Circuit Analysis, Physics tutorial

Introduction to Circuit analysis:

In usual English, a circuit is a way or path which begins at one place and finally returns to that similar place. The engineering sense of the term is similar. In electronics, a circuit is a closed path via which electrical currents can flow. The flow of electrical current via a flashlight describes the idea.

The electrical circuits are built up of components like resistors, capacitors, transistors and inductors joined altogether through wires. You can make randomly amazing, complex devices by means of hooking such things up in different manners.

What is Circuit Analysis?

In an electrical circuit the procedure of studying and examining the different electrical quantities comprised, particularly the nodal voltages and currents via computations, is termed as circuit analysis.

To be capable to resolve the different problems usually comprised in practical electrical circuits, it would be first significant to learn the given fundamental units of electricity:

Current (I): The SI unit of current is Ampere (A) and is stated as the quantity of current forced by a pressure of one volt via a resistance of one Ohm.

Voltage (V): The SI unit of voltage is Volt (V) and is stated as the amount of pressure needed to force a current of one ampere via a resistance of one Ohm.

Electromotive Force (E): This is the voltage that exists across the terminals of a battery or dynamo which is not connected to any external circuit.

Potential Difference (U): It is the voltage that exists across the terminals of a dynamo or battery that is joined to an external circuit. It might be computed by utilizing the formula which is as follows:

U = (E - Internal resistance of battery or dynamo) × I

Resistance (?): The SI unit of resistance is Ohm (?) and is stated as the amount of resistance provided by a circuit in such a way that a current of one ampere is permitted to flow via it at a pressure of one volt.

Passive Components:

The component is stated to be passive when the net energy delivered to it from the rest of the circuit is for all time non-negative. There are usually three types of passive components:

a) Resistors:

A resistor is an electrical machine whose main function is to introduce resistance to the flow of electric current. The magnitude of opposition to the flow of current is termed as the resistance of the resistor. The resistance 'R' of a resistor is represented by:

R = ρl/A

Here, 'ρ' is the resistivity of the material; 'l' is the length of the resistor and 'A' is the cross-sectional area perpendicular to the current flow.

b) Capacitors:

It is a passive electrical device which is intentionally designed to store energy in an electric field is termed as a capacitor. The capability of a capacitor to store energy in an electric field is termed as the capacitance. A capacitor is physically built up of two conducting plates separated through a dielectric medium and its capacitance 'C' is represented by:

C = εA/d

Here, 'A' is the area of the plate; 'ε' is the dielectric constant of an insulating material and 'd' is the separation between plates. 'C' is measured in terms of Farads (F).

c) Inductors:

An inductor is the electrical device designed to store energy in the magnetic field. Inductance is basically a measure of the capability of an inductor to store energy in the magnetic field. An inductor can be built up by winding a coil of wire around a toroidal core and its inductance 'L', is measured in Henrys (H), and is represented by:

L = N2Aμ/2πr

Here, 'N' is the number of turns of wires; 'A' is the cross-sectional area of the torus; 'μ' is the permeability of the material and 'r' is the radius of the torus.

Ohm's law:

The Ohm's Law is a mathematical relationship among the electric current, resistance and voltage. The principle is termed after the German scientist Georg Simon Ohm. The relationship defines that: The potential difference (or voltage) across an ideal conductor is proportional to the current via it.

V = IR

Here, 'V' is voltage measured in volts; 'I' is the current measured in amperes and 'R' is the resistance measured in ohms.

Kirchhoff's law:

In the year 1845, a German physicist, Gustav Kirchhoff introduced a pair or set of rules or laws which mainly deal by the conservation of current and energy in the Electrical Circuits. These two rules are generally termed as: Kirchhoff's Circuit Laws mainly dealing with the current flowing around a closed circuit whereas the other law deals with the voltage sources present in the closed circuit, generally termed as Kirchhoff's Voltage Law (or KVL).

1) Kirchhoff's First Law - The Current Law (KCL):

Kirchhoff's Current Law or KCL, defines that the total current or charge entering a junction or node is precisely equivalent to the charge leaving the node as it consists of no other place to go apart from to leave, as no charge is lost in the node. In another words the algebraic sum of all the currents entering and leaving a node should be equivalent to zero, I(exiting) + I(entering) = 0. This theory proposed by Kirchhoff is generally termed as the Conservation of Charge.

