274:
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3265:
3252:
3249:
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3222:
3219:
3216:
3213:
3210:
3207:
3204:
3193:
3192:
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3177:
3169:
3165:
3161:
3158:
3155:
3150:
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3142:
3137:
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3126:
3122:
3116:
3112:
3109:
3104:
3100:
3096:
3091:
3087:
3061:
3057:
3034:
3030:
3020:. The current
3010:Main article:
3007:
3004:
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3002:
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2965:
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2922:
2919:
2914:
2910:
2906:
2901:
2897:
2871:
2867:
2844:
2840:
2830:. The voltage
2820:Main article:
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2814:
2809:
2806:
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2048:
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2040:
2036:
2035:
2014:
2013:
1998:
1994:
1990:
1983:
1978:
1975:
1972:
1968:
1962:
1958:
1952:
1948:
1944:
1938:
1935:
1930:
1894:
1889:
1886:
1881:
1850:
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1828:Main article:
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1407:
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1393:
1390:
1385:
1378:
1375:
1370:
1363:
1360:
1358:
1354:
1350:
1346:
1345:
1336:
1333:
1328:
1324:
1318:
1315:
1310:
1306:
1300:
1297:
1292:
1283:
1280:
1275:
1268:
1265:
1260:
1253:
1250:
1248:
1244:
1240:
1236:
1235:
1226:
1223:
1218:
1214:
1208:
1205:
1200:
1196:
1190:
1187:
1182:
1173:
1170:
1165:
1158:
1155:
1150:
1143:
1140:
1138:
1134:
1130:
1126:
1125:
1113:
1110:
1084:
1079:
1076:
1071:
1036:Main article:
1033:
1030:
1029:
1028:
1027:
1026:
1015:
1007:
1003:
999:
994:
990:
982:
978:
972:
968:
961:
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952:
947:
924:
917:
913:
909:
904:
900:
896:
889:
885:
881:
876:
869:
865:
861:
856:
848:
845:
840:
836:
821:Impedances in
819:
808:
803:
799:
795:
791:
787:
782:
778:
774:
769:
765:
761:
755:
752:
747:
733:Impedances in
723:Main article:
720:
717:
695:
677:
673:
669:
664:
660:
637:
633:
629:
624:
620:
550:Main article:
540:
537:
534:
533:
525:
519:
518:
515:
507:
506:
499:
491:
490:
487:
479:
478:
475:
467:
466:
463:
457:
456:
453:
445:
444:
437:
426:
423:
379:
378:
376:
375:
368:
361:
353:
351:
348:
347:
312:
311:
305:
304:
284:
283:
279:
278:
259:
239:
238:
232:
228:
227:
202:
201:
195:
194:
159:
158:
152:
151:
126:
125:
119:
118:
52:
51:
45:
44:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5776:
5765:
5762:
5760:
5757:
5756:
5754:
5745:
5742:
5741:
5737:
5724:
5718:
5715:
5712:
5711:0-12-170960-4
5708:
5704:
5698:
5695:
5690:
5688:9780470538715
5684:
5680:
5673:
5671:
5669:
5667:
5663:
5658:
5654:
5650:
5646:
5643:(4): 045101.
5642:
5638:
5631:
5628:
5623:
5617:
5613:
5612:
5604:
5602:
5598:
5593:
5587:
5584:. CRC Press.
