Basic Electronics on the Go - PN Junction Theory

From http://www.physics-and-radio-electronics.com/electronic-devices-and-circuits/semiconductor-diodes/depletion-region.html

If p-type semiconductor is joined with n-type semiconductor, a p-n junction is formed. The region in which the p-type and n-type semiconductors are joined is called p-n junction. This p-n junction separates n-type semiconductor from p-type semiconductor.

In n-type semiconductors, large number of free electrons is present - due to this they get repelled from each other and try to move from a high concentration region (n-side) to a low concentration region (p-side). Moreover, near the junction free electrons and holes are close to each other. According to coulombs law there exist a force of attraction between opposite charges.

Hence, the free electrons from n-side attracted towards the holes at p-side. Thus, the free electrons move from n-side to p-side. Similarly, holes move from p-side to n-side.

Positive and negative charge at p-n junction

The free electrons that are crossing the junction from n-side provide extra electrons to the atoms on the p-side by filling holes in the p-side atoms. The atom that gains extra electron at p-side has more number of electrons than protons. We know that, when the atom gains an extra electron from the outside atom it will become a negative ion. 

 

Thus, each free electron that is crossing the junction from n-side to fill the hole in p-side atom creates a negative ion at p-side. Similarly, each free electron that left the parent atom at n-side to fill the hole in p-side atom creates a positive ion at n-side.

Negative ion has more number of electrons than protons. Hence, it is negatively charged. Thus, a net negative charge is build at the p-side of p-n junction. Similarly, positive ion has more number of protons than electrons. Hence, it is positively charged. Thus, a net positive charge is build at n-side of the p-n junction.

The net negative charge at p-side of the p-n junction prevents further flow of free electrons crossing from n-side to p-side because the negative charge present at the p-side of the p-n junction repels the free electrons. Similarly, the net positive charge at n-side of the p-n junction prevents further flow of holes from p-side to n-side.

Thus, immobile positive charge at n-side and immobile negative charge at p-side near the junction acts like a barrier or wall and prevent the further flow of free electrons and holes. The region near the junction where flow of charges carriers are decreased over a given time and finally results in empty charge carriers or full of immobile charge carriers is called depletion region.

The depletion region is also called as depletion zone, depletion layer, space charge region, or space charge layer. The depletion region acts like a wall between p-type and n-type semiconductor and prevents further flow of free electrons and holes.

Basic Electronics on the Go - Semiconductor Basics

From http://www.electronics-tutorials.ws/diode/diode_1.html

 If Resistors are the most basic passive component in electrical or electronic circuits, then we have to consider the Signal Diode as being the most basic “Active” component.

However, unlike a resistor, a diode does not behave linearly with respect to the applied voltage as it has an exponential I-V relationship and therefore can not be described simply by using Ohm’s law as we do for resistors.

Diodes are basic unidirectional semiconductor devices that will only allow current to flow through them in one direction only, acting more like a one way electrical valve, (Forward Biased Condition). But, before we have a look at how signal or power diodes work we first need to understand the semiconductors basic construction and concept.

Diodes are made from a single piece of Semiconductor material which has a positive “P-region” at one end and a negative “N-region” at the other, and which has a resistivity value somewhere between that of a conductor and an insulator. But what is a “Semiconductor” material?, firstly let’s look at what makes something either a Conductor or an Insulator.

Resistivity

The electrical Resistance of an electrical or electronic component or device is generally defined as being the ratio of the voltage difference across it to the current flowing through it, basic Ohm´s Law principals. The problem with using resistance as a measurement is that it depends very much on the physical size of the material being measured as well as the material out of which it is made. For example, if we were to increase the length of the material (making it longer) its resistance would also increase proportionally.

 Likewise, if we increased its diameter (making it fatter) its resistance value would decrease. So we want to be able to define the material in such a way as to indicate its ability to either conduct or oppose the flow of electrical current through it no matter what its size or shape happens to be.

The quantity that is used to indicate this specific resistance is called Resistivity and is given the Greek symbol of ρ, (Rho). Resistivity is measured in Ohm-metres, ( Ω-m ). Resistivity is the inverse of conductivity.

If the resistivity of various materials is compared, they can be classified into three main groups, Conductors, Insulators and Semi-conductors as shown below.

