Linear property

Linear property 
             is the linear relationship between cause and effect of an element. This property gives linear and nonlinear circuit definition. The property can be applied in various circuit elements. The homogeneity (scaling) property and the additivity 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 additivity 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 additivity and the homogeneity properties.
We can tell a circuit is linear if the circuit both the additive and the homogeneous. A linear circuital ways consists of linear elements, linear independent and dependent sources.
What is linear circuit?
A circuit is linear if the output is linearly related with its input.
The relation between power and voltage is nonlinear. So this theorem cannot be applied in power.
See a circuit in figure 1. The box is linear circuit. We cannot see any independent source inside the linear circuit.
 

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

Suppose v_s = 5V and i = 1A. According to linearity property if the voltage is multiplied by 2 then the voltage v_s = 10V and then the current also will be multiplied by 2 hence i = 2A.

The power relation is nonlinear. For example, if the current i₁ flows through the resistor R, the power p₁ = i₁²R and when current i₂ flows through the resistor R then power p₂ = i₂²R.
If the current (i₁ + i₂) flows through R resistor the power absorbed

P₃ = R(i₁ + i₂)² = Ri₁² + Ri₂² + 2Ri₁i₂ ≠ p₁ + p₂

So the power relation is nonlinear.

Source Transformations

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.

 

The circuits in this set of problems consist of independent sources, resistors and a meter. In particular, these circuits do not contain dependent sources. Each of these circuits has a series-parallel structure that makes it possible to simplify the circuit by repeatedly

 

Performing source transformations. 

 

Replacing series or parallel resistors by an  equivalent resistor. 

 

Replacing series voltage sources by an equivalent voltage source.

Replacing parallel current sources by an equivalent source source. 

 

Source transformations are easy to perform as long as there is a familiarity with Ohm's Law. If there is a voltage source in series with an impedance, it is possible to find the value of the equivalent current source in parallel with the impedance by dividing the value of the voltage source by the value of the impedance. The converse also applies here: if a current source in parallel with an impedance is present, multiplying the value of the current source with the value of the impedance will result in the equivalent voltage source in series with the impedance. 

 

Superposition

Superposition theorem is one of those strokes of genius that takes a complex subject and simplifies it in a way that makes perfect sense. A theorem like Millman's certainly works well, but it is not quite obvious why it works so well. Superposition, on the other hand, is obvious.
The strategy used in the Superposition Theorem is to eliminate all but one source of power within a network at a time, using series/parallel analysis to determine voltage drops (and/or currents) within the modified network for each power source separately. Then, once voltage drops and/or currents have been determined for each power source working separately, the values are all “superimposed” on top of each other (added algebraically) to find the actual voltage drops/currents with all sources active.

 Let's look at our example circuit again and apply Superposition Theorem to it: 

 

Since we have two sources of power in this circuit, we will have to calculate two sets of values for voltage drops and/or currents, one for the circuit with only the 28 volt battery in effect. . .  

 

. . . and one for the circuit with only the 7 volt battery in effect: 

 

When re-drawing the circuit for series/parallel analysis with one source, all other voltage sources are replaced by wires (shorts), and all current sources with open circuits (breaks). Since we only have voltage sources (batteries) in our example circuit, we will replace every inactive source during analysis with a wire.

 Analyzing the circuit with only the 28 volt battery, we obtain the following values for voltage and current: 

Analyzing the circuit with only the 7 volt battery, we obtain another set of values for voltage and current: 

 

Applying these superimposed voltage figures to the circuit, the end result looks something like this:  

Currents add up algebraically as well, and can either be superimposed as done with the resistor voltage drops, or simply calculated from the final voltage drops and respective resistances (I=E/R). Either way, the answers will be the same. 

Here I will show the superposition method applied to current:

 Quite simple and elegant, don't you think? It must be noted, though, that the Superposition Theorem works only for circuits that are reducible to series/parallel combinations for each of the power sources at a time (thus, this theorem is useless for analyzing an unbalanced bridge circuit), and it only works where the underlying equations are linear (no mathematical powers or roots). The requisite of linearity means that Superposition Theorem is only applicable for determining voltage and current, not power!!! Power dissipations, being nonlinear functions, do not algebraically add to an accurate total when only one source is considered at a time. The need for linearity also means this Theorem cannot be applied in circuits where the resistance of a component changes with voltage or current. Hence, networks containing components like lamps (incandescent or gas-discharge) or varistors could not be analyzed.

Mesh Analysis

What is 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.
 


Mesh analysis works by arbitrarily assigning mesh currents in the essential meshes (also referred to as independent meshes). An essential mesh is a loop in the circuit that does not contain any other loop. Figure 1 labels the essential meshes with one, two, and three.
A mesh current is a current that loops around the essential mesh 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. Figure 2 shows the same circuit from Figure 1 with the mesh currents labeled.

Solving for mesh currents instead of directly applying Kirchhoff's current law and Kirchhoff's voltage law can greatly reduce the amount of calculation required. This is because there are fewer mesh currents than there are physical branch currents. In figure 2 for example, there are six branch currents but only three mesh currents.

Figure 1: 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.

 

Figure 2: Circuit with mesh currents labeled as i₁, i₂, and i₃. The arrows show the direction of the mesh current.

Each mesh produces one equation. These equations are the sum of the voltage drops in a complete loop of the mesh current. For problems more general than those including current and voltage sources, the voltage drops will be the impedance of the electronic component multiplied by the mesh current in that loop.
If a voltage source is present within the mesh loop, the voltage at the source is either added or subtracted depending on if it is a voltage drop or a voltage rise in the direction of the mesh current. For a current source that is not contained between two meshes, the mesh current will take the positive or negative value of the current source depending on if the mesh current is in the same or opposite direction of the current source. The following is the same circuit from above with the equations needed to solve for all the currents in the circuit.

 
Once the equations are found, the system of linear equations can be solved by using any technique to solve linear equations.

Supermesh

Figure 3: Circuit with a supermesh. Supermesh occurs because the current source is in between the essential meshes.

A supermesh 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 supermesh.

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.        

The Y-Δ transform is known by a variety of other names, mostly based upon the two shapes involved, listed in either order. The Y, spelled out as wye, can also be called T or star; the Δ, spelled out as delta, can also be called triangle, Π (spelled out as pi), or mesh. Thus, common names for the transformation include wye-delta or delta-wye, star-delta, star-mesh, or T-Π.

The transformation is used to establish equivalence for networks with three terminals. Where three elements terminate at a common node and none are sources, the node is eliminated by transforming the impedances. For equivalence, the impedance between any pair of terminals must be the same for both networks. The equations given here are valid for complex as well as real impedances.

 Transformation from Delta-load to Wye-load 

The general idea is to compute the impedance Ry at a terminal node of the Y circuit with impedances R', R'' to adjacent node in the Δ circuit by

where R are all impedances in the Δ circuit. This yields the specific formula

 Transformation from Wye-load to Delta-load 

The general idea is to compute an impedance R in the Δ circuit by 

where Rp = R1R2 + R2R3 + R3R1 is the sum of the products of all pairs of impedances in the Y circuit and Ropposite is the impedance of the node in the Y circuit which is opposite the edge with R. The formula for the individual edges are thus