2) Kirchhoff's Second Law - The Voltage Law (KVL):

Kirchhoff's Voltage Law or KVL defines that in any closed loop network, the net voltage around the loop is equivalent to the sum of all the voltage drops in the similar loop, which is as well equivalent to zero. In another words the algebraic sum of all voltages in the loop should be equivalent to zero. This theory proposed by Kirchhoff is termed as the Conservation of Energy.

Node and Mesh analysis:

Node analysis:

This is a process of analyzing the linear electric networks, which is, a process of finding out the currents in the branches of such a network and the voltages between the terminals of the passive elements and active elements (that is, sources of energy) in the network.

In examining a circuit by using Kirchhoff's circuit laws, one can either do nodal analysis by using Kirchhoff's current law (or KCL) or mesh analysis by using Kirchhoff's voltage law (or KVL). Nodal analysis represents an equation at each and every electrical node, requiring that the branch currents incident at a node should sum to zero. The branch currents are written in terms of the circuit node voltages. As an effect, each and every branch constitutive relation should provide current as a function of voltage; an admittance representation. For illustration, for a resistor, Ibranch = Vbranch * G, here G (=1/R) is the admittance (or conductance) of the resistor.

Mesh analysis:

A mesh analysis is the set of mathematical equations which help in finding out the paths of electrical currents. The analysis procedure employs Ohm's law and Kirchhoff's voltage law to inspect how dissimilar points on a circuit board form the communication bonds. It comprises isolating a circuit board's loops, recognizing opposite voltage charges and replacing values into equations to resolve the unknown direction of the currents. The mesh analysis is one of many processes employed to examine electrical currents and is as well termed as the loop current method.

The most complex portion of the mesh analysis method is making the equations which reveal whether the supposed current direction is right or wrong. Kirchhoff's voltage law takes the first circuit's resistor's voltage value and then adds it to the amount of the unknown value of the first current and the second current, which is then multiplied by two. This outcome is as well added to the unknown value of the first current multiplied by four. The value of first side of the equation is set to equivalent zero and is resolved mathematically.

Equivalent circuit:

The electric circuit is built up of the fundamental elements resistance, inductance and capacitance in a simple arrangement in such a way that its performance would duplicate that of a more complex circuit or network.

275_Equivalent circuit.jpg

The behavior of the circuit represented in the first part of the figure can be represented in terms of a simplified equivalent circuit represented in the second part of the figure, where we state:

Vout = (VinR2)/(R1 + R2)

R' = (R1R2)/(R1 + R2)

In result, we are treating the unloaded output, Vout, as a sort of 'internal voltage' (or electromotive force) generated through the combination of the real input and the volume control. R' is the efficient output resistance of the volume control that sits among the internal voltage and the output load.

It is noted that this equivalent circuit is an illustration of applying a general rule termed as Thevenin's Law that defines that any circuit built up of lots of resistors and voltage sources can be simplified into the internal voltage in series having an output resistance. As all the current flowing via R' should as well flow through RL we can state that,

Vout = I (R' + RL)

V'out = I RL

On combining both the equation:

V'out = IRL

V'out = Vout x [RL/(RL + R')]

Thevenin's and Norton equivalents:

One of the most astonishing properties of linear circuit theory associates to the capability to treat any two-terminal circuit no matter how complicated as behaving as just a source and an impedance that encompass either of two simple equivalent circuit forms:

Thevenin's equivalent: Any linear two-terminal circuit can be substituted through single voltage source and series impedance.

Norton equivalent: Any linear two-terminal circuit can be substituted through a current source and parallel impedance.

Though, the single impedance can be of arbitrary complexity (that is, as a function of frequency) and might be irreducible to the simpler form.

DC and AC equivalent circuits:

In the linear circuits, due to the superposition principle, the output of a circuit is equivalent to the sum of the output due to its DC sources alone, and the output from its AC sources alone. Thus, the DC and AC response of a circuit is frequently examined independently, by employing separate DC and AC equivalent circuits that encompass the similar response as the original circuit to DC and AC currents correspondingly. The composite response is computed by adding the DC and AC responses.

A DC equivalent of a circuit can be made by substituting all capacitances having open circuits, inductances having short circuits, and decreasing AC sources to zero (substituting AC voltage sources through short circuits and AC current sources through open circuits.)

An AC equivalent circuit can be made by decreasing all the DC sources to zero (by substituting DC voltage sources by means of short circuits and DC current sources by means of open circuits).

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