autoCircuits
PoliTO

The Circuit Solution Tree

The solution of an electric circuit can be seen as a sequence of steps, where each step corresponds to application of a single and well-defined method. Series-parallel reduction or Superposition Theorem are just examples of methods that do not provide the value of a given circuit variable, but rather approaches that need to be combined with other methods in order to reach the final desired solution. It is straightforward to note that the process of applying such step-by-step solutions can be organized in a graph, where nodes represent circuits and edges/connectors represent methods that trasform a circuit into another circuit. This graph is here called the Circuit Solution Tree. A key feature of the iCircuits service is the interactive visualization and modification of the Circuit Solution Tree as the circuit is being solved by selecting the desired methods. This document provides some basic information on how to interpret the various components of a Circuit Solution Tree, namely the nodes, the connectors and in particular the templates associated to each circuit analysis method.

Circuit Solution Tree: Nodes

Each node in the Circuit Solution Tree corresponds to a well-defined circuit, equipped with data and problem statements. A given node may correspond to a circuit to be computed or derived, or to the final solution to be computed, in which case it is a placeholder to highlight graphically future operations. Hovering with the mouse over a node displays a tooltip with a minimal explanation. Clicking on a node will display the corresponding circuit in the main page.

Each node is depicted with a border and a fill color. The border color defines whether the corresponding circuit is completely available or still undefined. The fill color defines whether the solution of the corresponding circuit has been computed or is still pending. The table below summarizes the various cases that may occur.

Root node

This is the first node of the Circuit Solution Tree, which corresponds to the initial problem statement, when solution has not been started yet. As soon as the first solution step is completed, this node will be redefined to become a "Completed node".

Result node

This is the last node the Circuit Solution Tree. This node corresponds to the final solution, when the solution has not been computed yet. At the end of the solution process, this node will be redefined to become a "Completed node".

Ready node

Any node of the Circuit Solution Tree, for which the solution has not been computed yet, and which is ready for performing one solution step (no dependency on other operations, both circuit and associated data are available). The same graphical notation of the Root node at the beginning of a solution process is used, since a Ready node is the root of the subtree that originates from its solution.

Active node

This is the single node of the Circuit Solution Tree that is currently selected for performing an operation. This particular circuit must be processed by some analysis method in order to advance the solution process.

Future node

Any node of the Circuit Solution Tree, for which the solution has not been computed yet, and which is NOT ready for performing a solution step. This node will have to be processed in a future step, but either circuit or associated data are NOT yet available. To be considered as a placeholder in the Circuit Solution Tree, for later steps that are expected. The same graphical notation of the Result node is used, since the latter is in fact a Future node (with no children nodes).

Completed node

Any node of the Circuit Solution Tree, for which the solution has already been computed. This is to be considered as a pointer to a completed operation, for which initial circuit and corresponding solution are available.

Circuit Solution Tree: Connectors

Each connector corresponds to all steps to be performed by applying a specific method to a circuit node in the tree, in order to obtain another circuit node. Possibly, some methods map a circuit node to several circuit nodes (e.g., superposition) or many circuit nodes to a single circuit node (e.g., when collecting results obtained from superposition). Hovering with the mouse over a connector displays a tooltip with a minimal explanation of the corresponding method. Clicking on a connector will display the solution process resulting from application of the corresponding method.

Each connector is depicted with a line connecting two nodes, with a line color and style. The line color defines whether the corresponding operation has been performed or is still pending. The line style defines whether the corresponding operation is already determined or is still undefined. The table below summarizes the various cases that may occur.

Completed path

One-to-one connector between two nodes, corresponding to an operation that has already been performed. The source node is completed, the target node can be either completed or ready.

Future path

One-to-one connector between two nodes, corresponding to an operation that has not been performed yet but has been decided already. The source node is ready, the target node is future.

Undefined path

One-to-one connector between two nodes, corresponding to an operation that has not been defined yet. The source node can be ready or future, the target node is future.

Completed path (splitting)

One-to-many connectors between two node sets, corresponding to an operation that splits a node into multiple circuits, to be solved independently. The source node must be in completed state, since the only operation involved in this case is the actual splitting. Individual operations are also marked as completed following the splitting. The target nodes may be in completed or ready state, depending on whether they have been processed by later steps or not.

Future path (collection)

Many-to-one connectors between two node sets, corresponding to an operation that collects results from multiple nodes originated by a splitting. The source nodes can be in any state. The target node can be in future state (in which case the connectors are also in future state, since collection of results has not been performed yet) or ready/completed state, in which case also connectors are in completed state.

Circuit Solution Tree: Methods and Templates

Each circuit analysis method involves a number of well-defined operations. Such operations can be formalized as a sequence of individual steps, which once again can be represented in form of a graph. This idea leads to the abstract templates that represent all steps required by a circuit analysis method in the Circuit Solution Tree. Templates are subgraphs of the global Circuit Solution Tree, that are inserted by replacing an undefined connector whenever a given method is selected by the user or by the solution process.

