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1 NIRMA UNIVERSITY INSTITUTE OF TECHNOLOGY B.Tech. Semester III (Electrical), July 2017 EE02: NETWORK ANALYSIS & SYNTHESIS (LPW) INDEX SR. NO. TITLE PAGE DATE SIGN REMARKS FROM TO 1. To verify the Superposition Theorem. 2. To verify the Thevenin Theorem. 3. To verify the Norton Theorem. 4. To verify the Maximum Power Transfer Theorem. 5. To verify the Reciprocity Theorem and the Tellegen`s Theorem 6. (i) To determine the z parameters of a two port resistive network. (ii) To determine the z parameters of Series connection of two port resistive networks. 7. (i) To determine the y parameters of a two port resistive network. (ii) To determine the y parameters of the parallel connection of two port resistive networks. 8. (i) To determine the ABCD parameters of a two port resistive network. (ii) To determine the ABCD parameters of the cascade connection of two port resistive networks. 9. (i) To determine the h parameters of a two port resistive network. (ii) To determine the g parameters of a two port resistive network 10. To study the response in R-L-C series circuit and determine various time response specifications. 11. To study the step response of first order R-C circuit and cascaded R-C sections 12. To design and test a passive constant-k High Pass Filter and measure its cut-off frequency

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  • 1

    NIRMA UNIVERSITY INSTITUTE OF TECHNOLOGY

    B.Tech. Semester III (Electrical), July 2017

    EE02: NETWORK ANALYSIS & SYNTHESIS (LPW)

    INDEX

    SR.

    NO. TITLE

    PAGE DATE SIGN REMARKS FROM TO

    1. To verify the Superposition Theorem.

    2. To verify the Thevenin Theorem.

    3. To verify the Norton Theorem.

    4. To verify the Maximum Power Transfer

    Theorem.

    5. To verify the Reciprocity Theorem and

    the Tellegen`s Theorem

    6.

    (i) To determine the z – parameters of a two port resistive network.

    (ii) To determine the z – parameters of Series connection of two port

    resistive networks.

    7.

    (i) To determine the y – parameters of a two port resistive network.

    (ii) To determine the y – parameters of the

    parallel connection of two port

    resistive networks.

    8.

    (i) To determine the ABCD parameters of a two port resistive network.

    (ii) To determine the ABCD parameters

    of the cascade connection of two port

    resistive networks.

    9.

    (i) To determine the h – parameters of a two port resistive network.

    (ii) To determine the g – parameters of a

    two port resistive network

    10.

    To study the response in R-L-C series

    circuit and determine various time

    response specifications.

    11. To study the step response of first order

    R-C circuit and cascaded R-C sections

    12.

    To design and test a passive constant-k

    High Pass Filter and measure its cut-off

    frequency

  • 2

    EXPERIMENT NO: 1 DATE:

    AIM : To verify Superposition Theorem.

    APPARATUS:

    (1) Regulated power supply (D.C) 0 - 30V 2

    (2) Board for containing the network 1

    (3) Ammeters 0 - 250 mA 3

    (4) Voltmeter 0 - 30 V 1

    THEORY:

    “The superposition theorem states that the response in any element of a linear bilateral

    network containing two or more sources is the algebraic sum of the responses obtained by

    each source acting separately at a time and with all the other sources set equal to zero,

    leaving behind their internal resistance in the network”.

    According to this theorem, if there are a number of emfs acting simultaneously in any linear

    bilateral network, each emf acts independently of the others i.e. as if the other emfs doesn't

    exist. The value of current in any element of the network is the algebraic sum of the currents

    due to each emf. Similarly voltage across any element/branch is the algebraic sum of the

    voltages which each emf would have produced while acting separately at a time. In

    other words, current through or voltage across any conductor of the network is obtained

    by superimposing the currents and voltages due to each e.m.f. in the network .It is

    important to note that this theorem is applicable only to linear networks. The superposition

    theorem is applied to determine currents and voltages which are linearly related to the

    sources acting on the network.

    In Fig(a) I1, I2 and I3 represent values of currents due to simultaneous action of the two

    sources of e.m.fs in the network. In fig(b) I1', I2' and I' represent values of currents due to

    source of e.m.f E1 alone. In fig (c) I1", I2" and I" represent values of currents due to source

    of e.m.f E2 alone. By superimposing the current values of fig (b) and fig (c) the actual

    values of currents due to both the sources can be obtained as under:

    I1 = I1' + I1" (algebraic)

    I2 = I2" + I2' (algebraic)

    I = I' + I" (algebraic)

  • 3

    PROCEDURE:

    1. Connect the circuit diagram as shown in the fig (1)

    2. Connect the network with two e.m.f sources and adjust the source voltages such that

    current values are not exceeded beyond the ranges and ratings of the resistance and note

    down the meter readings.

