Tellegen Theorem

This theorem has been introduced in the year of 1952 by Dutch Electrical Engineer Bernard D.H. Tellegen. This is very useful theorem in network analysis. According to Tellegen theorem the summation of instantaneous powers for the n number of branches in an electrical network is zero.

Suppose n number of branches in an electrical network have i1, i2, i3, .............in respective instantaneous currents through them. These currents satisfy Kirchhoff 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,

In the network shown in the figure, let v1, v2 and v3 be 7, 2 and 3 volts respectively. Applying Kirchhoff voltage law around loop ABCDEA. We see that v4 = 2 volt is required. Around loop CDFC, v5 is required to be 3 volt and around loop DFED, v6 is required to be 2. We next applyKirchhoff current law successively to nodes B, C and D.

At node B let ii = 5 A, then it is required that i2 = − 5 A. At node C let i3 = 3 A and then i5 is required to be − 8. At node D assume i4 to be 4 then i6 is required to be − 9. Carrying out the operation of equation,

we get, 7 X 5 + 2 X ( − 5) + 3 X 3 + 2 X 4 + 3 X ( − 8) + 2 X ( − 9) = 0
Hence Tellegen theorem is verified.

Maximum Power Transfer Theorem

Suppose we have a voltage source or battery whose internal electrical resistance is 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 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.

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 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 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

Compensation theorem

This theorem based on one basic concept. When electric current flows through anyresistor, 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.

Series Resonance

Consider a RLC circuit in which resistor, inductor and capacitor are connected in series across a voltage supply. This series RLC circuit has an distinguishing property of resonating at a specific frequency called resonant frequency. In this circuit containing inductor and capacitor the energy is stored in two different ways.

• When a current flows in a inductor ,energy is stored in magnetic field.
• When a capacitor is charged, energy is stored in electric field.

The magnetic field in the inductor is build by the electric current, which is provided by the discharging capacitor. Similarly the capacitor is charged by the electric current produced by collapsing magnetic field of inductor and this process continue on and on, causing electrical energy to oscillate between the magnetic field and the electric field .In some cases at certain frequency called resonant frequency, the inductive reactance of the circuit becomes equal to capacitive reactance which cause the electrical energy to oscillate between the electric field of the capacitor and magnetic field of the inductor. This forms a harmonic oscillator for current. In RLC circuit, the presence of resistor causes these oscillation to die out over period of time and is called damping effect of resistor.

Variation in Inductive Reactance and Capacitive Reactance with Frequency

Variation of Inductive Reactance Vs Frequency

Variation of Inductive Reactance Vs Frequency

We know that inductive reactance XL = 2πfL it means that inductive reactance is directly proportional to frequency ( XL ∝ ƒ ). When the frequency is zero or in case of DC, inductive reactance is also zero, the circuit act as a short circuit but when frequency increases inductive reactance also increases. At infinite frequency inductive reactance become infinity and circuit behave as open circuit. It means that when frequency increases inductive reactance also increase and when frequency decrease inductive reactance also decrease. So if we plot a graph between inductive reactance and frequency, it is a straight line linear curve passing through origin as shown in figure above

Variation of Capacitive Reactance Vs Frequency

Variation of Capacitive Reactance Vs Frequency

It is clear from the formula of capacitive reactance, XC = 1 / 2πfC that frequency and capacitive reactance are inversely proportional to each other. In case of DC or when frequency is zero, capacitive reactance becomes infinity and circuit behave as open circuit and when frequency increases and become infinite, capacitive reactance decrease and become zero at infinite frequency, at that point the circuit act as short circuit so, the capacitive reactance increases with decease in frequency and if we plot a graph between capacitive reactance and frequency it is an hyperbolic curve as shown in figure above.

Inductive Reactance and Capacitive Reactance Vs Frequency

Inductive Reactance and Capacitive Reactance Vs Frequency

From the above discussion it is concluded the inductive reactance is directly proportional to frequency and capacitive reactance is inversely proportional to frequency i.e at low frequency XL is low and XC is high but there must be a frequency, where the value of inductive reactance becomes equal to capacitive reactance. Now if we plot a single graph of inductive reactance vs frequency and capacitive reactance vs frequency then there must occur a point where these two graph cut each other. At that point of intersection the inductive and capacitive reactance becomes the equal and the frequency at which these two reactances become equal is called resonant frequency, fr
At resonant frequency, XL = XL