5583:
5576:
5574:
5572:
5570:
5566:
5559:
5555:
5551:
5547:
5542:
5535:
5531:
5527:
5523:
5515:
5511:
5507:
5503:
5499:
5495:
5491:
5485:
5482:
5476:
5472:
5469:
5467:
5464:
5462:
5459:
5457:
5454:
5452:
5449:
5447:
5444:
5442:
5439:
5437:
5434:
5433:
5429:
5427:
5422:
5414:
5412:
5410:
5409:Wilhelm Cauer
5406:
5402:
5398:
5394:
5390:
5382:
5380:
5376:
5373:
5369:
5367:
5356:
5349:
5347:
5339:
5335:
5331:
5321:
5316:
5312:
5307:
5299:
5297:
5293:
5289:
5285:
5283:
5274:
5272:
5270:
5266:
5262:
5253:
5251:
5249:
5244:
5240:
5236:
5234:
5225:
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5214:
5211:
5202:
5200:
5198:
5194:
5190:
5171:
5168:
5162:
5159:
5156:
5150:
5143:
5129:
5126:
5120:
5117:
5114:
5108:
5101:
5100:
5099:
5082:
5079:
5073:
5070:
5067:
5061:
5054:
5053:
5052:
5050:
5046:
5038:
5036:
5029:
5023:
5020:
5014:
5011:
5007:
5004:
5003:
5002:
4984:
4980:
4977:
4969:
4965:
4959:
4954:
4949:
4944:
4938:
4934:
4930:
4927:
4920:
4919:
4918:
4916:
4908:
4906:
4904:
4900:
4892:
4890:
4888:
4884:
4880:
4876:
4871:
4862:
4842:
4839:
4836:
4832:
4825:
4821:
4817:
4812:
4809:
4806:
4802:
4795:
4787:
4784:
4781:
4777:
4769:
4766:
4757:
4738:
4736:
4703:
4699:
4695:
4692:
4689:
4684:
4680:
4664:
4659:
4657:
4653:
4646:
4638:
4636:
4631:
4623:
4621:
4619:
4615:
4611:
4602:
4600:
4596:
4582:
4579:
4576:
4554:
4547:
4544:
4538:
4534:
4525:
4522:
4507:
4499:
4495:
4485:
4481:
4474:
4467:
4459:
4451:
4448:
4442:
4435:
4427:
4424:
4418:
4409:
4401:
4398:
4392:
4385:
4377:
4374:
4368:
4362:
4357:
4352:
4344:
4340:
4330:
4326:
4319:
4308:
4302:
4294:
4292:
4278:
4271:
4267:
4261:
4257:
4251:
4247:
4241:
4235:
4227:
4223:
4218:
4201:
4197:
4191:
4187:
4181:
4175:
4172:
4166:
4158:
4154:
4145:
4122:
4119:
4112:
4109:
4108:
4091:
4088:
4085:
4078:
4075:
4074:
4057:
4054:
4051:
4044:
4041:
4040:
4037:
4015:
4012:
4009:
4005:
4000:
3994:
3991:
3985:
3978:
3975:
3974:
3957:
3954:
3951:
3948:
3942:
3939:
3933:
3926:
3923:
3922:
3905:
3902:
3896:
3893:
3887:
3880:
3877:
3876:
3873:
3871:
3847:
3844:
3840:
3835:
3829:
3823:
3816:
3813:
3812:
3795:
3792:
3789:
3783:
3777:
3770:
3767:
3766:
3749:
3746:
3740:
3734:
3727:
3724:
3723:
3720:
3718:
3710:
3708:
3706:
3702:
3697:
3695:
3691:
3687:
3682:
3674:
3669:
3665:
3662:
3659:
3658:Superposition
3656:
3653:
3649:
3648:Mesh analysis
3646:
3643:
3640:
3639:
3638:
3631:
3629:
3625:
3620:
3613:Superposition
3612:
3607:
3604:
3600:
3597:
3596:
3595:
3593:
3588:
3587:mesh analysis
3577:
3567:
3563:
3559:
3555:
3551:
3545:
3544:
3539:This section
3537:
3528:
3527:
3522:Mesh analysis
3521:
3516:
3513:
3510:
3507:
3503:
3502:
3501:
3498:
3496:
3491:
3486:
3475:
3465:
3461:
3457:
3453:
3449:
3443:
3442:
3437:This section
3435:
3426:
3425:
3419:
3403:
3399:
3391:
3387:
3383:
3378:
3374:
3367:
3363:
3357:
3353:
3348:
3344:
3336:
3322:
3318:
3310:
3306:
3302:
3297:
3293:
3286:
3282:
3276:
3272:
3267:
3263:
3255:
3254:
3250:
3248:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3179:
3175:
3167:
3163:
3159:
3156:
3153:
3148:
3144:
3140:
3135:
3131:
3124:
3120:
3114:
3110:
3107:
3102:
3098:
3094:
3089:
3085:
3077:
3076:
3075:
3059:
3055:
3032:
3028:
3019:
3013:
3005:
2989:
2985:
2977:
2973:
2969:
2966:
2963:
2958:
2954:
2950:
2945:
2941:
2934:
2930:
2924:
2920:
2917:
2912:
2908:
2904:
2899:
2895:
2887:
2886:
2885:
2869:
2865:
2842:
2838:
2829:
2823:
2815:
2813:
2807:
2802:
2799:
2796:
2793:
2792:
2775:
2764:
2758:
2747:
2716:
2712:
2709:
2698:
2690:
2689:
2688:
2673:
2666:
2664:
2650:
2647:
2635:
2632:
2613:
2591:
2587:
2583:
2578:
2574:
2570:
2562:
2558:
2552:
2548:
2537:
2533:
2529:
2524:
2520:
2511:
2507:
2501:
2497:
2490:
2486:
2480:
2474:
2471:
2465:
2460:
2454:
2451:
2444:
2438:
2434:
2428:
2424:
2420:
2406:
2398:
2397:
2396:
2392:
2364:
2360:
2353:
2349:
2343:
2339:
2335:
2330:
2326:
2320:
2316:
2312:
2307:
2303:
2297:
2293:
2286:
2284:
2271:
2267:
2261:
2257:
2251:
2247:
2236:
2232:
2226:
2222:
2218:
2213:
2209:
2203:
2199:
2195:
2190:
2186:
2180:
2176:
2167:
2163:
2157:
2153:
2146:
2142:
2136:
2130:
2127:
2121:
2116:
2110:
2107:
2101:
2096:
2090:
2087:
2080:
2074:
2070:
2064:
2060:
2056:
2054:
2038:
2026:
2025:
2024:
2020:
1996:
1992:
1988:
1981:
1976:
1973:
1970:
1966:
1960:
1956:
1950:
1946:
1942:
1928:
1920:
1919:
1918:
1917:is given by:
1887:
1884:
1866:
1865:
1860:
1853:
1843:
1831:
1823:
1799:
1795:
1788:
1784:
1778:
1774:
1770:
1765:
1761:
1755:
1751:
1747:
1742:
1738:
1732:
1728:
1721:
1719:
1703:
1691:
1687:
1680:
1676:
1670:
1666:
1662:
1657:
1653:
1647:
1643:
1639:
1634:
1630:
1624:
1620:
1613:
1611:
1595:
1583:
1579:
1572:
1568:
1562:
1558:
1554:
1549:
1545:
1539:
1535:
1531:
1526:
1522:
1516:
1512:
1505:
1503:
1487:
1475:
1474:
1470:
1436:
1432:
1418:
1414:
1400:
1383:
1368:
1361:
1359:
1352:
1348:
1326:
1322:
1308:
1304:
1290:
1273:
1258:
1251:
1249:
1242:
1238:
1216:
1212:
1198:
1194:
1180:
1163:
1148:
1141:
1139:
1132:
1128:
1116:
1115:
1111:
1109:
1105:
1103:
1099:
1077:
1074:
1044:
1039:
1038:Y-Δ transform
1031:
1013:
1005:
1001:
997:
992:
988:
980:
976:
970:
966:
959:
945:
936:
935:
922:
915:
911:
907:
902:
898:
894:
887:
883:
879:
874:
867:
863:
859:
854:
838:
834:
824:
820:
806:
801:
797:
793:
789:
785:
780:
776:
772:
767:
763:
759:
745:
736:
732:
731:
730:
726:
718:
716:
713:
708:
694:
675:
671:
667:
662:
658:
635:
631:
627:
622:
618:
608:
606:
602:
598:
593:
591:
587:
583:
579:
575:
571:
567:
563:
558:
553:
545:
538:
531:
526:
524:
521:
516:
514:
513:
509:
504:
500:
498:
497:
493:
488:
486:
485:
481:
476:
474:
473:
469:
464:
462:
459:
454:
452:
451:
447:
442:
438:
436:
435:
431:
424:
422:
420:
416:
412:
408:
404:
400:
396:
395:
390:
386:
374:
369:
367:
362:
360:
355:
354:
352:
350:
349:
346:
344:
339:
334:
329:
324:
319:
313:
310:
306:
303:
301:
296:
291:
285:
280:
277:
275:
270:
265:
260:
258:
256:
251:
246:
241:
240:
236:
233:
230:
229:
226:
224:
219:
214:
209:
203:
200:
196:
193:
191:
186:
181:
176:
171:
166:
160:
157:
153:
150:
148:
143:
138:
133:
127:
124:
120:
117:
115:
110:
105:
100:
95:
90:
84:
79:
74:
69:
64:
59:
53:
50:
46:
43:
38:
33:
19:
5717:
5702:
5697:
5678:
5640:
5636:
5630:
5610:
5581:
5549:
5545:
5525:
5521:
5497:
5493:
5484:
5424:
5396:
5386:
5377:
5374:
5370:
5357:
5353:
5344:
5332:
5317:
5313:
5309:
5294:
5292:achieve it.