The User has the possibility to select the desired method by clicking on the corresponding button

A special feature of the iCircuits service is the ability to present the User with a list of analysis methods that are applicable to the circuit under processing. Methods that are not appropriate or not applicable (or not yet implemented!) are not listed. Method that have a high priority (i.e. likely to significantly simplify the overall solution process) are highlighted with a green color, whereas methods with a lower priority (i.e. potentially more complicated or requiring more efforts) are highlighted with a red color.

If the User selects the desired method by clicking on the corresponding button, this method will be applied when submitting the request to the iCircuits server. If no selection is performed, the first (leftmost) method will be automatically chosen. Pressing the Run to end button will iteratively apply the first listed (highest priority) method in batch-mode, without User interaction, until the circuit solution is obtained.

The following table lists all (implemented or planned) methods with the corresponding templates, by showing the portion of the Circuit Solution Tree that is involved before and after applying the methods.

Initial circuit Solution

Initial Configuration

This is the default configuration that is initialized when a new problem is generated. Source node is the original circuit, target node is the solution to be computed. The connector is undefined since no method has been chosen yet.

Before

Initial circuit Solution

After

Initial circuit Obtained by series-parallel reduction Solution Series-parallel reduction

Series-Parallel Reduction

This method performs one step of circuit reduction by:

  1. Redrawing the circuit with highlighted pairs or groups of resistances/impedances connected in series or parallel;
  2. Evaluating and typesetting the values of the equivalent resistances/impedances for each group;
  3. Redrawing the circuit with the equivalents replacing the original groups;
  4. Stating again the problem assignment.

The above steps correspond to the completed edge in the template.

Problem statement of the new active circuit is the same as the original circuit.

Before

Initial circuit Solution

After

Initial circuit Obtained by Star-Delta reduction Solution Star-Delta reduction

Star-Delta Reduction

This method performs one step of circuit reduction by:

  1. Redrawing the circuit with highlighted groups of resistances/impedances connected in a star or delta configuration;
  2. Evaluating and typesetting the values of the equivalent resistances/impedances for each group;
  3. Redrawing the circuit with the equivalents replacing the original groups;
  4. Stating again the problem assignment.

The above steps correspond to the completed edge in the template.

Problem statement of the new active circuit is the same as the original

Before

Initial circuit Solution

After

Initial circuit Superposition circuit n.1 Superposition circuit n.2 Superposition circuit n.3 Solution of superposition circuit n.1 Solution of superposition circuit n.2 Solution of superposition circuit n.3 Solution Splitting by superposition Splitting by superposition Splitting by superposition Collection of superposition results Collection of superposition results Collection of superposition results

Superposition

This method applies superposition to evaluate one or more circuit variables. The various steps are:

  1. Splitting the original circuit into a set of disjoint circuits, each with a single independent source; each of these circuits will have the same problem statement of the original circuit. The first (leftmost) circuit is marked as active;
  2. Solving each independent circuit. At the time of splitting, the method is still not known, so that the corresponding edges are left undefined;
  3. Collect individual results and apply superposition (at the time of splitting, this step is known but cannot be applied until all individual circuits have been solved).

This template only operates the splitting when invoked, but it initializes all data structures also for those future steps that are known. See the template "Superposition (collect)", which defines and implements the above step 3.

Before

Contribution n.1 Contribution n.2 Contribution n.3 Solution Collect results and sum contributions Collect results and sum contributions Collect results and sum contributions

After

Contribution n.1 Contribution n.2 Contribution n.3 Solution Collect results and sum contributions Collect results and sum contributions Collect results and sum contributions

Superposition (collect)

This method collects results of a previous splitting by superposition, evaluates the sum of individual contributions, and presents the result.

This template only exists in combination with (following) a "Superposition" template. It is activated when all contributions are available, i.e. when the corresponding nodes are in ready state.

Before

Initial circuit Solution

After

Initial circuit Circuit for initial conditions Circuit for asymptotic (DC) solution Circuit for time constant Initial conditions Asymptotic (DC) solution Time constant Solution Derive circuit for initial conditions Derive circuit for asymptotic (DC) solution Derive circuit for time constant Collect partial results and write solution Collect partial results and write solution Collect partial results and write solution

First-order circuits with DC sources

This method is invoked when the problem statement is to solve a I order circuit with DC sources and switches. The solution process involves:

  1. Splitting the original circuit into a set of disjoint circuits; from left to right:
    1. Find initial conditions. This step is further split into the following consecutive operations:
      1. Find initial conditions on state variables, for \(t=0-\). This requires to set all switches to their position for \(t<0\), set all sources to their values for \(t<0\), replace capacitors with open circuits, inductors with short circuits, and redraw the circuit by declaring as problem statement to solve at DC steady-state the values of the state variables.
      2. Freeze the circuit at \(t=0+\) by setting the switches and the sources to their values for \(t>0\), and replacing the capacitors and the inductors with voltage and current sources whose values are equal to the initial conditions at \(t=0-\). For this operation, the template "Fix State Variables" is used. The problem statement is redefined to evaluate all initial conditions on the desired variables for \(t=0+\).
    2. Find asymptotic solution for \(t\to\infty\). This operation requires to set switches and sources to their configuration for \(t>0\), and evaluating the desired variables. The problem statement is redefined to compute the DC steady state values of the desired variables.
    3. Find time constant \(\tau\). This operation requires to switch off all sources, set the switches to their position for \(t>0\), remove the dynamic element, and set the problem statement to evaluate the equivalent resistance \(R_{eq}\). After this operation, the result is collected to evaluate the time constant. To this end, method "Find time constant" is applied.
  2. Solving each independent circuit (at the time of splitting, the methods are still undefined, except for the particular step of generating circuit at \(t=0+\));
  3. Collect individual results and write solution. This is achieved by dedicated method "First-order (collect)".

Before

Initial circuit State variable values Solution Construct circuit with fixed state variables Construct circuit with fixed state variables

After

Initial circuit State variable values Solution Construct circuit with fixed state variables Construct circuit with fixed state variables

Fix state variables (collect)

This method starts with a given dynamic circuit and with a set of given values of state variables, and replaces the corresponding dynamic elements with independent sources whose value matches the provided values.

This template only exists in combination with (following) other templates, such as "First-order circuits with DC sources" and "State variables". It is activated when all contributions are available, i.e. when the corresponding nodes are in ready state.

Before

Initial circuit Solution

After

Initial circuit Circuit for equivalent resistance Equivalent resistance Time constant Derive circuit for equivalent resistance Evaluate time constant Evaluate time constant

Find Time Constant

Consider a generic I order circuit with arbitrary sources and find the time constant. First, compute equivalent resistance seen by the dynamic element with all sources switched off and all switches in their position for \(t>0\). Then, evaluate the time constant and return the result.

Before

Value of Req Value and type of dynamic element Time constant Evaluate time constant Evaluate time constant

After

Value of Req Value and type of dynamic element Solution Evaluate time constant Evaluate time constant

Evaluate time constant (collect)

This method starts with a known value of \(R_{eq}\) and a known type/value of dynamic element, and evaluates the time constant of the corresponding I order circuit.

This template only exists in combination with (following) the template "Find Time Constant". It is activated when all contributions are available, i.e. when the corresponding nodes are in ready state.

Before

Initial conditions Asymptotic (DC steady-state) solutions Time constant Solution Collect results and write solution Collect results and write solution Collect results and write solution

After

Initial conditions Asymptotic (DC steady-state) solutions Time constant Solution Collect results and write solution Collect results and write solution Collect results and write solution

First-order (collect)

This method collects partial results obtained after splitting the solution of a I order circuit with DC sources and switches into initial conditions, steady-state (asymptotic solution), and time constant.

This template only exists in combination with (following) a "First-order circuits with DC sources" template. It is activated when all contributions are available, i.e. when the corresponding nodes are in ready state.

Before

Initial circuit Solution

After

Initial circuit Solution Solution via KCL/KVL

Kirchhoff Laws

Solution of circuit through a system of fundamental KCL/KVL equations based on a randomly-generated fundamental tree. Only the equations that are strictly necessary to compute the required output variables are formulated and solved.

The last node could be both ready (if the solution is part of a more complex method, i.e. superposition) or completed (if the solution is the objective of the exercise).

Before

Initial circuit Solution

After

Initial circuit Obtained by generalized series reduction Solution Generalized series reduction

Generalized series of resistances/impedances

As series reduction, but applied to identify and simplify groups of resistances or impedances connected pairwise in cutsets (generalized series).

Before

Initial circuit Solution

After

Initial circuit Obtained by generalized series reduction Solution Generalized series reduction

Generalized series of voltage sources

As source reduction of voltage sources connected in series, but applied to identify and simplify groups of voltage sources connected pairwise in cutsets (generalized series).

Before

Initial circuit Solution

After

Initial circuit Obtained by source reduction Solution Source reduction

Source reduction

Circuit simplification by identification of groups of series-connected voltage sources and/or parallel-connected current sources, and replacement with suitable equivalents.

Before

Initial circuit Solution

After

Initial circuit Obtained by Substitution Theorem Solution Substitution Theorem

Substitution Theorem applied to single sources

Substitution theorem is applied to remove all elements that are connected in parallel to voltage sources or in series with current sources.

Before

Initial circuit Solution

After

Initial circuit Solution Solution via Simplified Nodal Analysis

Simplified Nodal Analysis

Simplified Nodal Analysis writes a minimal number of KCL at circuit nodes or supernodes, by eliminating unknowns that are not strictly necessary (voltages at nodes connected by voltage sources, and currents through voltage sources).