At resonance f = fr and on solving above equation we get,

Variation of Impedance Vs Frequency

Variation of Impedance Vs Frequency

At resonance in series RLC circuit, two reactances become equal and cancel each other. So in resonant series RLC circuit, the opposition to the flow of current is due to resistance only. At resonance the total impedance of series RLC circuit is equal to resistance i.e Z = R, impedance has only real part but no imaginary part and this impedance at resonant frequency is called dynamic impedance and this dynamic impedance is always less than impedance of series RLC circuit. Before series resonance i.e before frequency, fr capacitive reactance dominate and after resonance inductive reactance dominate and at resonance the circuit act as purely resistive circuit causing a large amount of current to circulate through the circuit.

Resonant Current

Resonant Current

In series RLC circuit the total voltage is the phasor sum of voltage across resistor, inductor and capacitor. At resonance in series RLC circuit both inductive and capacitive reactance cancel each other and we know that in series circuit the current flowing through all the elements is same So, the voltage across inductor and capacitor is equal in magnitude and opposite in direction and thereby they cancel each other. So, in a series resonant circuit voltage across resistor is equal to supply voltage i.e V = Vr
In series RLC circuit current, I = V / Z but at resonance current I = V / R , therefore the current at resonant frequency is maximum as at resonance in , impedance of circuit is resistance only and is minimum.

The above graph shows the plot between circuit current and frequency. At starting when the frequency increases, the impedance Zc decrease and hence the circuit current increases. After some time frequency becomes equal to resonant frequency at that point inductive reactance become equal to capacitive reactance and the impedance of circuit reduces and is equal to circuit resistance only. So at this point the circuit current becomes maximum I = V / R. Now when the frequency is further increased, ZL increases and with increase in ZL , the circuit current reduces and then the current drop finally to zero as frequency becomes infinite.

Power Factor at Resonance

Power Factor at Resonance

At resonance, the inductive reactance is equal to capacitive reactance and hence the voltage across inductor and capacitor cancel each other. The total impedance of circuit is resistance only. So, the circuit behave like a pure resistive circuit and we know that in pure resistive circuit voltage and the circuit current are in same phase i.e Vr , V and I are in same phase direction . Therefore the phase angle between voltage and current is zero and the power factor is unity.

Application of Series RLC Resonant circuit

Since resonance in Series RLC Circuit occurs at particular frequency so, it is used for filtering and tuning purpose as it does not allow unwanted oscillations that would otherwise cause signal distortion , noise and damage to circuit to pass through it.

Summary
For a series RLC circuit at certain frequency called resonant frequency, the following points must be remembered. So at resonance :

Inductive reactance XL is equal to capacitive reactance XC

Total impedance of circuit becomes minimum which is equal to R i.e Z = R

Circuit current becomes maximum as impedance reduces, I = V / R

voltage across inductor and capacitor cancels each other, so voltage across resistor Vr = V, supply voltage

Since net reactance is zero, circuit becomes purely resistive circuit and hence the voltage and the current are in same phase , so the phase angle between them is zero

Power factor is unity

Frequency at which resonance in series RLC circuit occur is given by

Theorems-Thevenin, Norton, Superposition and Millman

Thevenin's Theorem

Consider the figure below which schematically represents the two-terminal network of constant emf's and resistances; a high-resistance voltmeter, connected to the accessible terminals, will indicate the so called open circuit voltage v_oc. If an extremely low-resistance ammeter is next connected to the same terminals, as in fig.(b), which is so called the short-circuit current i_sc will be measured. 
 

Test circuits for Thevenin's Theorem

Now the two quantities determined above may be used to represent an equivalent simple network consisting of the single resistance R_TH, which is equal to v_oc/i_sc. If the resistor R_L is connected to the two terminals, the load current of the circuit will be

I_L = v_oc / R_TH+R_L---------------> equation no.1 

The analysis leading to the equation no.1 above was first proposed by M.L. Thevenin the latter part of the nineteenth century, and has been recognized as an important principle in electric circuit theory. His theory was stated as follows: In any two-terminal network of fixed resistances and constant sources of emf, the current in the load resistor connected to the output terminals is equal to the current that would exist in the same resistor if it were connected in series with (a) a simple emf whose voltage is measured at the open-circuited network terminals and (b) a simple resistance whose magnitude is that of the network looking back from the two terminals into the network with all sources of emf replaced by their internal resistances.

