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- A network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, every component in the network. There are a number of different techniques for achieving this. However, for the most part, they assume that the components of the network are all linear. The methods described in this article are only applicable to linear network analysis except where explicitly stated. Definitions Equivalent circuits A useful procedure in network analysis is to simplify the network by reducing the number of components. This can be done by replacing the actual 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 in to 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 sources, and ideal voltage sources. If the sources are constant sources, the result is a DC circuit. The analysis of a circuit refers to the process of solving for the voltages and currents present in the circuit. The solution principles outlined here also apply to phasor analysis of AC circuits. Two circuits are said to be equivalent with respect to a pair of terminals if the voltage across the terminals and current through the terminals for one network have the same relationship as the voltage and current at the terminals of the other network. If V_2V_1 implies I_2I_1 for all (real) values of V_1, then with respect to terminals ab and xy, circuit 1 and circuit 2 are equivalent. 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 Any two terminal network of impedances can eventually be reduced to a single impedance by successive applications of impendances in series or impendances in parallel. Impedances in series: Z_\mathrm{eq} Z_1 + Z_2 + \,\cdots\, + Z_n. Impedances in parallel: \frac{1}{Z_\mathrm{eq}} \frac{1}{Z_1} + \frac{1}{Z_2} + \,\cdots\, + \frac{1}{Z_n} . The above simplified for only two impedances in parallel: Z_\mathrm{eq} \frac{Z_1Z_2}{Z_1 + Z_2} . Delta-wye transformation A network of impedances with more than two terminals cannot be reduced to a single impedance equivalent circuit. An n-terminal network can, at best, be reduced to n impedances. For a three terminal network, the three impedances can be expressed as a three node delta (Δ) network or a 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 terminals 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. It can be shown that this is sufficient to find the minimal network for any arbitrary network with successive applications of series, parallel, Y-Δ and Δ-Y; no more complex transformations are required. 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 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}} Star-to-delta transformation equations R_\mathrm{ac} \frac{R_aR_b + R_bR_c + R_cR_a}{R_b} R_\mathrm{ab} \frac{R_aR_b + R_bR_c + R_cR_a}{R_c} R_\mathrm{bc} \frac{R_aR_b + R_bR_c + R_cR_a}{R_a} General form of network node elimination The star-to-delta and series-resistor transformations are special cases of the general resistor network node elimination algorithm. Any node connected by N resistors (R_1 .. R_N) to nodes 1 .. N can be replaced by {N \choose 2} resistors interconnecting the remaining N nodes. The resistance between any two nodes x and y is given by: R_\mathrm{xy} R_xR_y\sum_{i1}^N \frac{1}{R_i} For a star-to-delta (N3) this reduces to: R_\mathrm{ab} R_aR_b(\frac 1 R_a+\frac 1 R_b+\frac 1 R_c) \frac{R_aR_b(R_aR_b+R_aR_c+R_bR_c)}{R_aR_bR_c}\frac{R_aR_b + R_bR_c + R_cR_a}{R_c} For a series reduction (N2) this reduces to: R_\mathrm{ab} R_aR_b(\frac 1 R_a+\frac 1 R_b) \frac{R_aR_b(R_a+R_b)}{R_aR_b} R_a+R_b For a dangling resistor (N1) it results in the elimination of the resistor because {1 \choose 2} 0. Source transformation A generator with an internal impedance (ie 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 V and I must be identical for both networks. Thus, V_\mathrm{s} RI_\mathrm{s}\,\! or I_\mathrm{s} \frac{V_\mathrm{s}}{R} Norton's theorem states that any two-terminal network can be reduced to an ideal current generator and a parallel impedance. Thévenin's theorem states that any two-terminal network can be reduced to an ideal voltage generator plus a series impedance. Simple networks Some very simple networks can be analysed without the need to apply the more systematic approaches. Voltage division of series components Consider n impedances that are connected in series. The voltage V_i across any impedance Z_i is V_i Z_iI \left(\frac{Z_i}{Z_1 + Z_2 + \cdots + Z_n} \right)V Current division of parallel components Consider n impedances that are connected in parallel. The current I_i through any impedance Z_i is I_i \left(\frac{\left}{ \left + \left + \,\cdots\, + \left} \right)I for i 1,2,... ,n. Special case: Current division of two parallel components I_1 \left(\frac{Z_2}{Z_1 + Z_2} \right)I I_2 \left(\frac{Z_1}{Z_1 + Z_2} \right)I 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 KCL equation for every node except the reference. 