Circuit Theory-Tellegen Theorem

Tellegen Theorem

 According toTellegen theorem the summation of instantaneous powers for the n number of branches in an electrical network is zero. Are you confused ? Let's explain. Suppose n number of branches in an electrical network have i1, i2, i3, .............in respective instantaneous currents through them. These currents satisfy Kirchhoff's current law. Again, suppose these branches have instantaneous voltages across them are v1, v2, v3, ........... vn respectively. If these voltages across these elements satisfy Kirchhoff Voltage law then,

Where vk is the instantaneous voltage across the kth branch and ik is the instantaneous current flowing through this branch. Tellegen theorem is applicable mainly general class of lumped networks consists of linear, non-linear, active, passive, time variant and time variant elements. This theorem can easily be explained by the following example.

In the network shown, arbitrary reference directions have been selected for all of the branch currents, and the corresponding branch voltages have been indicated, with positive reference direction at the tail of the current arrow. For this network, we will assume a set of branch voltages satisfy the Kirchhoff voltage law and a set of branch current satisfy Kirchhoff current law at each node. We will then show that these arbitrary assumed voltage and currents satisfy the equation.

And it is the condition of Tellegen theorem

Circuit Theory-Compensation Theorem

Compensation Theorem

This theorem based on one basic concept. When electric current flows through any resistor, there would be a voltage drop across the resistor according to Ohm's law. This dropped voltage opposes the source voltage. Hence voltage drop across an electric resistance in any network can be assumed as a voltage source acting opposite to the source voltage. The compensation theorem depends upon this concept.

According to this theorem, any resistance in a network may be replaced by a voltage source that has zero internal resistance and a voltage equal to the voltage drop across the replace resistance due to the current which was flowing through it. This imaginary voltage source is directed opposite to the voltage source of that replaced resistance. Think about a resistive branch of any complex network whose resistance value is R. Let's assume current I flowing through that resistor R and voltage drop due to this current across the resistor is V = I.R. According to compensation theorem this resistor can be replaced by a voltage source whose generated voltage will be V ( = IR) and directed against the direction of network voltage or direction of current I.

The compensation theorem can easily be understood by this following example.

Compensation Theorem

Here in the network for 16 V source, all the currents flowing through the different resistive branches are shown in the first figure. The current through the right most branch in the figure is 2A and its resistance is 2 Ω. If this right most branch of the network is replaced by a voltage source V = 2ΩX2A = 4V directed as shown in the second figure, then current through the other branches of the network remain same as shown in the second figure.

CIRCUIT THEORY-Reciprocity Theorem

Reciprocity Theorem

In many electrical network it is found that if positions of voltage source and ammeter are interchanged, the reading of ammeter remains same. It is not clear to you. Let's explain in details. Suppose a voltage source is connected to a passive network and an ammeter is connected to other part of the network to indicate the response. Now any one interchanges the positions of ammeter and voltage source that means he or she connects the voltage source at the part of the network where the ammeter was connected and connects ammeter to that part of the network where the voltage source was connected. The response of the ammeter means electric current through the ammeter would be same in both cases. This is where the property of reciprocity comes in circuit. The particular circuit which has this reciprocal property is called reciprocal circuit. This type of circuit perfectly obeys reciprocity theorem.

The voltage source and the ammeter used in this theorem must be ideal. That means the internal electrical resistance of both voltage source and ammeter must be zero. The reciprocal circuit may be a simple or complex network. But every complex reciprocal passive network can be simplified to a simple network. As per reciprocity theorem in a linear passive network, supply voltage V and output current I are mutually transferable. The ratio of V and I is called the transfer resistance. The theorem can easily be understood by this following example.

Circuit theory- Maximum power transfer theorem

Maximum Power Transfer Theorem

Suppose we have a voltage source or battery whose internal electrical resistanceis Ri and a load resistance RL is connected across this battery. Maximum power transfer theorem determines the value of resistance RL for which the maximum power will be transferred from source to it. Actually the maximum power, drawn from the source, depends upon the value of the load resistance. There may be some confusion let us clear it.

Power delivered to the load resistance,

To find the maximum power, differentiate the above expression with respect to electrical resistance RL and equate it to zero. Thus,

Thus in this case, the maximum power will be transferred to the load when load resistance is just equal to internal resistance of the battery.