Resistivity Chart

 

 

Conductors

From above we now know that Conductors are materials that have very low values of resistivity, usually in the micro-ohms per metre. This low value allows them to easily pass an electrical current due to many free electrons floating about,within their basic atom structure. But these electrons will only flow through a conductor if there is something to spur their movement, and that something is an electrical voltage.

 When a positive voltage potential is applied to the material these “free electrons” leave their parent atom and travel together through the material forming an electron drift, more commonly known as a current. How “freely” these electrons can move through a conductor depends on how easily they can break free from their constituent atoms when a voltage is applied. Then the amount of electrons that flow depends on the amount of resistivity the conductor has.

Examples of good conductors are generally metals such as Copper, Aluminium, Silver or non metals such as Carbon because these materials have very few electrons in their outer “Valence Shell” or ring, resulting in them being easily knocked out of the atom’s orbit and become "free".

This allows them to flow freely through the material until they join up with other atoms, producing a “Domino Effect” through the material thereby creating an electrical current. Copper and Aluminium is the main conductor used in electrical cables.

Generally speaking, most metals are good conductors of electricity, as they have very small resistance values, usually in the region of micro-ohms per metre. While metals such as copper and aluminium are very good conducts of electricity, they still have some resistance to the flow of electrons and consequently do not conduct perfectly. Also the conductivity of conductors increases with ambient temperature because metals are also generally good conductors of heat.

Insulators

Insulators on the other hand are the exact opposite of conductors. They are made of materials, generally non-metals, that have very few or no “free electrons” floating about within their basic atom structure because the electrons in the outer valence shell are strongly attracted by the positively charged inner nucleus.

In other words, the electrons are stuck to the parent atom and can not move around freely so if a potential voltage is applied to the material, no current will flow as there are no “free electrons” available to move and which gives these materials their insulating properties.

Insulators also have very high resistances, millions of ohms per metre, and are generally not affected by normal temperature changes. Examples of good insulators are marble, fused quartz, p.v.c. plastics, rubber etc.

Insulators play a very important role within electrical and electronic circuits, because without them electrical circuits would short together and not work. For example, insulators made of glass or porcelain are used for insulating and supporting overhead transmission cables while epoxy-glass resin materials are used to make printed circuit boards, PCB’s etc. while PVC is used to insulate electrical cables.

Semiconductor Basics

Semiconductors materials such as silicon (Si), germanium (Ge) and gallium arsenide (GaAs), have electrical properties somewhere in the middle, between those of a “conductor” and an “insulator”. They are not good conductors nor good insulators (hence their name “semi”-conductors). They have very few “free electrons” because their atoms are closely grouped together in a crystalline pattern called a “crystal lattice” but electrons are still able to flow, but only under special conditions.

The ability of semiconductors to conduct electricity can be greatly improved by replacing or adding certain donor or acceptor atoms to this crystalline structure thereby, producing more free electrons than holes or vice versa. That is by adding a small percentage of another element to the base material, either silicon or germanium.

On their own Silicon and Germanium are classed as intrinsic semiconductors, that is they are chemically pure, containing nothing but semi-conductive material. But by controlling the amount of impurities added to this intrinsic semiconductor material it is possible to control its conductivity. Various impurities called donors or acceptors can be added to this intrinsic material to produce free electrons or holes respectively.

This process of adding donor or acceptor atoms to semiconductor atoms (the order of 1 impurity atom per 10 million (or more) atoms of the semiconductor) is called Doping. As the doped silicon is no longer pure, these donor and acceptor atoms are collectively referred to as “impurities”.

The most commonly used semiconductor basics material by far is silicon. Silicon has four valence electrons in its outermost shell which it shares with its neighbouring silicon atoms to form full orbital’s of eight electrons. The structure of the bond between the two silicon atoms is such that each atom shares one electron with its neighbour making the bond very stable.

As there are very few free electrons available to move around the silicon crystal, crystals of pure silicon (or germanium) are therefore good insulators, or at the very least very high value resistors.
Silicon atoms are arranged in a definite symmetrical pattern making them a crystalline solid structure. A crystal of pure silica (silicon dioxide or glass) is generally said to be an intrinsic crystal (it has no impurities) and therefore has no free electrons.

 But simply connecting a silicon crystal to a battery supply is not enough to extract an electric current from it. To do that we need to create a “positive” and a “negative” pole within the silicon allowing electrons and therefore electric current to flow out of the silicon. These poles are created by doping the silicon with certain impurities.