4. Solve the resulting system of equations. Mesh analysis Mesh — 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 KVL equation for every mesh whose current is unknown. 3. Solve the resulting equations Superposition In this method, the effect of each generator in turn is calculated. All the generators other than the one being considered are removed; 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 across, a particular branch is then calculated by summing all the individual currents or voltages. There is an underlying assumption to this method that the total current or voltage is a linear superposition of its parts. The method cannot, therefore, be used if non-linear components are present. Note that mesh analysis and node analysis also implicitly use superposition so these too, are only applicable to linear circuits. Choice of method 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. 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. 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. Nodal analysis is thus best for voltage sources. 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 is thus best for current sources. Mesh analysis, however, cannot be used with networks which cannot be drawn as a planar network, that is, with no crossing components. Transfer function A transfer function 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 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 Laplace transform on them first and then express the result in terms of the Laplace parameter s, which in general is complex. This is described as working in the s-domain. 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. This approach is standard in control theory and is useful for determining stability of a system, for instance, in an amplifier with feedback. Two terminal component transfer functions For two terminal components the transfer function 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; For a network to which only steady ac signals are applied, s is replaced with jω and the more familiar values from ac network theory result. Finally, for a network to which only steady dc is applied, s is replaced with zero and dc network theory applies. Two port network transfer function 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), A(jω), so that; A(j\omega)\frac{V_o}{V_i} 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 magnitiude 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; A(\omega)\left|{\frac{V_o}{V_i}}\right| Two port parameters 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 (ie, 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; \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} The matrix may be abbreviated to a representative element; \left [z(j\omega) \right] or just \left [z \right] These concepts are capable of being extended to networks of more than two ports. However, this is rarely done in reality as 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 decomposed into a number of two-port networks. Distributed components 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 distributed components, such as in the case of a transmission line, 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 lumped components. Image analysis 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 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. Non-linear networks Most electronic designs are, in reality, non-linear. There is very little that does not include some semiconductor devices. These are invariably non-linear, the transfer function of an ideal semiconductor pn junction is given by the very non-linear relationship; i I_o (e^{\frac{v}{V_T}}-1) where; i and v are the instantaneous current and voltage. Io is an arbitrary parameter called the reverse leakage current whose value depends on the construction of the device. VT 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. 