Maximum power transfer theorem can be applicable in complex network as follows-

A resistive load in a resistive network will abstract maximum power when the load resistance is equal to the resistance viewed by the load as it looks back to the network. Actually this is nothing but the resistance presented to the output terminals of the network. This is actually Thevenin equivalent resistance as we explained in Thevenin theorem if we consider the whole network as a voltage source. Similarly if we consider the network as current source, this electrical resistance will be Norton equivalent resistance as we explained in Norton theorem.

CIRCUIT THEORY-Norton theorem

Norton Theorem

 In this theorem the circuit network is reduced into a single constant current source in which the equivalent internal resistance is connected in parallel with it. Every voltage source can be converted to equivalent current source.

Suppose in complex network we have to find out the electrical current through a particular branch. If the network has one of more active sources then it will supply current through the said branch. As the said branch current comes from the network, it can be considered the network itself is a current source. So in Norton theorem the network with different active sources is reduced to single current source whose internal resistance is nothing but the looking back resistance connected in parallel to the derived source. The looking back resistance of a network is the equivalent electrical resistance of the network when someone looks back into the network from the terminals where said branch is connected. During calculating this equivalent resistance, all sources are removed leaving their internal resistances in the network. Actually in Norton theorem, the branch of the network through which we have to find out the current, is removed from the network. After removing the branch we short circuit the terminals where the said branch was connected. Then we calculate the short circuit current flows between the terminals. This current is nothing but Norton equivalent current IN of the source. The equivalent resistance between the said terminals with all sources removed leaving their internal resistances in the circuit is calculated and say it is RN. Now we will form a current source whose current is IN A and internal shunt resistance is RN Ω.

For getting more clear concept of this theorem, we have explained it by the following example,

In the example two resistors R1 and R2 are connected in series and this series combination is connected across one voltage source of emf E with internal resistance Ri as shown. Series combination of one resistive branch of RL and another resistance R3 is connected across the resistance R2 as shown. Now we have to find out the current through RL by applying Norton theorem.

First we have to remove the resistor RL from terminals A and B and make the terminals A and B short circuited by zero resistance.

Second we have to calculate the short circuit current or Norton equivalent current IN through the points A and B.

For determining of internal resistance or Norton equivalent resistance RN of the network under consideration. Remove the the branch between A and B and also replace the voltage source by its internal resistance. Now the equivalent resistance as viewed from open terminals A and B is RN,

As per Norton theorem when electrical resistance RL is reconnected across terminals A and B, the network behaves as a source of constant current IN with shunt connected internal resistance RN and This is Norton equivalent circuit.

Norton Equivalent Circuit

CIRCUIT THEORY-Thevenin theorem

Thevenin Theorem

This theorem is very conceptual. If we think deeply about an electrical circuit, we can visualize the statements made in Thevenin theorem. Suppose we have to calculate the electric current through any particular branch in a circuit. This branch is connected with rest of the circuit at its two terminals. Due to active sources in the circuit there is one potential difference between the points where the said branch is connected. The current through the said branch is caused by this potential difference appears across the terminals. So rest of the circuit can be considered as a single voltage source, whose voltage is nothing but the open circuit voltage between the terminals where the said branch is connected and the internal resistance of the source is nothing but the equivalent electrical resistance of the circuit looking back into the terminals where the branch is connected. So the Thevenin theorem can be stated as follows,

1) When a particular branch is remove from a circuit, the open circuit voltage appears across the terminals of the circuit, is Thevenin equivalent voltage and,

2) The equivalent resistance of the circuit network looking back into the terminals, isThevenin equivalent resistance.

3) If we replace the rest of the circuit network by a single voltage source, then the voltage of the source would be Thevenin equivalent voltage and internal resistance of the voltage source would be Thevenin equivalent resistance which would be connected in series with the source as shown in the figure below.

For better understanding Thevenin theorem, we have shown the circuit below,

Here two resistors R1 and R2 are connected in series and this series combination is connected across one voltage source of emf E with internal resistance Ri as shown. One resistive branch of RL is connected across the resistance R2 as shown. Now we have to calculate the current through RL.

First we have to remove the resistor RL from the terminals A and B.

Second we have to calculate the open circuit voltage or Thevenin equivalent voltage VT across the terminals A and B.

