Linearity 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 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.

Suppose vs = 5V and i = 1A. According to linearity property if the voltage is multiplied by 2 then the voltage vs = 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.

Nodal Analysis

What is Nodal Analysis

Nodal Analysis provide a general procedure for analyzing circuits using node voltages as the circuit variables.

Steps to Determine Node Voltages:

1. Select a node as the reference node, Assign voltages v1, v2, . . . . . , 

vn-1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node.

2.Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express currents in terms of node voltages.

3. Solve the resulting simultaneous equations to obtain the unknown node voltages.

Nodal Analysis with Voltage Sources

Case 1: If the voltage source (dependent or independent) is connected between two non-reference nodes, the two non-reference nodes form a generalized node or super node, we apply both KCL and KVL to determine the node voltages.

Case 2: if a voltage source is connected between the reference node and a non-reference node, we simply set the voltage at the non-reference node equal to the voltage of the voltage source.

~ In every different kind of circuits when in comes to analyzing and solving, we should practice the steps of determining the node voltages and must able to observe if what kind of case the circuit is.

There are some instances that a voltage source is in between to the two non-reference node in a loop, that's SUPERNODE. 
In circuit theo, a super-node is a theoretical construct that can be used to solve a circuit. This is done by viewing a voltage source on a wire as a point source voltage in relation to other point voltages located at various nodes in the circuit, relative to a ground node assigned a zero or negative charge.

The application that needs to study:
- KVL/KCL (on my CHAPTER 2 blog)
- Cramer's rule

Mesh Analysis

Mesh Analysis

A Mesh is a loop that does not contain any other loop within it.

~ a loop can be a mesh, but a mesh can't be a loop.

Same as the nodal analysis, a mesh analysis have also a steps in getting the equation but they differ for some aspects which is the Nodal analysis talks about the nodal voltages while the Mesh analysis talks about mesh currents. 

~ a mesh current is quite similar to the Branch Current method in that it uses simultaneous equations, Kirchhoff's Voltage Law, and Ohm's Law to determine unknown currents in a network.

Why is it QUITE SIMILAR TO THE BRANCH CURRENT? 
-  it is quite similar, then of course it is also quite different but it depends on how the mesh is being ISOLATED.


There are steps in determining mesh currents same as nodal analysis, there are steps to determine nodal voltage in order to form an equation.

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.

Example: 

As you observe the figure, the current flow counter clockwise but wherever the mesh currents will flow, it's direction is arbitrary and does not affect the validity of the solution.

~ as a class, we are more prefer to have the mesh currents direction clockwise because for us it is more convenient and easier to analyze.

Mesh analysis with current sources

Case 1:

When a current source exists only in one mesh

Case 2:

When a current source exists between two meshes and that is SUPERMESH.

What is supermesh?

A SUPERMESH results when two meshes have current source in common.

Superposition Theorem

The superposition theorem for electrical circuit 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.

Example:

 Find R2;

The figure has two(2) voltage source, in order to get the voltage across R2, we need to get first the voltage that being supplied on the other loop by deactivating/turning off the other source.

As you can see, the B2 is already turned off, therefore we can solve it using the voltage division principle.
-The Voltage division can be seen on the past blog for more information!

After that, the B1 must be turned off and B2 is now on in order to get the voltage on the other loop.

 

 I'm sure, we can get now the total voltage that R2 have.

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)).