Before

Initial circuit Solution

After

Initial circuit Obtained by short/open circuit removal Solution Removal of short/open circuits

Simplify by removing short/open circuits

This method simplifies the circuit graph by collapsing nodes connected by short circuits and removing open circuits. This method is useful in combination with superposition (which switches off sources leaving open/short circuits), transients with switches and piecewise constant sources including the Heaviside step function.

This method should be given highest priority and should be applied whenever there is a short or open-circuit that can be eliminated.

Before

Initial circuit Solution

After

Initial circuit Obtained by pruning circuit Solution Removal of unnecessary circuit parts

Simplify by pruning connected subgraphs

This method simplifies the circuit graph by detecting the biconnected subgraphs that do not include output variables and that are not coupled through element characteristics to circuit parts that include output variables. This method is useful in combination with superposition (which switches off sources leaving open/short circuits), transients with switches and piecewise constant sources including the Heaviside step function.

This method should be given highest priority and should be applied whenever there is a short or open-circuit that can be eliminated. In fact, due to its advanced nature, this method is applied automatically as a post-processing without user interaction.

Before

Initial circuit Solution

After

Initial circuit Circuit for Req calculation: sources are switched off Circuit for Veq calculation: open terminals Solution Solution for Req Solution for open-circuit voltage Switch off independent sources Leave terminals open Construct Thevenin equivalent Construct Thevenin equivalent

Thevenin equivalent: two-step

Evaluation of Thevenin equivalent by computing separately the equivalent resistance and the open-circuit voltage. Two dedicated circuits are prepared, whose solution is undefined until in the last step the results are collected and the equivalent is constructed.

Before

Initial circuit Solution

After

Initial circuit Obtained by connecting external current source Solution Solution for voltage across current source Connect external current source Extract Req and Veq from solution

Thevenin equivalent: one-step

Evaluation of Thevenin equivalent by excitation through external current source. This method only constructs the basic circuit that includes the original one-port plus an externally-connected current source. The new active circuit will have to be solved for the voltage across the current source. The last (future) solution step will extract \(R_{eq}\) and \(V_{eq}\) from the solution.

Before

Initial circuit Solution

After

Initial circuit Circuit for Geq calculation: sources are switched off Circuit for Ieq calculation: open terminals Solution Solution for Geq Solution for short-circuit current Switch off independent sources Connect short-circuit at output terminals Construct Norton equivalent Construct Norton equivalent

Norton equivalent: two-step

Evaluation of Norton equivalent by computing separately the equivalent conductance and the short-circuit current. Two dedicated circuits are prepared, whose solution is undefined until in the last step the results are collected and the equivalent is presented.

Before

Initial circuit Solution

After

Initial circuit Obtained by connecting external voltage source Solution Solution for current through voltage source Connect external voltage source Extract Geq and Ieq from solution

Norton equivalent: one-step

Evaluation of Norton equivalent by excitation through external voltage source. This method only constructs the basic circuit that includes the original one-port plus an externally-connected voltage source. The new active circuit will have to be solved for the current through the voltage source. The last (future) solution step will extract \(G_{eq}\) and \(I_{eq}\) from the solution.

Before

Solution for Req Solution for open-circuit voltage Thevenin equivalent Construct Thevenin equivalent Construct Thevenin equivalent

After

Solution for Req Solution for open-circuit voltage Thevenin equivalent Construct Thevenin equivalent Construct Thevenin equivalent

Assemble Thevenin equivalent: two-step

This method combines partial results obtained in the two-step evaluation of Thevenin equivalent and assembles the Thevenin equivalent.

This template only exists in combination with the other template, "Thevenin equivalent: two-step". It is activated when all contributions are available, i.e. when the corresponding nodes are in ready state.

Before

Solution for Geq Solution for short-circuit current Norton equivalent Construct Norton equivalent Construct Norton equivalent

After

Solution for Geq Solution for short-circuit current Norton equivalent Construct Norton equivalent Construct Norton equivalent

Assemble Norton equivalent: two-step

This method combines partial results obtained in the two-step evaluation of Norton equivalent and assembles the Norton equivalent.

This template only exists in combination with the other template, "Norton equivalent: two-step". It is activated when all contributions are available, i.e. when the corresponding nodes are in ready state.

Before

Characteristic of one-port Thevenin or Norton equivalent

After

Characteristic of one-port Thevenin or Norton equivalent Direct single-loop circuit solution

Assemble Thevenin/Norton equivalent: one-step

This method processes the symbolic solution obtained by the methods "Thevenin equivalent: one-step" or "Norton equivalent: one step", available as the characteristic of a one-port element, and assembles the corresponding Thevenin or Norton equivalent circuit. This method handles also the particular case of evaluation of \(R_{eq}\) or \(G_{eq}\) for a homogeneous one-port, resulting e.g. from application of methods "Equivalent resistance/conductance (using definition)".