A Silicon Atom Structure

 

 

The diagram above shows the structure and lattice of a ‘normal’ pure crystal of Silicon.

N-type Semiconductor Basics

In order for our silicon crystal to conduct electricity, we need to introduce an impurity atom such as Arsenic, Antimony or Phosphorus into the crystalline structure making it extrinsic (impurities are added). These atoms have five outer electrons in their outermost orbital to share with neighbouring atoms and are commonly called “Pentavalent” impurities.

 This allows four out of the five orbital electrons to bond with its neighbouring silicon atoms leaving one “free electron” to become mobile when an electrical voltage is applied (electron flow). As each impurity atom “donates” one electron, pentavalent atoms are generally known as “donors”.

 Antimony (symbol Sb) as well as Phosphorus (symbol P), are frequently used as a pentavalent additive to silicon. Antimony has 51 electrons arranged in five shells around its nucleus with the outermost orbital having five electrons. The resulting semiconductor basics material has an excess of current-carrying electrons, each with a negative charge, and is therefore referred to as an N-type material with the electrons called “Majority Carriers” while the resulting holes are called “Minority Carriers”.


When stimulated by an external power source, the electrons freed from the silicon atoms by this stimulation are quickly replaced by the free electrons available from the doped Antimony atoms. But this action still leaves an extra electron (the freed electron) floating around the doped crystal making it negatively charged.
Then a semiconductor material is classed as N-type when its donor density is greater than its acceptor density, in other words, it has more electrons than holes thereby creating a negative pole as shown.

Antimony Atom and Doping

 

 

 The diagram above shows the structure and lattice of the donor impurity atom Antimony.

 

P-Type Semiconductor Basics

If we go the other way, and introduce a “Trivalent” (3-electron) impurity into the crystalline structure, such as Aluminium, Boron or Indium, which have only three valence electrons available in their outermost orbital, the fourth closed bond cannot be formed. Therefore, a complete connection is not possible, giving the semiconductor material an abundance of positively charged carriers known as holes in the structure of the crystal where electrons are effectively missing.

As there is now a hole in the silicon crystal, a neighbouring electron is attracted to it and will try to move into the hole to fill it. However, the electron filling the hole leaves another hole behind it as it moves. This in turn attracts another electron which in turn creates another hole behind it, and so forth giving the appearance that the holes are moving as a positive charge through the crystal structure (conventional current flow).


This movement of holes results in a shortage of electrons in the silicon turning the entire doped crystal into a positive pole. As each impurity atom generates a hole, trivalent impurities are generally known as “Acceptors” as they are continually “accepting” extra or free electrons.

Boron (symbol B) is commonly used as a trivalent additive as it has only five electrons arranged in three shells around its nucleus with the outermost orbital having only three electrons. The doping of Boron atoms causes conduction to consist mainly of positive charge carriers resulting in a P-type material with the positive holes being called “Majority Carriers” while the free electrons are called “Minority Carriers”.

Then a semiconductor basics material is classed as P-type when its acceptor density is greater than its donor density. Therefore, a P-type semiconductor has more holes than electrons.

Boron Atom and Doping

 

 The diagram above shows the structure and lattice of the acceptor impurity atom Boron.

 

Basic Electronics on the Go - Star Delta Transformation

From http://www.electronics-tutorials.ws/dccircuits/dcp_10.html

Star Delta Transformation

 We can now solve simple series, parallel or bridge type resistive networks using Kirchhoff´s Circuit Laws, mesh current analysis or nodal voltage analysis techniques but in a balanced 3-phase circuit we can use different mathematical techniques to simplify the analysis of the circuit and thereby reduce the amount of math’s involved which in itself is a good thing.

 Standard 3-phase circuits or networks take on two major forms with names that represent the way in which the resistances are connected, a Star connected network which has the symbol of the letter, Υ (wye) and a Delta connected network which has the symbol of a triangle, Δ (delta).

 If a 3-phase, 3-wire supply or even a 3-phase load is connected in one type of configuration, it can be easily transformed or changed it into an equivalent configuration of the other type by using either the Star Delta Transformation or Delta Star Transformation process.
A resistive network consisting of three impedances can be connected together to form a T or “Tee” configuration but the network can also be redrawn to form a Star or Υ type network as shown below.

