Superposition Theorem

Superposition Theorem

The superposition theorem for electrical circuits states that for a linear system the response (voltage or current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by their internal impedances.

To ascertain the contribution of each individual source, all of the other sources first must be "turned off" (set to zero) by:

1. Replacing all other independent voltage sources with a short circuit (thereby eliminating difference of potential i.e. V=0; internal impedance of ideal voltage source is zero (short circuit)).
2. Replacing all other independent current sources with an open circuit (thereby eliminating current i.e. I=0; internal impedance of ideal current source is infinite (open circuit)).

This procedure is followed for each source in turn, then the resultant responses are added to determine the true operation of the circuit. The resultant circuit operation is the superposition of the various voltage and current sources.

Simple Steps to Apply Superposition Principle:

1.     Turn off all independent sources except one source. Find the output (voltage or current) due to that active source using nodal or mesh analysis.

2.     Repeat step 1 for each of the other independent sources.

3.     Find the total contribution by adding algebraically all the contributions due to the independent sources.

Another point that should be considered is that superposition only works for voltage and current but not power. In other words the sum of the powers of each source with the other sources turned off is not the real consumed power. To calculate power we should first use superposition to find both current and voltage of each linear element and then calculate the sum of the multiplied voltages and currents.

Overview and Insights

           In this week we discussed about the superposition theorem, where one of the sources is removed and the other is solved using voltage division or current division. In using the Superposition, we replace the voltage source with a short circuit on its place, and an open circuit when we replace the current source. We turn off other independent sources except one source. After converting the circuit by superposition, we can use nodal and mesh analysis or any other techniques to get the value of the output voltage or current. We add the two results to get the answer.

Example: Vo = Va + Vb  or  Io = Ia + Ib

Linearity Property and Source Transformation

Linearity property

This property gives linear and nonlinear circuit definition. The property can be applied in various circuit elements. The homogeneity (scaling) property and the additive property are both the combination of linearity property.The homogeneity property is that if the input is multiplied by a constant k then the output is also multiplied by the constant k. Input is called excitation and output is called response here. As an example if we consider ohm’s law. Here the law relates the input i to the output v.

Mathematically,              

v= iR

If we multiply the input current  i by a constant k then the output voltage also increases correspondingly by the constant k. The equation stands,     

kiR = kv

The additive property is that the response to a sum of inputs is the sum of the responses to each input applied separately.

Using voltage-current relationship of a resistor if

v1 = i1R       and   v2 = i2R

Applying (i1 + i2) gives

V = (i1 + i2) R = i1R+ i2R = v1 + v2

We can say that a resistor is a linear element. Because the voltage-current relationship satisfies both the additive and the homogeneity properties.

We can tell a circuit is linear if the circuit both the additive and the homogeneous. A linear circuit always consists of linear elements, linear independent and dependent sources.

What is a linear circuit?

The linear circuit is excited by another outer voltage source vs. Here the voltage source vs acts as input. The circuit ends with a load resistance R. we can take the current I through R as the output.

source transformation

Source transformation is simplifying a circuit solution, especially with mixed sources, by transforming a voltage into a current source, and vice-versa.  Finding a solution to a circuit can be difficult without using methods such as this to make the circuit appear simpler. Source transformation is another tool for simplifying circuits. Basic to these tools is the concept of equivalence.

 A Source Transformation is the process of replacing a voltage source Vs in series with a resistor R by a current source Is in parallel with a resistor R, or vice-versa.

Source transformation requires that,

Vs = IsR     or     Is = Vs/R

Source transformation also applies to dependent sources, provided we carefully handle the dependent variable. As shown in below, a dependent voltage source in series with a resistor can be transformed to a dependent current source in parallel with the resistor or vice versa.

OVERVIEW AND INSIGHTS

Linearity Property talks about the voltage (v) and current (i), whereas voltage is directly proportional to the current, that is, when the voltage is increasing the current also increasing and vice-versa. I also learned that Source Transformation is only applicable to simple circuits and not in a complicated circuits.

Source Transformation can be applied if and only if the voltage is in series with the resistor and/or the current is in parallel with the resistor.

In Source Transformation you need to check the polarity of the source, when the polarity of the voltage source has a positive on top therefore the resulting current source is pointing upward and vice-versa. Also when the voltage source is dependent therefore the resulting current is also dependent, same calculations with independent sources.

Mesh Analysis

Mesh Analysis

Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. A more general technique, called loop analysis (with the corresponding network variables called loop currents) can be applied to any circuit, planar or not. Mesh analysis and loop analysis both make use of Kirchhoff’s voltage law to arrive at a set of equations guaranteed to be solvable if the circuit has a solution. Mesh analysis is usually easier to use when the circuit is planar, compared to loop analysis.

Steps to determine mesh currents:

1.)    Assign mesh currents i1,i2, . . . . in to the n meshes.

2.)    Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents.

3.)    Solve the resulting n simultaneous equations to get the mesh currents

Essential meshes of the planar circuit labeled 1, 2, and 3. R₁, R₂, R₃, 1/sc, and Ls represent the impedance of the resistors, capacitor, and inductor values in the s-domain. V_s and i_s are the values of the voltage source and current source, respectively.

 Circuit with mesh currents labeled as i₁, i₂, and i₃. The arrows show the direction of the mesh current.

What is Super mesh?

A super mesh occurs when a current source is contained between two essential meshes. The circuit is first treated as if the current source is not there. This leads to one equation that incorporates two mesh currents. Once this equation is formed, an equation is needed that relates the two mesh currents with the current source. This will be an equation where the current source is equal to one of the mesh currents minus the other. The following is a simple example of dealing with a super mesh.

 

Circuit with a super mesh. Super mesh occurs because the current source is in between the essential meshes.

OVERVIEW AND INSIGHTS

A mesh current is a current that loops around and the equations are set solved in terms of them. A mesh current may not correspond to any physically flowing current, but the physical currents are easily found from them. It is usual practice to have all the mesh currents loop in the same direction. This helps prevent errors when writing out the equations. The convention is to have all the mesh currents looping in a clockwise direction.

in  Mesh analysis it is a method that is used to solve planar circuits for the currents at any place in the circuit. Planar circuits are circuits that can be drawn on a plane.

From my own understanding a mesh I a loop that does not contain any other loop within it.

Wye-Delta Transformation

The Y-Δ transform, also written wye-delta and also known by many other names, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like the letter Y and the Greek capital letter Δ. This circuit transformation theory was published by Arthur Edwin Kennelly in 1899. It is widely used in analysis of three-phase electric power circuits.

A delta-wye transformer is a type of three-phase electric power transformer design that employs delta-connected windings on its primary and wye/star connected windings on its secondary. A neutral wire can be provided on wye output side. It can be a single three-phase transformer, or built from three independent single-phase units. An equivalent term is delta-star transformer. Delta-wye transformers are common in commercial, industrial, and high-density residential locations, to supply three-phase distribution systems.

 

Δ and Y circuits with the labels

Transformation Formulas

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Learnings and overview:

A bridge circuit can be simplified to a series/parallel circuit by converting half of it from a Δ to a Y network. After voltage drops between the original three connection points (A, B, and C) have been solved for, those voltages can be transferred back to the original bridge circuit, across those same equivalent points.

In solving the wye-delta and delta-wye transformation it is easier to determine the labels by creating a free body diagram of it with labels and transformed circuit. In this way you can easily solve for the total resistance of the circuit, and by the help of my past learnings in the series-parallel connections I can now understand how to solve the problems given. I have sometimes difficulties in determining the labels but I will try to research and study more about it :)