    3. Set the e.m.f E2 to zero and note down the readings. due to E1 alone. If any meter

    indicates negative, interchange the connection of that meter and consider that reading as

    negative. Refer fig(2) .

    4. Adjust E2 as before (as per step. 2) and set E1 to zero and note down the meter readings

    If any meter indicates negative, interchange the connection of that meter and note

    down the reading of that meter with opposite sign w.r.t. the step 3. Refer fig(3).

    5. Verify the superposition theorem and tabulate the results.

    OBSERVATION TABLE:

    SR.

    NO

    E1

    Volts

    E2

    Volts

    I1

    mA

    I2

    mA

    I3

    mA

    V1

    Volts

    V2

    Volts

    V3

    Volts

    1

    2 0

    3 0

    CALCULATION:

    I1 = I1' + I1" (Algebraic) V1 = V1' + V1" (Algebraic)

    I2 = I2' + I2" (Algebraic) V2 = V2' + V2" (Algebraic)

    I3 = I3' + I3" (Algebraic) V3 = V3' + V3" (Algebraic)

    RESULT TABLE:

    SR.

    NO

    I1

    mA

    I2

    mA

    I3

    mA

    V1

    Volts

    V2

    Volts

    V3

    Volts

    Practical

    Theoretical

  • 4

    CONCLUSION:

    QUIZ :

    1. Superposition theorem can be applied only to circuits having ________.

    2. Superposition theorem requires as many circuits to be solved as there are

    (a) sources , nodes and meshes (b) sources and nodes

    (c) sources (d) nodes.

    3. Total resistance of a parallel circuit is _______ the smallest branch resistance.

    4. Is superposition theorem applicable to POWER as it is applicable to voltage and current? Why?

    5. Calculate the voltage across 5 A source in the given circuit

    6. The potential of the point A in the given network

    7. The current through 30 Ω branch in the given circuit is

    8. The current in 1 Ω resistor is

  • 5

    EXPERIMENT NO: 2 DATE :

    AIM : To verify Thevenin’s Theorem

    APPARATUS:

    (1) Board containing network 1

    (2) Ammeter 0 - 50 mA. 1

    (3) Voltmeter 0 - 10V 1

    (4) Regulated power supply 0-30V 1

    THEORY:

    Thevenin’s theorem state that any two terminal network whether simple or complex can

    be replaced by a single source of voltage Vth in series with a single resistance Rth (in case

    of d.c) or impedance Zth(in case of a.c) Hence Thevenin's equivalent circuit consists of Vth

    in series with Rth (or Zth) as shown in fig(B). Once a Thevenin's circuit is obtained it is

    connected across the resistance RL in which current is to be determined. Once the

    current value in RL is known, potential difference across it can be calculated if required.

    For obtaining Thevenin’s circuit, proceed as follows:

    1. Remove the resistance RL and measure (or calculate) voltage Eth between the terminals

    from where RL has been removed.

    2. Replace all the e.m.f sources by their internal resistance (or impedances) and measure

    (or calculate) Rth (or Zth) between the terminals from where RL has been disconnected.

    3. Draw the Thevenin's equivalent network.

    4. For calculating current in RL, connect RL which was removed earlier across this

    Thevenin's circuit.

    5. Current through RL is given by

    Vth

    IL = -----------

    Rth + RL

    PROCEDURE:

    1. Connect the circuit as shown in the fig(1).

    2. Switch on the supply and adjust the supply voltage such that meter readings are not

    exceeded their ranges and ratings of the resistances. Note down the current through the

    load resistance RL.

  • 6

    3. Disconnect the resistance RL from the circuit and measure the voltage across the terminals

    from where the resistance RL is disconnected. This voltage is known as Eth. Refer fig(2).

    4. Replace source of e.m.f. by its internal resistance and measure the total resistance (or

    impedance) of the network between the terminals from where the resistance RL is

    disconnected. This resistance (or impedance) is known as Rth (or Zth). Refer fig(3).

    5. Calculate the current through RL using the formula.

    Vth

    IL = -----------

    Rth + RL

    6. Compare it with the value obtained in step (2)

    OBSERVATION TABLE:

    SR.