T-connected and Equivalent Star Network

 

As we have already seen, we can redraw the T resistor network above to produce an electrically equivalent Star or Υ type network. But we can also convert a Pi or π type resistor network into an electrically equivalent Delta or Δ type network as shown below.

Pi-connected and Equivalent Delta Network.

 

  Having now defined exactly what is a Star and Delta connected network it is possible to transform the Υ into an equivalent Δ circuit and also to convert a Δ into an equivalent Υ circuit using a the transformation process. This process allows us to produce a mathematical relationship between the various resistors giving us a Star Delta Transformation as well as a Delta Star Transformation.

 

Delta Star Transformation

To convert a delta network to an equivalent star network we need to derive a transformation formula for equating the various resistors to each other between the various terminals. Consider the circuit below.

Delta to Star Network.

 

 Compare the resistances between terminals 1 and 2.

 Resistance between the terminals 2 and 3.

 

 

 Resistance between the terminals 1 and 3.

 This now gives us three equations and taking equation 3 from equation 2 gives:

 

Then, re-writing Equation 1 will give us:

Adding together equation 1 and the result above of equation 3 minus equation 2 gives:

 From which gives us the final equation for resistor P as:

Similarly, resistor Q and R  can be found  :-

When converting a delta network into a star network the denominators of all of the transformation formulas are the same: A + B + C, and which is the sum of ALL the delta resistances. 

Star Delta Transformation

Star Delta transformation is simply the reverse of above. We have seen that when converting from a delta network to an equivalent star network that the resistor connected to one terminal is the product of the two delta resistances connected to the same terminal, for example resistor P is the product of resistors A and B connected to terminal 1.

The value of the resistor on any one side of the delta, Δ network is the sum of all the two-product combinations of resistors in the star network divide by the star resistor located “directly opposite” the delta resistor being found. For example, resistor A is given as:

 Resistor B is given as:

Resistor C given as:

Basic Electronics on the Go - Maximum Power Transfer

From http://www.electronics-tutorials.ws/dccircuits/dcp_9.html

We have seen in the previous tutorials that any complex circuit or network can be replaced by a single energy source in series with a single internal source resistance, R_S.

 When we connect a load resistance, R_L across the output terminals of the power source, the impedance of the load will vary resulting in the power being absorbed by the load becoming dependent on the impedance of the actual power source. Then for the load resistance to absorb the maximum power possible it has to be “Matched” to the impedance of the power source and this forms the basis of Maximum Power Transfer.

The Maximum Power Transfer Theorem is another useful circuit analysis method to ensure that the maximum amount of power will be dissipated in the load resistance when the value of the load resistance is exactly equal to the resistance of the power source. The relationship between the load impedance and the internal impedance of the energy source will give the power in the load. Consider the circuit below.
 

Thevenins Equivalent Circuit.

In our Thevenin equivalent circuit above, the maximum power transfer theorem states that “the maximum amount of power will be dissipated in the load resistance if it is equal in value to the Thevenin or Norton source resistance of the network supplying the power“.

In other words, the load resistance resulting in greatest power dissipation must be equal in value to the equivalent Thevenin source resistance, then R_L = R_S but if the load resistance is lower or higher in value than the Thevenin source resistance of the network, its dissipated power will be less than maximum.

For example, find the value of the load resistance, R_L that will give the maximum power transfer in the following circuit.

Maximum Power Transfer Example No1.

 

  Where:
  R_S = 25Ω
  R_L is variable between 0 – 100Ω
  V_S = 100v

Then by using the following Ohm’s Law equations:

We can now complete the following table to determine the current and power in the circuit for different values of load resistance.

Table of Current against Power

RL (Ω) I (amps) P (watts) 0 4.0 0 5 3.3 55 10 2.8 78 15 2.5 93 20 2.2 97

RL (Ω) I (amps) P (watts) 25 2.0 100 30 1.8 97 40 1.5 94 60 1.2 83 100 0.8 64

Using the data from the table above, we can plot a graph of load resistance, R_L against power, P for different values of load resistance. Also notice that power is zero for an open-circuit (zero current condition) and also for a short-circuit (zero voltage condition).

Graph of Power against Load Resistance

 From the above table and graph we can see that the Maximum Power Transfer occurs in the load when the load resistance, R_L is equal in value to the source resistance, R_S that is: R_S = R_L = 25Ω. This is called a “matched condition”.