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 obtain. Boolean analysis of switching networks 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 the device is actually switching. Here the non-linearity is designed to be extreme, and the analyst can actually take advantage of that fact. These kinds of networks can be analysed using Boolean algebra by assigning the two states ("on"/"off", "positive"/"negative" or whatever states are being used) to the boolean constants "0" and "1". The transients are ignored in this analysis, along with any slight discrepancy between the actual 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 actually be +4.5V but the analyst still considers this to be boolean "1". Device manufacturers will usually specify a range of values in their data sheets that are to be considered undefined (ie the result will be unpredictable). 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 analysis can be applied to obtain at least an approximate answer. It is mathematically possible to derive boolean algebras which have more than two states. There is not too much use found for these in electronics, although three-state devices are passingly common. 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. A transistor amplifier is an example of this kind of network. The essence of this technique is to separate the analysis in to two parts. Firstly, the dc biases are analysed using some non-linear method. This establishes the quiescent operating point of the circuit. Secondly, the small signal characteristics of the circuit are analysed using linear network analysis. Examples of methods that can be used for both these stages are given below. 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). 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. 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 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. 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. 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 achieve it. 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 consequently more tedious to do. Small signal equivalent circuit 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. 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 through the origin. With more terminals, more complicated equivalent circuits are required. A popular form of specifying the small signal equivalent circuit amongst transistor manufacturers is to use the two-port network parameters known as h] parameters. These are a matrix of four parameters as with the [z] parameters but in the case of the [h] 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 [h] 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, h21, in the common emitter configuration. This is designated hfe on data sheets. 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 [z] parameter model leads to dependent voltage generators as shown in this diagram; File:Z-equivalent two port. png [z] parameter equivalent circuit showing dependent voltage generators There will always be dependent generators in a two-port parameter equivalent circuit. This applies to the [h] parameters as well as to the [z] and any other kind. These dependencies must be preserved when developing the equations in a larger linear network analysis. Piecewise linear method 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 again be linear but with a different slope. A well known application of this method is the approximation of the transfer function of a pn junction diode. The actual 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 used instead. It can be seen that the diode current rapidly diminishes to -Io 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 bulk resistance of the semiconducting material. 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. 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. These can therefore both be analysed in the same operations and will be linearly superimposable. 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 amplifiers, and variable 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 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 Foster and Wilhelm Cauer used for analysing linear circuits. See also Bartlett's bisection theorem Circuit theory Equivalent impedance transforms Kirchhoff's circuit laws Mesh analysis Millman's Theorem Ohm's law Reciprocity theorem Resistive circuit Series and parallel circuits Tellegen's theorem Two-port network Wye-delta transform References External links Circuit Analysis Techniques — includes node/mesh analysis, superposition, and thevenin/norton transformation Nodal Analysis of Op Amp Circuits Analysis of Resistive Circuits Circuit Analysis Related Laws, Examples and Solutions