5290:
5286:
5278:
5269:small signal
5257:
5245:
5241:
5237:
5229:
5215:
5210:tunnel diode
5206:
5196:
5192:
5188:
5186:
5097:
5042:
5033:
5024:
5015:
5009:
5005:
5000:
4915:p-n junction
4912:
4896:
4872:
4739:
4660:
4648:
4633:
4606:
4597:
4526:
4523:
4304:
4225:
4221:
4219:
4159:), so that;
4156:
4152:
4149:
4035:
3869:
3867:
3714:
3698:
3678:
3635:
3626:
3622:
3590:
3574:October 2022
3571:
3548:Please help
3540:
3505:
3499:
3492:
3488:
3472:October 2022
3469:
3446:Please help
3438:
3194:
3017:
3015:
2827:
2825:
2811:
2677:
2611:
2608:
2390:
2387:
2018:
2015:
1863:
1862:
1858:
1848:
1841:
1833:
1106:
1102:star-polygon
1048:
728:
709:
692:
609:
596:
594:
588:analysis of
572:, and ideal
559:
555:
522:
510:
494:
482:
470:
460:
448:
432:
418:
402:
392:
382:
315:
287:
261:
242:
205:
162:
129:
55:
41:
5490:Belevitch V
5421:Spintronics
5399:component.
3558:Wikiversity
3456:Wikiversity
590:AC circuits
425:Definitions
389:electronics
5753:Categories
5591:1420037277
5500:(5): 849.
5477:References
5419:See also:
5393:equalisers
5320:parameters
4643:See also:
3566:Wikivoyage
3504:Label all
3464:Wikivoyage
1838:resistors
597:equivalent
582:DC circuit
399:components
123:Components
5456:Ohm's law
5282:load line
5265:quiescent
5121:φ
4978:−
4818:−
4796:≈
4696:≤
4690:≤
4548:ω
4452:ω
4428:ω
4402:ω
4378:ω
4307:black box
4242:ω
4176:ω
4126:∞
4110:Capacitor
4013:ω
3995:ω
3976:Capacitor
3955:ω
3943:ω
3897:ω
3814:Capacitor
3705:stability
3562:Wikibooks
3541:contains
3490:N nodes.
3460:Wikibooks
3439:contains
3157:⋯
2967:⋯
1967:∑
1856:to nodes
899:⋯
790:⋯
566:resistors
503:generator
441:terminals
434:Component
237:theorems
5552:(1): 4.
5514:51666316
5430:See also
5035:obtain.
4885:such as
4863:, where
4859:for the
4770:′
4735:adaptive
4568:or just
4076:Inductor
4042:Resistor
3924:Inductor
3878:Resistor
3768:Inductor
3725:Resistor
3694:s-domain
3601:Write a
3018:parallel
823:parallel
712:one-port
650:implies
568:, ideal
411:currents
407:voltages
49:Elements
5645:Bibcode
5221:Methods
5001:where;
3690:complex
605:current
601:voltage
496:Circuit
394:network
235:Network
40:Linear
5729:
5709:
5685:
5618:
5588:
5518:cites
5512:
5397:linear
5261:biases
5187:where
3652:planar
3556:it to
3454:it to
2828:series
735:series
586:phasor
461:Branch
419:linear
415:linear
5510:S2CID
5045:diode
3668:graph
3564:, or
3506:nodes
3462:, or
5707:ISBN
5683:ISBN
5616:ISBN
5586:ISBN
5407:and
5043:The
5008:and
4220:The
3592:Mesh
3554:move
3452:move
3195:for
2683:and
1915:x, y
703:and
484:Port
472:Mesh
450:Node
391:, a
387:and
372:edit
365:talk
358:view
5653:doi
5641:122
5554:doi
5530:doi
5502:doi
4867:n+1
4744:n+1
4725:to
3603:KVL
3074:is
2884:is
2739:or
2614:= 1
2393:= 2
2021:= 3
610:If
532:).