Before

Initial circuit Solution

After

Initial circuit Solution Direct single-loop circuit solution

Single-loop circuit solution

For single-loop circuits, the solution is obtained directly through a simple formula, provided that only basic elements are present.

Before

Initial circuit Solution

After

Initial circuit Solution Direct two-node circuit solution

Two-node circuit solution

For two-node circuits, the solution is obtained directly through a simple formula, provided that only basic elements are present.

Before

Initial circuit Solution

After

Initial circuit Solution Direct solution via Millman's Theorem

Millman's theorem

For circuits with Millman topology, the solution is obtained directly through a simple formula.

Before

Initial circuit Solution

After

Initial circuit Obtained by connecting external current source Solution Solution for voltage across current source Connect external current source Divide computed voltage by current source

Equivalent resistance (using definition)

Evaluation of equivalent resistance or impedance by excitation through external current source. This method only constructs the basic circuit that includes the original one-port plus an externally-connected current source. The new active circuit will have to be solved for the voltage across the current source. The last (future) solution step will evaluate the equivalent resistance/impedance by dividing the computed voltage by the external current source. The last operation is a particular case of method "Assemble Thevenin/Norton equivalent: one-step", which is here applied with a simplified characteristic having only the homogeneous contribution.

Before

Initial circuit Solution

After

Initial circuit Obtained by connecting external voltage source Solution Solution for current through voltage source Connect external voltage source Divide computed current by voltage source

Equivalent conductance (using definition)

Evaluation of equivalent conductance or admittance by excitation through external voltage source. This method only constructs the basic circuit that includes the original one-port plus an externally-connected voltage source. The new active circuit will have to be solved for the current through the voltage source. The last (future) solution step will evaluate the equivalent conductance/admittance by dividing the computed current by the external voltage source. The last operation is a particular case of method "Assemble Thevenin/Norton equivalent: one-step", which is here applied with a simplified characteristic having only the homogeneous contribution.

Before

Initial circuit Solution

After

Initial circuit Solution Direct solution via voltage divider

Voltage divider

Basic voltage divider applied to single-loop circuit with voltage sources and resistances/impedances.

Before

Initial circuit Solution

After

Initial circuit Solution Direct solution via current divider

Current divider

Basic current divider applied to two-node circuit with current sources and resistances/impedances.

Before

Initial circuit Solution

After

Initial circuit Obtained by basic Thevenin-Norton conversion Solution Thevenin-Norton conversion

Basic Thevenin to Norton conversion

Basic conversion of Thevenin to Norton topology. This method involves identifying elementary two-element sources in Thevenin configuration, highlighting them in the circuit, providing the conversion to Norton form in a new circuit. This is only performed if this operation is detected to be useful, i.e., if the resulting circuit can be simplified by further operations (series/parallel of resistances/impedances/sources).

Before

Initial circuit Solution

After

Initial circuit Obtained by basic Norton-Thevenin conversion Solution Norton-Thevenin conversion

Basic Norton to Thevenin conversion

Basic conversion of Norton to Thevenin topology. This method involves identifying elementary two-element sources in Norton configuration, highlighting them in the circuit, providing the conversion to Thevenin form in a new circuit. This is only performed if this operation is detected to be useful, i.e., if the resulting circuit can be simplified by further operations (series/parallel of resistances/impedances/sources).

Before

Initial circuit Solution

After

Initial circuit Original circuit with highlighted one-port to be replaced One-port to characterize through Thevenin or Norton equivalent Circuit after reduction Solution Thevenin or Norton equivalent Identify subcircuit to replace with simplified equivalent Extract subcircuit to characterize via Thevenin/Norton equivalent Replace equivalent in original circuit Replace equivalent in original circuit

Circuit reduction

Circuit reduction via Thevenin/Norton application: identify well-defined subgraph with two interface nodes, find the corresponding Thevenin/Norton equivalent, and replace in original circuit. The replacement step that is scheduled is performed by method "Substitute Subgraph".

Before

Original circuit with highlighted subgraphs Subgraphs to be used for replacement Circuit with replaced subgraphs Replace multiple subgraphs Replace multiple subgraphs

After

Original circuit with highlighted subgraphs Subgraphs to be used for replacement Circuit with replaced subgraphs Replace multiple subgraphs Replace multiple subgraphs

Substitute subgraph

This method starts with an original circuit that includes a set of disjoint subgraphs to be replaced, and a corresponding set of replacement subgraphs (number of interface nodes must match in source and target subgraphs, of course). A new circuit is generated where each subgraph is replaced.