One good example of impedance matching is between an audio amplifier and a loudspeaker. The output impedance, Z_OUT of the amplifier may be given as between 4Ω and 8Ω, while the nominal input impedance, Z_IN of the loudspeaker may be given as 8Ω only.

Then if the 8Ω speaker is attached to the amplifiers output, the amplifier will see the speaker as an 8Ω load. Connecting two 8Ω speakers in parallel is equivalent to the amplifier driving one 4Ω speaker and both configurations are within the output specifications of the amplifier.

Improper impedance matching can lead to excessive power loss and heat dissipation. But how could you impedance match an amplifier and loudspeaker which have very different impedances. Well, there are loudspeaker impedance matching transformers available that can change impedances from 4Ω to 8Ω, or to 16Ω’s to allow impedance matching of many loudspeakers connected together in various combinations such as in PA (public address) systems.

Transformer Impedance Matching

 

 The maximum power transfer can be obtained even if the output impedance is not the same as the load impedance. This can be done using a suitable “turns ratio” on the transformer with the corresponding ratio of load impedance, Z_LOAD to output impedance, Z_OUT matches that of the ratio of the transformers primary turns to secondary turns as a resistance on one side of the transformer becomes a different value on the other.

 If the load impedance, Z_LOAD is purely resistive and the source impedance is purely resistive, Z_OUT then the equation for finding the maximum power transfer is given as:

 Where: N_P is the number of primary turns and N_S the number of secondary turns on the transformer. Then by varying the value of the transformers turns ratio the output impedance can be “matched” to the source impedance to achieve maximum power transfer.

Basic Electronics on the Go - Norton’s Theorem

From http://www.electronics-tutorials.ws/dccircuits/dcp_8.html

Nortons Theorem states that “Any linear circuit containing several energy sources and resistances can be replaced by a single Constant Current generator in parallel with a Single Resistor“.

As far as the load resistance, R_L is concerned this single resistance, R_S is the value of the resistance looking back into the network with all the current sources open circuited and I_S is the short circuit current at the output terminals as shown below.

The value of this “constant current” is one which would flow if the two output terminals where shorted together while the source resistance would be measured looking back into the terminals, (the same as Thevenin).
For example, consider our now familiar circuit from the previous section

To find the Nortons equivalent of the above circuit we firstly have to remove the centre 40Ω load resistor and short out the terminals A and B to give us the following circuit.

When the terminals A and B are shorted together the two resistors are connected in parallel across their two respective voltage sources and the currents flowing through each resistor as well as the total short circuit current can now be calculated as (done with Mesh Current Analysis, it seems):

If we short-out the two voltage sources and open circuit terminals A and B, the two resistors are now effectively connected together in parallel. The value of the internal resistor Rs is found by calculating the total resistance at the terminals A and B giving us the following circuit.

Find the Equivalent Resistance (Rs)

 

Having found both the short circuit current, Is and equivalent internal resistance, Rs this then gives us the following Nortons equivalent circuit.

Nortons equivalent circuit.

 

 We now have to solve with the original 40Ω load resistor connected across terminals A and B as shown below.

Again, the two resistors are connected in parallel across the terminals A and B which gives us a total resistance of:

The voltage across the terminals A and B with the load resistor connected is given as:

Then the current flowing in the 40Ω load resistor can be found as:

Nortons Theorem Summary

The basic procedure for solving a circuit using Nortons Theorem is as follows:

• 1. Remove the load resistor R_L or component concerned.
• 2. Find R_S by shorting all voltage sources or by open circuiting all the current sources.
• 3. Find I_S by placing a shorting link on the output terminals A and B.
• 4. Find the current flowing through the load resistor R_L.

In a circuit, power supplied to the load is at its maximum when the load resistance is equal to the source resistance. In the next tutorial we will look at Maximum Power Transfer. The application of the maximum power transfer theorem can be applied to either simple and complicated linear circuits having a variable load and is used to find the load resistance that leads to transfer of maximum power to the load.

Basic Electronics on the Go - Thevenin’s Theorem

From http://www.electronics-tutorials.ws/dccircuits/dcp_7.html

In this tutorial we will look at one of the more common circuit analysis theorems (next to Kirchhoff´s) that has been developed, Thevenin’s Theorem.