383:In
5755::
5665:^
5651:.
5639:.
5600:^
5568:^
5550:31
5548:.
5526:48
5524:.
5508:.
5498:50
5496:.
5328:fe
5324:21
4889:.
4737:.
4620:.
4460:22
4436:21
4410:12
4386:11
4226:jω
4157:jω
3870:jω
3679:A
3560:,
3458:,
2663:.
1861:…
1847:…
825::
737::
705:xy
701:ab
592:.
578:DC
560:A
401:.
5691:.
5659:.
5655::
5647::
5624:.
5594:.
5560:.
5556::
5536:.
5532::
5516:.
5504::
5361:o
5197:q
5193:φ
5189:f
5172:0
5169:=
5166:)
5163:q
5160:,
5157:v
5154:(
5151:f
5130:0
5127:=
5124:)
5118:,
5115:v
5112:(
5109:f
5083:0
5080:=
5077:)
5074:i
5071:,
5068:v
5065:(
5062:f
5027:T
5025:V
5018:o
5016:I
5010:v
5006:i
4985:)
4981:1
4970:T
4966:V
4960:/
4955:v
4950:e
4945:(
4939:o
4935:I
4931:=
4928:i
4865:h
4843:1
4840:+
4837:n
4833:h
4826:n
4822:x
4813:1
4810:+
4807:n
4803:x
4793:)
4788:1
4785:+
4782:n
4778:t
4774:(
4767:x
4751:n
4749:t
4742:t
4729:f
4727:t
4722:0
4720:t
4704:f
4700:t
4693:t
4685:0
4681:t
4669:0
4667:t
4583:]
4580:z
4577:[
4555:]
4551:)
4545:j
4542:(
4539:z
4535:[
4508:]
4500:0
4496:I
4486:1
4482:I
4475:[
4468:]
4456:)
4449:j
4446:(
4443:z
4432:)
4425:j
4422:(
4419:z
4406:)
4399:j
4396:(
4393:z
4382:)
4375:j
4372:(
4369:z
4363:[
4358:=
4353:]
4345:0
4341:V
4331:1
4327:V
4320:[
4279:|
4272:i
4268:V
4262:o
4258:V
4252:|
4248:=
4245:)
4239:(
4236:A
4222:A
4202:i
4198:V
4192:o
4188:V
4182:=
4179:)
4173:j
4170:(
4167:A
4155:(
4153:A
4123:=
4120:Z
4092:0
4089:=
4086:Z
4058:R
4055:=
4052:Z
4016:C
4010:j
4006:1
4001:=
3998:)
3992:j
3989:(
3986:Z
3958:L
3952:j
3949:=
3946:)
3940:j
3937:(
3934:Z
3906:R
3903:=
3900:)
3894:j
3891:(
3888:Z
3848:C
3845:s
3841:1
3836:=
3833:)
3830:s
3827:(
3824:Z
3796:L
3793:s
3790:=
3787:)
3784:s
3781:(
3778:Z
3750:R
3747:=
3744:)
3741:s
3738:(
3735:Z
3576:)
3572:(
3568:.
3546:.
3474:)
3470:(
3466:.
3444:.
3404:I
3400:)
3392:2
3388:Z
3384:+
3379:1
3375:Z
3368:1
3364:Z
3358:(
3354:=
3349:2
3345:I
3323:I
3319:)
3311:2
3307:Z
3303:+
3298:1
3294:Z
3287:2
3283:Z
3277:(
3273:=
3268:1
3264:I
3236:.
3233:n
3230:,
3227:.
3224:.
3221:.