Thevenin's Theorem has been applied to many network solutions which considerably simplify the calculations as well as reduce the number of computations.

Norton's Theorem

From the previous topic above, it was learned that a somewhat modified approach of Thevenin was formulated. This modified approach is to convert the original network into a simple circuit in which a parallel combination of constant-current source and looking-back resistance "feeds" the load resistor. Take a look on the figure below

 

Norton's equivalent circuit

Take note that Norton's theory also make use of the resistance looking back into the network from the load resistance terminals, with all potential sources replaced by the zero-resistance conductors. It also employs a fictitious source which delivers a constant current, which is equal to the current that would pass into a short circuit connected across the output terminals of the original circuit. 

From the fig (b) above of Norton's equivalent circuit, the load current would be

I_L = I_N R_N / R_N+R_L ---------------> equation no.2

Superposition Theorem

The theorem states like this: In the network of resistors that is energized by two or more sources of emf, (a) the current in any resistor or (b) the voltage across any resistor is equal to: (a) the algebraic sum of the separate currents in the resistor or (b) the voltages across the resistor, assuming that each source of emf, acting independently of the others, is applied separately in turn while the others are replaced by their respective internal values of resistance. 

This theorem is illustrated in the given circuit below:

Illustration of Superposition Theorem

The original circuit above ( left part ) have one emf source and a current source. If you like to obtain the current I which is equal to the sum of I' + I"using the superposition theorem, we need to do the following steps:

a. Replace the current source I_o by an open circuit. Therefore, an emf source v_o will act independently having a current I' as the first value obtained when the circuit computed.

b. Replace emf source v_o by a short circuit. This time I_o will act independently and I" now will be obtained when the circuit computed.

c. The two values obtained ( I' and I") with emf and current source acting independently will be added to get I = I' + I"
Millman's Theorem

Any combination of parallel-connected voltage sources can be represented as a single equivalent source using Thevenin's and Norton theorems appropriately. This can be illustrated as :

This is Millman's Theorem

The formula above can be written as:

V_L = V₁/R₁ + V₂/R₂ + .....Vn / Rn  
       -------------------------------
        1/R₁ + 1/R₂ + ......1/Rn +  1/R_L

where:

V1, V2, V3... Vn  are the voltages of the individual voltage sources.
R1, R2, R3... Rn  are the internal resistances of the individual voltage sources.

Vout or V_L= load voltage
RL    = load resistor 

Current Division in Parallel Circuit of Resistors

Consider a parallel circuit of tow resistor and connected across a source of V volts.

       Current through R₁ is I₁ and R₂ is I₂  , while total current drawn from source is I_T.

.^..                                I_T = I₁  + I₂ 

But                             I₁  = V/R₁   , I₂  = V/R₂

i.e.                              V =I₁ R₁  = I₂  R₂

.^..                                 I₁  = I₂   (R₂/R₁)

       Substituting value of I₁ in I_T,

Key point : In general, the current in any branch is equal to the ratio of opposite branch resistance to the total resistance value, multiplied by the total current in the circuit.

Example : Find the magnitude of total current, current through and if , R₁ = 10 Ω , R₂= 20 Ω and V = 50V.

Solution :

      The equivalent resistance of tow is,

                            R_eq =(R₁ R₂ ) / ( R₁ + R₂ ) = (10 x 20)/ (10 + 20) = 6.67 .

                            I_T = V/R_eq = 50/6.67 = 7.5 A

       As per the current distribution in parallel circuit,

It can be verified that  :         I_T = I₁  + I₂ .

Voltage Division in Series Circuit of Resistors

 Consider a series circuit of tow resistors R₁ and R₂ connected to source V volts.

       As tow resistors are connected in series, the current flowing through both the resistors is same, i.e. I. Then applying KVL, we get,

                                                                 V= I R₁ +I R₂ 

       Total voltage applied is equal to the sum of voltage drops V_R1 and V_R2 across R₁and R₂ respectively.

.^..                     V_R1 = I R₁ 

Similarly,             V_R2  = I R₂ 

 

       So this circuit is a voltage divider circuit.

Key point : So in general, voltage drop across any resistor, or combination of resistors, in a series circuit is equal to the ratio of that resistance value to the total resistance, multiplied by the source voltage.

Example : Find the voltage across the three resistances shown in the Fig.

 Solution : 

Key point : It can be seen that voltage across any resistance of series circuit is ratio of that resistance to the total resistance, multiplied by the source voltage.