- Als Netzwerkanalyse bezeichnet man in der Elektrotechnik die Vorgehensweise in einem Netzwerk (siehe Bild) aus den bekannten Werten der Schaltelemente sowie den vorgegebenen Quellgrößen alle Ströme und Spannungen zu berechnen. Von Hand und mit analytischen Methoden können mit realistischem Aufwand nur lineare Systeme untersucht werden. Die rechnergestützte Schaltungssimulation dagegen beruht vorwiegend auf iterativen Näherungsverfahren, benötigt sehr viele Rechenschritte, kann aber auch mit nichtlinearen Bauelementen umgehen. Allgemein In einem Netzwerk sind die Zusammenhänge zwischen allen auftretenden Strömen bzw. allen auftretenden Spannungen durch die nach dem deutschen Physiker Gustav Robert Kirchhoff benannten kirchhoffschen Regeln beschrieben. Der Zusammenhang zwischen Strom und Spannung ist durch das ohmsche Gesetz beschrieben, welche die Bauelementegleichung von Widerständen beschreibt. Voraussetzung sind reelle lineare Schaltelemente, d. h. kein kapazitiver oder induktiver Anteil und eine gerade Kennlinie im Gegensatz z. B. zur Diode. Bei nicht reellen Widerständen wird die sog. komplexe Rechnung erforderlich. So lässt sich auch eine Analyse für Wechselspannung durchführen, wobei jede betrachtete Frequenz einzeln zu berechnen ist. Damit eine Netzwerkanalyse möglich ist, werden in dem Netzwerk Knotenpunkte, Zweige und Maschen definiert. Mithilfe der kirchhoffschen Regeln können ihnen dann Gleichungen zugeordnet werden. Damit die mathematischen Gleichungssysteme zu einer eindeutigen Lösung führen, müssen die jeweiligen Gleichungen voneinander unabhängig sein. Ein Knotenpunkt ist dabei ein Punkt im Netzwerk, in dem eine Stromverzweigung auftritt. Ein Netzwerk beinhaltet dann k</math> Knotenpunkte. Insgesamt gibt es (k-1)</math> unabhängige Knotengleichungen, eine hiervon in dem gezeigten Beispiel ist I_1 - I_2 + I_3 0</math>. Ein Zweig ist die Verbindung zweier Knoten durch Zweipolelemente. Insgesamt gibt es z</math> unabhängige Zweiggleichungen. Im dargestellten Beispiel sind die Zweiggleichungen nach Ausnutzung der Bauelementgleichungen: U_{z1}R_1 \cdot I_1 - U_{q1}</math> U_{z2}R_2 \cdot I_2</math> U_{z3}R3 \cdot I_3 + R4 \cdot I_3 - U_{q2}</math> Als Baum bezeichnet man ein Gerüst aus Zweigen, welches alle Knoten verbindet, wobei kein Knoten zweimal berührt werden darf. Anschaulich ausgedrückt, die gebildete Struktur darf keine Möglichkeiten bieten, um im Kreis zu gehen. Für den Baum sind verschiedene Varianten möglich. Insgesamt sind bei einem vollständig vermaschten Netzwerk (jeder Knoten hat einen Zweig zu jedem anderen Knoten) k^{k-2}</math> Varianten denkbar. Im vorliegenden Beispiel ergeben sich drei unterschiedliche Bäume, da mehr als ein Zweig zwei Knoten miteinander verbindet. Die einzelnen Zweige im Baum werden Baumzweige oder Äste genannt. Wegen des Aufbaus des Baumes gibt es a k-1</math> Äste. Alle Zweige, die nicht zum Baum gehören, bezeichnet man als Sehnen oder auch Verbindungszweige. Dies entspricht der Zahl unabhängiger Maschengleichungen (z-)</math>, mit denen ein Gleichungssystem aufgestellt werden kann. Somit existieren k-1</math> unabhängige Knotengleichungen und m z-(k-1)</math> unabhängige Maschengleichungen. Eine Masche ist ein über Zweige geschlossener Umlauf. Für eine einfache Analyse sollte stets ein Umlauf über nur eine Sehne bzw. einen Verbindungszweig gewählt werden. Für den Fortlauf wird dieser Weg genutzt. Der Umlaufsinn der m</math> unabhängigen Maschen kann willkürlich festgelegt werden, ist jedoch relevant für spätere Berechnungen. Im Beispiel sind die folgenden zwei Maschenumläufe gewählt. M1:\ -U_{q1} + U_1 + U_2 0</math> M2:\ U_{q2} - U_3 - U_4 - U_2 0 </math> Zweigstromanalyse Zur Lösung mittels Zweigstromanalyse werden alle unabhängigen Knotengleichungen und die unabhängigen Maschengleichungen aufgestellt. Anschließend werden diese sortiert, indem man diese nach Strom/Widerstand auf der einen Seite der Gleichung und Spannungen auf der anderen Seite aufreiht. Als Ergebnis erhält man ein lineares Gleichungssystem. Im gezeigten obigen Beispiel folgt hiermit dann in geordneter Reihenfolge (Knotengleichung 1, Maschengleichung 1 und Maschengleichung 2): I_1 - I_2 + I_3 0,\quad\quad I_1 R_1+ I_2 R_2 U_{q1}, \quad\quad- I_2 R_2 - I_3 R_3 - I_3 R_4 -U_{q2}</math> In Matrixschreibweise lautet nun