3218:,
3215:2
3212:,
3209:1
3206:=
3203:i
3180:I
3176:)
3168:n
3164:Y
3160:+
3154:+
3149:2
3145:Y
3141:+
3136:1
3132:Y
3125:i
3121:Y
3115:(
3111:=
3108:V
3103:i
3099:Y
3095:=
3090:i
3086:I
3060:i
3056:Y
3033:i
3029:I
2990:V
2986:)
2978:n
2974:Z
2970:+
2964:+
2959:2
2955:Z
2951:+
2946:1
2942:Z
2935:i
2931:Z
2925:(
2921:=
2918:I
2913:i
2909:Z
2905:=
2900:i
2896:V
2870:i
2866:Z
2843:i
2839:V
2776:R
2770:s
2765:V
2759:=
2753:s
2748:I
2722:s
2717:I
2713:R
2710:=
2704:s
2699:V
2685:I
2681:V
2651:0
2648:=
2641:)
2636:2
2633:1
2628:(
2612:N
2592:b
2588:R
2584:+
2579:a
2575:R
2571:=
2563:b
2559:R
2553:a
2549:R
2543:)
2538:b
2534:R
2530:+
2525:a
2521:R
2517:(
2512:b
2508:R
2502:a
2498:R
2491:=
2487:)
2481:b
2475:R
2472:1
2466:+
2461:a
2455:R
2452:1
2445:(
2439:b
2435:R
2429:a
2425:R
2421:=
2415:b
2412:a
2407:R
2391:N
2365:c
2361:R
2354:a
2350:R
2344:c
2340:R
2336:+
2331:c
2327:R
2321:b
2317:R
2313:+
2308:b
2304:R
2298:a
2294:R
2287:=
2272:c
2268:R
2262:b
2258:R
2252:a
2248:R
2242:)
2237:c
2233:R
2227:b
2223:R
2219:+
2214:c
2210:R
2204:a
2200:R
2196:+
2191:b
2187:R
2181:a
2177:R
2173:(
2168:b
2164:R
2158:a
2154:R
2147:=
2143:)
2137:c
2131:R
2128:1
2122:+
2117:b
2111:R
2108:1
2102:+
2097:a
2091:R
2088:1
2081:(
2075:b
2071:R
2065:a
2061:R
2057:=
2047:b
2044:a
2039:R
2019:N
1997:i
1993:R
1989:1
1982:N
1977:1
1974:=
1971:i
1961:y
1957:R
1951:x
1947:R
1943:=
1937:y
1934:x
1929:R
1911:N
1893:)
1888:2
1885:N
1880:(
1864:N
1859:1
1854:)
1851:N
1849:R
1845:1
1842:R
1840:(
1836:N
1800:a
1796:R
1789:a
1785:R
1779:c
1775:R
1771:+
1766:c
1762:R
1756:b
1752:R
1748:+
1743:b
1739:R
1733:a
1729:R
1722:=
1712:c
1709:b
1704:R
1692:c
1688:R
1681:a
1677:R
1671:c
1667:R
1663:+
1658:c
1654:R
1648:b
1644:R
1640:+
1635:b
1631:R
1625:a
1621:R
1614:=
1604:b
1601:a
1596:R
1584:b
1580:R
1573:a
1569:R
1563:c
1559:R
1555:+
1550:c
1546:R
1540:b
1536:R
1532:+
1527:b
1523:R
1517:a
1513:R
1506:=
1496:c
1493:a
1488:R
1445:c
1442:b
1437:R
1433:+
1427:b
1424:a
1419:R
1415:+
1409:c
1406:a
1401:R
1392:c
1389:a
1384:R
1377:c
1374:b
1369:R
1362:=
1353:c
1349:R
1335:c
1332:b
1327:R
1323:+
1317:b
1314:a
1309:R
1305:+
1299:c
1296:a
1291:R
1282:c
1279:b
1274:R
1267:b
1264:a
1259:R
1252:=
1243:b
1239:R
1225:c
1222:b
1217:R
1213:+
1207:b
1204:a
1199:R
1195:+
1189:c
1186:a
1181:R
1172:b
1169:a
1164:R
1157:c
1154:a
1149:R
1142:=
1133:a
1129:R
1083:)
1078:2
1075:n
1070:(
1055:n
1051:n
1014:.
1006:2
1002:Z
998:+
993:1
989:Z
981:2
977:Z
971:1
967:Z
960:=
954:q
951:e
946:Z
923:.
916:n
912:Z
908:1
903:+
895:+
888:2
884:Z
880:1
875:+
868:1
864:Z
860:1
855:=
847:q
844:e
839:Z
835:1
807:.
802:n
798:Z
794:+
786:+
781:2
777:Z
773:+
768:1
764:Z
760:=
754:q
751:e
746:Z
696:1
693:V
676:1
672:I
668:=
663:2
659:I
636:1
632:V
628:=
623:2
619:V
34:.
20:)
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