The current through resistance R2,

Hence voltage appears across the terminals A and B i.e.

Third, for applying Thevenin theorem, we have to determine the Thevenin equivalent electrical resistance of the circuit and for that first we have to replace the voltage source from the circuit leaving behind only its internal resistance Ri. Now view the circuit inwards from the open terminals A and B. It is found the circuits now consists of two parallel paths - one consisting of resistance R2 only and the other consisting of resistance R1 and Ri in series.

Thus the Thevenin equivalent resistance RT as viewed from the open terminals A and B is given as. As per Thevenin theorem when resistance RL is connected across terminals A and B the network behaves as a source of voltage VT and internal resistance RT and this is called Thevenin equivalent circuit. The current through RL is given as,

Thevenin Equivalent Circuit

CIRCUIT THEORY-superposition theorem

Superposition Theorem

This theorem is very simple one. Suppose a branch of an electrical circuit is connected to numbers of voltage and current sources. As we can consider electrical current as electrical quantity, it can be easily assumed that total current flows through the branch is nothing but the summation of all individual currents, contributed by the each individualvoltage or current source. This simple conception mathematically represents in theSuperposition theorem.

If there are several sources acting simultaneously in an electrical circuit then the current through any branch of the circuit is summation of currents which would flow through the branch for each sources keeping all other sources dead. Suppose there are n number of sources acting in a circuit due to which I current flows through a particular branch of the circuit. If some one replaces all the sources from the circuit by their internal resistance except first source which is now acting along in the circuit and giving current I1 through the said branch. Then he or she reconnects the second source and replaces the first source by its internal resistance. Now the current through that said branch for this second source alone can be assumed I2. Similarly if he or she reconnect the third source and replaces the second source by its internal resistance. Now the current through that said branch for this third source alone is assumed I3. Similarly when nth source acts alone in the circuit and all other sources are replaced by their internal electrical resistances, then say In current flows through the said branch of the circuit. Now according toSuperposition theorem, current through the branch when all the sources are acting on the circuit simultaneously, is nothing but summation of these individual current caused by individual sources acting alone on the circuit.

Electrical sources may be of two kinds mainly, one is voltage source and other is current source. When we remove the voltage source from a circuit, the voltage, was contributed to the circuit becomes zero. So for getting zero potential difference between the points where the removed voltage source was connected, these two points must be short circuited by zero resistance path. Fore more accuracy one can replace the voltage source by its internal resistance. Now if we remove a current source from the circuit,electric current, is contributed by this source, will become zero. Zero current implies open circuit. So when we remove current sourcefrom a circuit, we just disconnect the source from the circuit terminals and keep both terminals open circuited. As the ideal internal resistance of a current source is infinitely large, removing a current source from a circuit can be alternatively referred as replacing the current source by its internal resistance. So for superposition theorem, the voltage sources are replaced by short circuits and current sources are replaced by open circuits.

This theorem is only applicable to linear circuit i.e. circuit consisting of resistances in which Ohm's law is valid. In the circuits having non - linear resistances such as thermionic valves, metallic rectifiers this theorem not applicable. This theorem is more laborious one than other many circuit theorems. But main advantage of this method is that it avoids solutions of two or more simultaneous equations. But after little practice with this method, equations can be written directly from the original circuit diagram and labor in drawing extra diagrams is saved. For better understanding the procedure, we have furnished the different steps of Superposition theorem as follows,

Step - 1

Replace all but one of the sources by their internal resistances.

Step - 2

Determine the currents in various branches using simple Ohm's law.

Step - 3

Repeat the process using each of the sources turn - by turn as the sole source each time.

Step - 4

Add all the currents in a particular branch due to each source. This the desired value of current at that branch when all the sources acting on the circuit simultaneously.

Example of Superposition Theorem

Suppose there are two voltage sources V1 and V2 acting simultaneously on the circuit.
Because of these two voltage sources, say current I flows through the resistance R.

Now replace V2 by short circuit, keeping V1 at its position and measure current through the resistance, R. Say it is I1.

Then replace, V1 by short circuit, reconnect V2 to its original position and measure current through the same resistance R and say it is I2.

Now if we add these two currents, I1 and I2 we will get the current which is equal to the current - was actually flowing through R, when both voltage sources V1 and V2 were acting on the circuit simultaneously. That is I1 + I2 = I.