    NO

    VOLTAGE

    ACROSS

    RL

    VL volts

    CURRENT

    THROUGH

    RL

    IL mA

    RL = VL/IL Eth Volts REMARKS

    1

    2 - - Disconnect the

    resistance RL

    (Measurement of Rth)

    SR

    NO

    SUPPLY

    VOLTAGE

    V volts

    CURRENT

    I mA

    Rth = V / I REMARKS

    1 Set source e.m.f to zero

    2 Set source e.m.f to zero

    CALCULATION :

    (1) RL = VL/IL = =

    (2) Rth = V/I = =

    Vth

    (3) IL = ----------- = =

    Rth + RL

  • 7

    RESULT TABLE:

    THEORETICAL PRACTICAL

    Vth

    Rth

    IL

    CONCLUSION: -

    QUIZ :

    1. While Thevenizing a circuit between two terminals, Vth is equal to __________. 2. Thevenin’s resistance is determined by _____________________. 3. While determining Rth in Thevenin’s and Norton’s equivalent

    (a) only current source are made dead (b) only voltage sources are made dead (c) all independent sources are made dead (d) all current and voltage sources ar emade dead

    4. In Thevenin’s theorem Z is determined by _____________. 5. Which theorem is applicable for both linear and nonlinear circuits? 6. The Thevenin impedance across the terminals AB of the given network is

    7. To find current in a resistance connected in one branch of a network thevenin’s

    theorem is used. VTH = 20V and RTH = 5 Ω. The current in the resistant is.

    (a) is 4 A (b) is 4A or less

    (c) is less than 4 A (d) equal to 4 A or less than 4 or more than 4 A.

    (e) none of these

    8 . In an ac network, the thevenin’s impedance and Norton’s impedance as seen from any

    two terminals are.

    (a) always the same (b) sometimes the same

    (c) generally the same (d) mostly the same

  • 8

    EXPERIMENT NO: 3 DATE:

    AIM : To verify Norton Theorem.

    APPARATUS:

    (1) Board containing network 1

    (2) Ammeter 0 - 50 mA. 1

    (3) Voltmeter 0 - 10V 1

    (4) Regulated power supply 0-30V 1

    THEORY :

    This theorem is an alternative to the Thevenin’s theorem. In fact, it is the dual of

    Thevenin's theorem. Whereas Thevenin’s theorem reduces a two - terminal active network

    to an equivalent constant voltage source and series resistance Norton's theorem replaces

    the network by an equivalent constant current source and a parallel resistance. It states

    that any two - terminal active network containing voltage/current sources and

    resistances/impedances when viewed from its output terminals is equivalent to a constant

    current source and a parallel resistance (or impedance). The constant current is equal to

    the current which would flow in a short - circuit placed across the terminals and parallel

    resistance (or impedance) is the resistance (or impedance) of the network when viewed

    from these open circuited terminals after all sources of e.m.fs have been supressed and

    replaced by their internal resistances (or impedances).

    PROCEDURE for analysis of network:

    1. Remove the resistance RL, short the terminals through an ammeter from where RL has

    been removed and observe (or calculate) the reading of the ammeter. This gives the

    value of the current of the Norton’s current source, Isc.

    2. Replace the source by its internal resistance (or impedance) and measure (or calculate)

    the resistance RN (or impedance ZN) between the terminals from where RL has been

    removed.

    3. Connect the RN (or ZN) in parallel with the current source and connect RL which

    was disconnected earlier across Norton's equivalent circuit.

    4. Current through the resistance RL is given by

  • 9

    Rth

    IL = Isc ----------

    Rth + RL

    PROCEDURE:

    1.Connect the circuit as shown in fig (4).

    2. Switch on the power supply and adjust the supply voltage such that meter readings are

    not exceeded their ranges and ratings of the resistances. Note down the current through

    the resistance RL.

    3. Disconnect the resistance RL and short the terminals through the ammeter from where

    RL has been removed and measure(or calculate) the current. This gives the value of the

    current (Isc) of the current source. Refer fig(5).

    4. Replace source of e.m.f by its internal resistance (or impedance) and measure the total

    resistance (or impedance) of the network between the terminals from where the resistance

    (RL) has been removed. This is known as RN (or ZN). Refer fig (6).

    1. Calculate the current through RL according to

    RN

    IL = Isc -----------

    RN + RL

    and compare its value obtained in step (2)

    OBSERVATION TABLE:

    SR.

    NO

    VOLTAGE

    ACROSS

    RL VL volts

    CURRENT

    THROUGH

    RL

    IL mA

    RL =

    VL/IL

    ISC

    mA

    REMARKS

    1

    -

    2 Disconnect RL and short

    the terminals through

    ammeter

    (Measurement of RN)

    SR.

    NO

    SUPPLY

    VOLTAGE

    V volts

    CURRENT

    I mA

    RN = V / I REMARKS

    1 Set source e.m.f. to zero

    2 Set source e.m.f. to zero

  • 10

    CALCULATION :

    (1) RL = VL/IL = =

    (2) RN = V/I = =

    RN

    (3) IL = Isc ----------- = =

    RN + RL

    RESULT TABLE:

    THEORETICAL PRACTICAL

    ISC

    RN

    IL

    CONCLUSION: -

    QUIZ: -

    1. For which type of network the Norton's theorem is applicable?

    2. The circuit whose parameters change with voltage or current is called a _______ circuit.

    3. _________ theorem is quite useful when the current in one branch of a network is to be

    determined or when the current in an added branch is to be calculated.