Before

Initial circuit Solution

After

Initial circuit Solution Solution via Modified Nodal Analysis

Modified Nodal Analysis (both as a formulation and as a solver)

Formulate Modified Nodal Analysis equations. The Modified Nodal Analysis method works as a solver and produces as output the values of the target variables to be computed, just as Kirchhoff Laws or Simplified Nodal Analysis. The methods MNA Formulation and MNA DC Formulation provide only the MNA matrices as output, in full DAE form or in DC form, respectively.

Before

Initial circuit Solution

After

Initial circuit Characteristics of dynamic elements Obtained by fixing state variables through independent sources Solution Solution for dual variables Extract characteristics of dynamic elements Replace dynamic elements with independent sources Combine with characteristics of dynamic elements Combine with characteristics of dynamic elements

State equations

Formulate the state equations of a dynamic circuit (regular, non-degenerate). Replace dynamic elements with independent sources corresponding to state variables, and compute the dual variables. State equations are obtained by eliminating the dual variables using the characteristics of dynamic elements.

Before

Characteristics of dynamic elements Solution for dual variables in terms of state variables State equations Assemble state equations Assemble state equations

After

Characteristics of dynamic elements Solution for dual variables in terms of state variables State equations Assemble state equations Assemble state equations

Assemble state equations

This method starts with a cardinal two-port characterization induced by extracted state variables (dual variables expressed in terms of state variables), together with a specification of the characteristics of the dynamic elements. The result are the state equations of the circuit. This method must follow the method "State Equations", of which it represents a sub-template.

Before

Initial circuit Solution

After

Initial circuit Circuit for element (1,1) Circuit for element (2,1) Circuit for element (1,2) Circuit for element (2,2) Solution for element (1,1) Solution for element (2,1) Solution for element (1,2) Solution for element (2,2) Solution Construct circuit for element (2,1) Construct circuit for element (1,1) Construct circuit for element (1,2) Construct circuit for element (2,2) Construct two-port matrix Construct two-port matrix Construct two-port matrix Construct two-port matrix

Two-port representation: elementwise

Find any cardinal representation of a two-port element by computing matrix elements separately. The collection of individual elements is performed through method "Assemble two-port matrix".

Before

Matrix element (1,1) Matrix element (2,1) Matrix element (1,2) Matrix element (2,2) Two-port matrix Construct two-port matrix Construct two-port matrix Construct two-port matrix Construct two-port matrix

After

Matrix element (1,1) Matrix element (2,1) Matrix element (1,2) Matrix element (2,2) Two-port matrix Construct two-port matrix Construct two-port matrix Construct two-port matrix Construct two-port matrix

Assemble two-port matrix: two-step

Collect partial results corresponding to individual matrix elements and collect them in the desired two-port matrix. In case of transmission matrix, the partial results are assumed to be reciprocal (see method "Transmission matrix: elementwise").

Before

Initial circuit Solution

After

Initial circuit Obtained by connecting external sources at the two ports Solution Solution for dual port variables Connect external sources at the two ports Construct two-port matrix

Two-port representation: one-step

Find any cardinal representation of a two-port element by computing all matrix elements at the same time. The two ports are excited by external sources, and the corresponding dual variables are computed. The results are finally collected to construct the desired two-port matrix using method "Assemble two-port representation: one step".

Before

Two-port characteristics (equations) Two-port matrix Extraction of two-port matrix

After

Two-port characteristics (equations) Two-port matrix Extraction of two-port matrix

Assemble two-port representation: one-step

This method assembles any two-port matrix starting from the symbolic expression that provides the output variables in terms of the input variables.

Before

Initial circuit Solution

After

Initial circuit Circuit to compute 1/A Circuit to compute 1/B Circuit to compute 1/C Circuit to compute 1/D Solution for 1/A) Solution for 1/B Solution for 1/C Solution for 1/D Solution Construct circuit to compute 1/B Construct circuit to compute 1/A Construct circuit to compute 1/C Construct circuit to compute 1/D Construct transmission matrix Construct transmission matrix Construct transmission matrix Construct transmission matrix

Transmission matrix: elementwise

Find the T matrix of a two-port by computing one matrix element at the time. Instead of computing \(A\), \(B\) ,\(C\), \(D\) evaluate their reciprocals \(1/A\), \(1/B\), \(1/C\), \(1/D\) by constructing individual circuits for which the latter are well-defined transfer functions. Finally, collect results and build the \(T\) matrix using method "Assemble two-port representation: two-step".

Before

Initial circuit Solution

After

Initial circuit Obtained by connecting nullator/norator + external sources at the two ports Solution Solution for unknown port variables Connect nullator/norator + external sources at the two ports Construct T matrix

Transmission matrix: one-step

Find the T matrix of a two-port by iterative application of KCL and KVL starting from the output nodes. The solution method implies that \(V_2\) and \(I_2\) are known port variables at port (2). This is accomplished by connecting a nullator with a shunt current source \(I_2\), and then connecting in series a voltage source \(V_2\) at the port (2). All this will have to be visualized graphically in a simpler way, by defining e.g. a new component whose voltage and current are known, visualized as a port (no element) but with voltage/current highlighted. On port (1), the dependent port, a norator is connected, which will be achieved through another new component with no symbol, and also with voltage and current suitably highlighted. The latter \(V_1\) and \(I_1\) will be the output variables of interest. The matrix \(T\) is assembled using method "Assemble two-port matrix: one-step".