das Gleichungssystem: \begin{pmatrix} R_1 & R_2 & 0 \\ 0 & - R_2 & - R_3 - R_4 \end{pmatrix} \cdot \begin{pmatrix} I_1 \\ I_2 \\ I_3 \end{pmatrix} \begin{pmatrix} 0 \\ U_{q1} \\ -U_{q2} \end{pmatrix}</math> Zur Lösung des linearen Gleichungssystems gibt es Standardmethoden die hierfür genutzt werden können. Kleinere Gleichungssysteme lassen sich analytisch „von Hand“ lösen, für umfangreichere Schaltkreise werden numerische Methoden (Computerprogramme) verwendet. (Ein Beispiel ist auf der Diskussionsseite) Überlagerungsverfahren nach Helmholtz Das Überlagerungsverfahren beruht auf dem Superpositionsprinzip bei linearen Systemen. Vorgehen Bis auf eine Quelle werden alle anderen entfernt. Spannungsquellen werden durch Kurzschlüsse ersetzt bzw. Stromquellen als Unterbrechung gesehen. Die Innenwiderstände der Quellen verbleiben jedoch in der Schaltung. Die gesuchten Teilströme mit der verbliebenen Quelle berechnen. Das Vorgehen für jede andere Quelle wiederholen. Zum Schluss die vorzeichenrichtige Addition der errechneten Teilströme für die betrachteten Zweige durchführen. Ergebnis Der gesuchte Teilstrom wurde ermittelt. Maschenstromverfahren Mit zunehmender Komplexität steigt der Aufwand zur Berechnung des Netzwerks mit der Zweigstromanalyse. Eine Reduzierung des Rechenaufwands ergibt sich durch das Maschenstromverfahren. Vorgehen (Kurzform) Netzwerk vereinfachen Baum wählen, ideale Stromquellen als Sehne Nicht Ideale Stromquellen in eine äquivalente Spannungsquelle umwandeln Maschen festlegen Matrix aufstellen: \begin{pmatrix} r_{11} & \dots & r_{1n} \\ \vdots & \ddots & \vdots \\ r_{n1} & \dots & r_{nn} \end{pmatrix} \cdot \begin{pmatrix} i_{M1} \\ \vdots \\ i_{Mn} \end{pmatrix} \begin{pmatrix} u_{qM1} \\ \vdots \\ u_{qMn} \end{pmatrix} </math> r_{ii} \sum R\ in\ M_i</math> r_{ij} r_{ji} \sum [R\ in\ M_i\ und\ M_j \cdot Umlaufsinn(M_i,M_j)]</math> u_{qMi} \sum [U_q\ in\ M_i \cdot -1 \cdot Pfeil(U_q,M_i)]</math> Sonderfall ideale Stromquellen (Iq) u_{qMi} \sum [U_q\ in\ M_i \cdot -1 \cdot Pfeil(U_q,M_i)] - I_q \cdot \sum [R\ in\ M_i\ und\ M_{Iq} \cdot Umlaufsinn(M_i,M_{Iq})]</math> Gleichungssystem lösen Zweigströme berechnen anhand der Summe der Maschenströme I_1 I_{M1} + I_{M2} - I_{M3}</math> Knotenpotentialverfahren Wie beim Maschenstromverfahren ergibt sich beim Knotenpotentialverfahren ein reduziertes lineares Gleichungssystem. Vorgehen (Kurzform) Spannungsquellen in äquivalente Stromquelle umwandeln Bezugsknotenpotential ("Masse") wählen, ideale Spannungsquellen an Bezugspotential angeschlossen (ansonsten mit dieser Anleitung nicht lösbar) Restliche Knoten durchnummerieren Matrix aufstellen, Knoten mit idealer Spannungsquelle weglassen \begin{pmatrix} g_{11} & \dots & g_{1n} \\ \vdots & \ddots & \vdots \\ g_{n1} & \dots & g_{nn} \end{pmatrix} \cdot \begin{pmatrix} u_{10} \\ \vdots \\ u_{n0} \end{pmatrix} \begin{pmatrix} i_{q1} \\ \vdots \\ i_{qn} \end{pmatrix} </math> g_{ii} \sum G\ mit\ Knoten\ i\ verbunden</math> g_{ij} g_{ji} -1 \cdot \sum G\ zwischen\ den\ Knoten\ i\ und\ j\ (Koppelleitwerte)</math> i_{qi} \sum I_q\ an\ Knoten{,}\ I_q\ vom\ Knoten\ weg\ flie \beta end\ sind\ negativ</math> Bei mit idealer Spannungsquelle gekoppelter Knoten, Spannungsquelle mit Koppelleitwert multiplizieren und zum Stromvektor addieren, wenn der Spannungspfeil zum Bezugsknotenpotential hin zeigt, andernfalls abziehen. Gleichungssystem lösen. Zweigspannung Uij ui0-uj0 aus den Knotenpotentialen ermitteln und daraus den Zweigstrom erreichen. Literatur Singhal, K. Vlach, J. : Computer Methods for Circuit Analysis and Design. Verlag Springer US, ISBN 978-0-442-01194-9, 2. Aufl. 1993 Führer, Heidemann, Nerreter – Grundgebiete der Elektrotechnik, Band 1: Stationäre Vorgänge, Carl Hanser Verlag München 1986, ISBN 3-446-13677-0 Elschner, H. , Moeschwitzer, A. , Reibiger, A. : Rechnergestuetzte Analyse in der Elektronik. Reihe Informationselektronik. Verlag Technik Berlin, 1977 Nagel, L.W. : Spice 2 - A Computer Programm to Simulate Semiconductor Circuits. University of Cal. , Berkeley, CA, USA, Erl-Memorandum, Nr. Erl-M520, May 1975 Nagel, L. W. , Rohrer, R. A. : Computer Analysis of Nonlinear Circuits, Excluding Radiation. IEEE Journal of Solid State Circuits SC-6: 166–182, Aug. 1971 Siehe auch Schaltungssimulation, rechnergestütze Netzwerkanalyse Netzwerk (Elektrotechnik) Norton-Theorem, Ersatzstromquelle Thévenin-Theorem, Ersatzspannungsquelle Netzwerkanalysator SPICE
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