Thevenin’s Theorem states that “Any linear circuit containing several voltages and resistances can be replaced by just one single voltage in series with a single resistance connected across the load“. In other words, it is possible to simplify any electrical circuit, no matter how complex, to an equivalent two-terminal circuit with just a single constant voltage source in series with a resistance (or impedance) connected to a load as shown below.

Thevenin’s equivalent circuit.

  

Consider the circuit from the previous section.

 

Firstly, to analyse the circuit we have to remove the centre 40Ω load resistor connected across the terminals A-B, and remove any internal resistance associated with the voltage source(s). This is done by shorting out all the voltage sources connected to the circuit, that is v = 0, or open circuit any connected current sources making i = 0. The reason for this is that we want to have an ideal voltage source or an ideal current source for the circuit analysis.

The value of the equivalent resistance, Rs is found by calculating the total resistance looking back from the terminals A and B with all the voltage sources shorted. We then get the following circuit.

Find the Equivalent Resistance (Rs)

 

 

The voltage Vs is defined as the total voltage across the terminals A and B when there is an open circuit between them. That is without the load resistor R_L connected.

We now need to reconnect the two voltages back into the circuit, and as V_S  =  V_AB the current flowing around the loop is calculated as:

This current of 0.33 amperes (330mA) is common to both resistors so the voltage drop across the 20Ω resistor or the 10Ω resistor can be calculated as:

V_AB  =  20  –  (20Ω x 0.33amps)  =   13.33 volts.

or

V_AB  =  10  +  (10Ω x 0.33amps)  =   13.33 volts, the same.

Then the Thevenin’s Equivalent circuit would consist or a series resistance of 6.67Ω’s and a voltage source of 13.33v. With the 40Ω resistor connected back into the circuit we get:

and from this the current flowing around the circuit is given as:

which again, is the same value of 0.286 amps, we found using Kirchoff´s circuit law in the previous circuit analysis tutorial.

Thevenin’s theorem can be used as another type of circuit analysis method and is particularly useful in the analysis of complicated circuits consisting of one or more voltage or current source and resistors that are arranged in the usual parallel and series connections.

While Thevenin’s circuit theorem can be described mathematically in terms of current and voltage, it is not as powerful as Mesh or Nodal analysis in larger networks because the use of Mesh or Nodal analysis is usually necessary in any Thevenin exercise, so it might as well be used from the start. However, Thevenin’s equivalent circuits of Transistors, Voltage Sources such as batteries etc, are very useful in circuit design.

Thevenin’s Theorem Summary

The basic procedure for solving a circuit using Thevenin’s Theorem is as follows:

• 1. Remove the load resistor R_L or component concerned.
• 2. Find R_S by shorting all voltage sources or by open circuiting all the current sources.
• 3. Find V_S by the usual circuit analysis methods.
• 4. Find the current flowing through the load resistor R_L.

In the next tutorial we will look at Nortons Theorem which allows a network consisting of linear resistors and sources to be represented by an equivalent circuit with a single current source in parallel with a single source resistance.

Basic Electronics on the Go - Nodal Voltage Analysis

From http://www.electronics-tutorials.ws/dccircuits/dcp_6.html

Nodal Voltage Analysis complements the previous mesh analysis in that it is equally powerful and based on the same concepts of matrix analysis.  Nodal Voltage Analysis uses the “Nodal” equations of Kirchhoff’s first law to find the voltage potentials around the circuit.
So by adding together all these nodal voltages the net result will be equal to zero. Then, if there are “n” nodes in the circuit there will be “n-1” independent nodal equations and these alone are sufficient to describe and hence solve the circuit.
At each node point write down Kirchhoff’s first law equation, that is: “the currents entering a node are exactly equal in value to the currents leaving the node” then express each current in terms of the voltage across the branch. For “n” nodes, one node will be used as the reference node and all the other voltages will be referenced or measured with respect to this common node.

Nodal Voltage Analysis Circuit

 

In the above circuit, node D is chosen as the reference node and the other three nodes are assumed to have voltages, Va, Vb and  Vc with respect to node D. For example;

 

 As Va = 10v and Vc = 20v , Vb can be easily found by:

In the next tutorial we will look at Thevenins Theorem which allows a network consisting of linear resistors and sources to be represented by an equivalent circuit with a single voltage source and a series resistance.