Thevenin's Theorem

• Thevenin’s Theorem is a very important and useful theorem.
• It is a method for the reduction of a portion of a complex circuit into a simple one.
• It reduces the need for repeated solutions of the same sets of equations.

Thevenin Equivalent Circuit
Any two-terminal linear network, composed of voltage sources, current sources, and resistors, can be replaced by an equivalent two-terminal network consisting of an independent voltage source in series with a resistor.

Superposition Theorem

Superposition Theorem statement:

The theorem states: “In a network with two or more sources, the current or voltage for any component is the algebraic sum of the effects produced by each source acting separately”

•This means that regardless of the source, we have to analyze them one at a time.
•Things to remember:
–Voltage sources will be shorted
–Current sources will be opened
•For these examples, we will stay with voltage sources


First circuit to consider:

Algebraically sum the results

•VP = VR1 + VR2 = 6V + -12V = 6V – 12V = -6V

Algebraically add the values

•VP = VR1 + VR2 = -30V + 20V = -10V

If the resistors were reversed, the overall value of VP would remain the same but the polarities would be reversed. This is due to R2 now being the larger in the voltage divider ratio when calculating the values.

Nodal & Mesh Analysis

NODAL ANALYSIS:



Steps to Determine Node Voltages

• Select a node as the reference node. Assign voltages v1, v2, · · · , vn − 1
to determine the remaining n − 1 nodes.
The voltages are referenced with respect to the reference
(datum) node (ground).
• Apply KCL to each of the n − 1 non-reference nodes. Use Ohm’s
law to express the branch currents in terms of node voltages.
• Solve the resulting simultaneous equations to obtain the unknown
node voltages.

Nodal Analysis with Voltage Sources

• Voltage source between reference node and a non reference node.
Set the voltage of the non reference node to that of the voltage
source.
• Voltage source between two non reference nodes.
Form a supernode (generalized node) and apply both KVL and
KCL to determine the node voltages.
A supernode is formed by enclosing a (dependent or independent)
voltage source connected between two non reference nodes
and any elements connected in parallel with it.
Properties of a supernode:




BASIC NODAL AND MESH ANALYSIS
– The voltage source inside the supernode provides a constraint
equation needed to solve for the node voltages.
– A supernode has no voltage of its own.

– A supernode requires the application of both KCL and KVL.




MESH ANALYSIS:
-A mesh is a loop that does not contain any other loops within it.

Steps to determine mesh currents:
• Assign mesh currents i1, i2, · · · , in to the meshes.
• Apply KVL to each of the n meshes. Use Ohm’s law to express
the voltages in terms of the mesh currents.

• Solve the resulting n simultaneous equations to get the mesh currents.



Mesh Analysis with Current Sources
• Current source exists only in one mesh. Here mesh current = ±
current source.
• Current source between two meshes, form ’Supermesh’ by excluding
the current source any element connected in series with it.
• A Supermesh results when two meshes have a (dependent or independent)
current source in common.

Properties of a supermesh
-The current source in the supermesh provides the constraint
equation necessary to solve for the mesh currents.
– A supermesh has no current of its own.

– A supermesh requires the application of both KVL and KCL.





Nodal and Mesh Analysis by Inspection

Gkk = sum of the conductances connected to node k
Gkj = Gjk= negative of the sum of the conductances directly connecting
vk = unknown voltage at node k
ik = sum of all independent current sources directly connected to no
In matrix form, G~v =~i

G is called the conductance matrix.

Rkk = sum of the resistances in mesh k
Rkj = Rjk= negative of the sum of the resistances in common with meshes
ik = unknown mesh current for mesh k in the clockwise direction.
vk = clockwise sum of all independent voltage sources in mesh k, with
In matrix form, R~i = ~v
R is called the resistance matrix.

Wye - Delta Transformation

Kirchoff's Laws

Kirchoff’s Voltage Law

Kirchoff’s Voltage Law (KVL) states that the algebraic sum of the voltages across any set of

branches in a closed loop is zero. i.e.;

Below is a single loop circuit. The KVL computation is expressed graphically in that voltages

around a loop are summed up by traversing (figuratively walking around) the loop.

The KVL equation is obtained by traversing a circuit loop in either direction and writing down unchanged the voltage of each element whose “+” terminal is entered first and writing down the negative of every element’s voltage where the minus sign is first met. The loop must start and end at the same point. It does not matter where you start on the loop. Note that a current direction must have been assumed. The assumed current creates a voltage across each resistor and fixes the position of the “+” and “-” signs so that the passive sign convention is obeyed. The assumed current direction and polarity of the voltage across each resistor must be in agreement with the passive sign convention for KVL analysis to work.

The voltages in the loop may be summed in either direction. It makes no difference except to change all the signs in the resulting equation. Mathematically speaking, its as if the KVL equation is multiplied by -1. See the illustration below.

Note that a current direction must have been assumed. The assumed current creates a voltage across each resistor and fixes the position of the “+” and “-” signs so that the passive sign convention is obeyed. The assumed current direction and polarity of the voltage across each resistor must be in agreement with the passive sign convention for KVL analysis to work.

The case on the right above will obviously result in negative result for the current. This is correct considering the current arrow is pointing in the opposite direction.

Kirchoff’s Current Law

The algebraic sum of all currents entering and leaving a node must equal zero

S (Entering Currents) = S (Leaving Currents)

As a direct consequence of the conservation of charge, namely charge can neither be created nor destroyed, the node, being of negligible physical size, holds no charge. For instance, referring to figure 1.6, the sum of ,  and must equal zero.