Before

Initial circuit Solution

After

Initial circuit Homogeneous circuit (no sources) Simplified homogeneous circuit Laplace-domain (symbolic) circuit Modfied Nodal Analysis matrix Natural frequencies Switch off all sources Simplify topology (remove open/short circuits, if any) Construct symbolic circuit (no initial conditions) Construct MNA matrix Find the zeros of the determinant of MNA matrix

Natural frequencies

Find the natural frequencies (poles) of a LTI circuit. These are obtained as the zeros of the determinant of the MNA matrix written in the symbolic Laplace domain, after switching off all sources and setting to zero all initial conditions.

Before

Initial circuit Solution

After

Initial circuit Steady-state circuit n.1 Steady-state circuit n.2 Steady-state circuit n.3 Solution of superposition circuit n.1 Solution of superposition circuit n.2 Solution of superposition circuit n.3 Solution Split into isofrequential circuit components Split into isofrequential circuit components Split into isofrequential circuit components Collection of superposition results Collection of superposition results Collection of superposition results

Steady-state analysis

Find the steady-state solution of a circuit including only DC or AC sources, possibly with multiple frequencies. Superposition is applied by collecting groups of sources with the same frequency (including DC as a particular case). Each individual circuit is solved independently, and finally all results are collected by applying superposition, using method "Superposition (collect)".

Before

Initial circuit Solution

After

Initial circuit DC circuit to be solved Solution Simplify circuit for DC analysis

DC Steady-state analysis

Find the DC steady-state solution of a generic LTI circuit with DC sources only. This method constructs the DC steady-state circuit by setting all switches and sources to their configuration for \(t>0\), all dynamic elements to open (capacitors) or short (inductors), and sets the problem statement to DC solve.

Before

Initial circuit Solution

After

Initial circuit Phasor (symbolic) circuit Solution (phasor) Solution Compute phasor (symbolic) circuit Conversion from Phasor to sinusoid

AC Steady-state analysis

Find the AC steady-state solution of a generic LTI circuit with AC sources only (same frequency). First step is to apply method "Convert to Phasor (AC) symbolic circuit", and to set up the problem statement to AC solve. After solution, conversion from phasor to sinusoid is performed by applying method "convert Phasor to sinusoid".

Before

Initial circuit Solution

After

Initial circuit Phasor (symbolic) circuit Solution for (phasor) voltages/currents Active/reactive power Compute phasor (symbolic) circuit Evaluate (complex, real, imag) power

AC Power

Find the AC power absorbed/delivered by given elements. Phasor circuit is first derived, by setting problem statement to AC solve for relevant voltages and currents. Once computed, the method "Evaluate AC power" is applied to process voltages and currents and derive active/reactive powers.

Before

Initial circuit Solution

After

Initial circuit Phasor (symbolic) circuit Solution Compute phasor (symbolic) circuit

Convert to Phasor (AC) symbolic circuit

Convert a generic LTI circuit with AC sources at the same frequency to the corresponding symbolic phasor circuit.

Before

Phasor solution Time-domain sinusoids Convert from phasor to sinusoid

After

Phasor solution Time-domain sinusoids Convert from phasor to sinusoid

Convert from Phasor to sinusoid

Convert phasor voltages or currents to time-domain sinusoids.

Before

Phasor solution Active/reactive power Evaluate active/reactive power

After

Phasor solution Active/reactive power Evaluate active/reactive power

Evaluate AC power

Process phasor voltages and/or currents to evaluate active and reactive power absorbed/delivered by desired elements.

Before

Initial circuit Solution

After

Initial circuit Circuit for initial conditions (t<0) Initial conditions on state variables Laplace-domain (symbolic) circuit Construct circuit for initial conditions Construct Laplace-domain (symbolic) circuit Construct Laplace-domain (symbolic) circuit

Convert to Laplace-domain symbolic circuit

Convert a generic LTI circuit with arbitrary sources to the corresponding Laplace-domain symbolic circuit. Initial conditions are computed first, and used to construct the Laplace-domain circuit. The latter is constructed using method "Assemble Laplace-domain symbolic circuit".

Before

Original circuit (time-domain) Initial conditions on state variables Laplace-domain symbolic circuit Assemble symbolic circuit Assemble symbolic circuit

After

Original circuit (time-domain) Initial conditions on state variables Laplace-domain symbolic circuit Assemble symbolic circuit Assemble symbolic circuit

Assemble Laplace-domain symbolic circuit

This method constructs the Laplace-domain symbolic circuit starting from the original circuit in the time-domain and knowledge of the initial conditions for the states at \(t=0-\).