 
Figure 1.6: Kirchhoff's current law 




Formally, KCL states that the algebraic sum of the currents in all the branches that converge in a common node is equal to zero. In mathematical form, for n branches converging into a node, KCL states that where  is the current flowing in the kth branch and its direction is assumed to be pointing towards the node.

Ohm's Law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:

where I is the current through the conductor in units of amperes, V is the potential difference measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms.

Series & Parallel Circuits

Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur very frequently.



SERIES CIRCUIT:

In a series circuit, the current through each of the components is the same, and the voltage across the circuit is the sum of the voltages across each component. In a parallel circuit, the voltage across each of the components is the same, and the total current is the sum of the currents through each component.

In my experience and observation, Series Circuits are the simplest to work with.Here we have three resistors of different resistances. They share a single connection point. When added together the total resistance is 90-Ohms.

Current

In a series circuit the current is the same for all elements.

Resistors

The total resistance of resistors in series is equal to the sum of their individual resistances:

Inductors

Inductors follow the same law, in that the total inductance of non-coupled inductors in series is equal to the sum of their individual inductances:

Capacitors

Capacitors follow the same law using the reciprocals. The total capacitance of capacitors in series is equal to the reciprocal of the sum of the reciprocals of their individual capacitances:

.

PARALLEL CIRCUIT

If two or more components are connected in parallel they have the same potential difference (voltage) across their ends. The potential differences across the components are the same in magnitude, and they also have identical polarities. The same voltage is applicable to all circuit components connected in parallel. The total current is the sum of the currents through the individual components.

A parallel circuit is shown here and it has TWO common connection points with another component. In this case another resistor. We cannot add the values of each resistor together like we can in the previous series circuit. So what do we need to do?

Calculating Total Resistance of a Parallel Circuit

Two methods can be used to calculate the total resistance of the parallel circuit. They are the Product Over Sum equation or the Reciprocal Formula.

Definition of Terms

Before discussing the basic analytical tools that determine the currents and voltages at different parts of the circuit, some basic definition of the following terms are considered.

Branch - A portion of a circuit that contains a load, and as a result, values for current, voltage, and resistance. The values of each branch of a circuit determine the total applied values for current, voltage, and resistance.

Capacity - The amount of electricity that can safely flow through a wire without the wire overheating.

Combination Circuit - A route for the flow of electricity that has elements of both series and parallel circuits.

Control - A part or component in a circuit that controls the flow of electricity.

Current - The flow of electricity. Current is measured in amps.

Current Adds - A Kirchoff Law for DC parallel circuits stating that the total current of the circuit In a parallel circuit is the sum of the currents through each individual branch, regardless of the number of branches.

Denominator - The expression in the bottom location of a fraction, below the fraction bar.

Direct Current - Current that travels in one direction. Direct current does not reverse the direction of flow.

Inversely Proportional- A relationship where a number either increases as another decreases or decreases as another increases. Inversely proportional is the opposite of directly proportional.

Kirchoff Laws - A set of universal truths established by scientist Gustav Kirchoff that govern circuit calculations.

Load - The part or component in a circuit that converts electricity into light, heat, or mechanical motion. Examples of loads are a light bulb, resistor, or motor.

Ohm's Law - The universal truth stating that it takes one volt to push one amp through one ohm.

Parallel Circuit - A route for the flow of electricity that has a branched path for each load. Parallel circuits do not require all loads to be switched on in order for the other loads in the circuit to function.

Path - A conductor that directs electricity in a circuit. The path is often copper wire.

Power - The rate at which a device converts electrical energy into another form, such as heat or light. Power is measured in watts.

Product Over Sum Method - An equation for determining the total resistance for a parallel circuit. The product over sum method divides a pair of resistors, and then divides the result by another resistor, over and over again until only one pair is left.

Reciprocal Formula - An equation for determining the total resistance for a parallel circuit. The reciprocal formula finds the total resistance of a parallel circuit by calculating the reciprocal of the sum of the reciprocals of the individual branches.

Resistance - The opposition to current flow. Resistance is measured in ohms.

Resistor - A device that restricts current flow and produces work, such as heat.

Resistors Of Equal Value Method  - An equation for determining the total resistance for a DC parallel circuit with resistors that have the same value. The resistors of equal value method finds the total resistance by dividing the value of one individual resistor by the number of branches.

Series Circuit - A route for the flow of electricity that has only one path. Series circuits are limited because, for any load to work, every load in the circuit must be switched on.

Source - The device that provides electrical power to a circuit. The source is the origin of electricity, such as a power plant.

Voltage - Electrical pressure that causes current flow. Voltage is measured in volts.

Voltage Drop - The amount of voltage needed to push a given amount of current through a given amount of resistance.

Introduction

The interconnection of various electric elements in a prescribed manner comprises as an electric circuit in order to perform a desired function. The electric elements include controlled and uncontrolled source of energy, resistors, capacitors, inductors, etc. Analysis of electric circuits refers to computations required to determine the unknown quantities such as voltage, current and power associated with one or more elements in the circuit. To contribute to the solution of engineering problems one must acquire the basic knowledge of electric circuit analysis and laws. Many other systems, like mechanical, hydraulic, thermal, magnetic and power system are easy to analyze and model by a circuit. To learn how to analyze the models of these systems, first one needs to learn the techniques of circuit analysis. We shall discuss briefly some of the basic circuit elements and the laws that will help us to develop the background of subject.