Before

Initial circuit Solution

After

Initial circuit Steady-state circuit for t<0 Steady-state solution of state variables for t<0 Initial conditions on state variables at t=0- Construct circuit for initial conditions Evaluate state variables at t=0-

Initial conditions (t=0-)

Consider a generic LTI circuit with arbitrary sources that operate at steady-state for \(t<0\), and evaluate the initial conditions on the state variables at \(t=0-\). First, derive a steady-state circuit that operates until \(t=0-\). Solve this circuit for state variables. Finally, evaluate the resulting expressions for \(t=0-\). The last step is not necessary if only DC sources are present. No steps are necessary if no sources are active for \(t<0\). The evaluation at \(t=0-\) is performed by method "Evaluate symbolic expression".

Before

Initial circuit Solution

After

Initial circuit Circuit for initial conditions (state variables) Initial conditions (state variables) Circuit for initial conditions (non-state variables, t=0+) Initial conditions at t=0+ (non-state variables) Derive circuit for initial conditions (state variables) Derive circuit at t=0+ for initial conditions (non-state variables) Derive circuit at t=0+ for initial conditions (non-state variables)

Initial conditions (t=0+)

Consider a generic LTI circuit with arbitrary sources that operate at steady-state for \(t<0\), and evaluate the initial conditions on arbitrary variables at \(t=0+\). First, derive a steady-state circuit that operates until \(t=0-\). Solve this circuit for state variables. Then, impose state continuity at \(t=0\), and solve the circuit at frozen time \(t=0+\) for the desired variables. The evaluation of the state variables at \(t=0-\) is performed using method "FindInitialConditions".

Before

Symbolic expression Value of independent variable Evaluated expression

After

Symbolic expression Value of independent variable Evaluated expression

Evaluate symbolic expression

This method processes a symbolic expression in a given variable \(x\) and a corresponding value \(x=x_*\). The result is the symbolic expression evaluated at \(x=x_*\).

Before

Initial circuit Solution

After

Initial circuit Laplace-domain (symbolic) circuit Solution for desired output variables Transfer functions Construct symbolic circuit (no initial conditions, single source) Extract transfer functions

Transfer function

Find the transfer function between a given source and one or more outputs. First, the symbolic circuit is derived by activating a single source (whose value is redefined to be symbolic) and deactivating any initial condition. The symbolic circuit is solved for the desired output variables, and finally the transfer functions are retrieved by extracting the coefficients that multiply the source.

Before

Symbolic solution Transfer function Extract transfer function

After

Symbolic solution Transfer function Extract transfer function

Extract transfer function

This method processes the symbolic solution obtained by the method "Transfer function" and extracts the desired transfer function(s) by typesetting the results.

Before

Initial circuit Solution

After

Initial circuit Laplace-domain (symbolic) circuit for impulse response Solution for desired output variables Impulse responses Construct symbolic circuit (no initial conditions, single impulsive source) Invert Laplace transform

Impulse response

Find the impulse response of one or more outputs due to a given source. First, the symbolic circuit is derived by activating a single source (whose value is redefined to be \(\delta(t)\)) and deactivating any initial condition. The symbolic circuit is solved for the desired output variables, and finally the inverse Laplace transform is applied to determine the impulse response(s). The latter operation is performed with method "Inverse Laplace transform".

Before

Initial circuit Solution

After

Initial circuit Circuit for initial conditions (t<0) Initial conditions on state variables Laplace-domain (symbolic) circuit Solution (symbolic) for desired output variables Solution Construct circuit for initial conditions Construct Laplace-domain (symbolic) circuit Construct Laplace-domain (symbolic) circuit Invert Laplace transform

General transients

Find the transient solution of a general LTI circuit. Initial conditions are computed first, by setting up a steady-state circuit for \(t<0\), whose solution is undefined since the flow will depend on the type of sources that are present (see method for Initial conditions). Once initial conditions are known, the Laplace-domain symbolic circuit is constructed, solved for the desired (symbolic) output variables, and finally the inverse Laplace transform is computed to obtained the desired transient solutions. The latter operation is performed with method "Inverse Laplace transform".

Before

Initial circuit Solution

After

Initial circuit (Laplace function to be inv-transformed) Solution Apply inverse Laplace transform

Inverse Laplace transform

Find the inverse Laplace transform of given rational function(s) \(H(s)\).

Before

Initial circuit Solution

After

Initial circuit (transfer function to be plotted) Solution Construct Bode diagrams

Bode diagrams

Draw the (asymptotic) Bode diagram corresponding to a given transfer function.

Copyright © 2018- Stefano Grivet-Talocia, Politecnico di Torino. All rights reserved. Contact: autocircuits - @ - polito.it
This service is still in a beta-testing phase. Please report any bug to autocircuits - @ - polito.it