    4. The circuit whose parameters are constant is called a linear circuit. (Yes/No)

    5. In Thevenin's theorem to find Zth, all independent ________ are set to zero and all

    independent_______ are open circuited.

    6. Thevenin’s equivalent circuit is preferred when the circuit is analyzed in terms of _________

    and __________.

    7. Norton equivalent circuit is preferred when the circuit is analyzed in terms of _________ and

    __________.

    8. Given the Thevenin`s equivalent of an electric circuit, how will you determine the Norton`s

    equivalent? Justify with detailed example.

  • 11

    EXPERIMENT NO: 4 DATE:

    AIM: To verify Maximum Power Transfer Theorem.

    APPARATUS:

    (1) Board for connecting network

    (2) Ammeter 0 - 10 mA 01

    (3) Voltmeter 0 - 10V 01

    (4) Regulated power supply 0-30V 01

    THEORY :

    Maximum power transfer theorem deals with transfer of maximum power from a source

    to load. This theorem in dc circuit states the relationship between the load resistance and

    the internal resistance of the source for maximum power transfer from source to load.

    This condition is also referred as resistance matching and it is very important

    in electronics and communication circuits for obtaining maximum output. Let

    us consider a circuit supplying a power to a load of resistance RL ohms.

    The circuit of fig (1) can be simplified to the circuit of fig (2) by using Thevenin's

    theorem, from fig (2) the current through RL is given by

    E

    I = -------

    Ri + RL

    Power transferred to the load

    PL = I2RL

    E 2

    = -------- RL

    Ri + RL

    E 2 RL

    = ----------- -----------(1)

    (Ri + RL)2

    In the above expression the resistance Rs and voltage E are constant. Hence PL varies

    with respect to only variable RL Power delivered to the load is a maximum if,

    d PL ------ = 0

    d RL

  • 12

    Differentiating the expression (1) wrt RL and equating to zero, we obtain the condition

    for maximum power i.e RL = Ri

    Hence for maximum power transfer the load resistance should be equal to the internal

    resistance of the source,

    E2 RL Pmax = ------------

    (RL + Ri )2

    E2

    = ------- watts (because RL = Ri)

    4RL

    PROCEDURE:

    (1) Connect the circuit as shown in the fig.(3)

    (2) Switch on the supply and adjust suitable voltage of the supply.

    (3) Vary the load resistance from zero onward in suitable steps. For each step take meter

    readings.

    (4) Calculate the power taken by the load for each value of the load resistance.

    (5) Draw the graph of PL v/s RL.

    OBSERVATION TABLE :

    SR.

    NO

    SUPPLY

    VOLTAGE

    Vs (Volts)

    LOAD

    CURRENT

    IL ( mA)

    VOLTAGE

    ACROSS

    LOAD

    VL(Volts)

    LOAD

    RESISTANCE

    RL = VL/IL

    POWER

    DELIVERED TO

    THE

    RESISTANCE,

    RL

    PL = IL2 X RL

    1.

    2.

    3.

    4.

    5.

    CALCUATION :

    VL (1) RL = -----

    IL

    (2) PL = IL2. RL

  • 13

    CONCLUSION: -

    QUIZ: -

    1. Maximum power transfer theorem is applicable when the sources are connected in

    _____________.

    2. Assuming that we can determine the Thevenin’s equivalent resistance of our wall socket,

    why don`t heater, microwave oven and TV manufacturer match each appliances Thevenin

    equivalent resistance of this value? Will it not is permit max power transfer to flow from

    the utility company to our household appliances?

    3. A black box with a circuit in it is connected to a variable resistor. An ideal ammeter and

    an ideal voltmeter are used to measure current and voltage respectively. The results are:

    R V I

    2 3 1.5 Determine the maximum power from the box.

    8 8 1.0

    14 10.5 0.75

    4. Maximum power transfer theorem is particularly useful for analyzing _______networks.

    5. The maximum power that can be distributed in the load in circuit shown is

    6. If Rg in the circuit shown is variable between 20Ω and 80Ω, then the maximum power

    transferred to the load RL will be

  • 14

    7. Which one of the following impedance values of load will cause maximum power to be

    transferred to the load for the network shown in the given figure?

    (a) 2+j2 (b) 2-j2 (c) –j2 (d)2

    8. The value of the resistance R in the circuit shown in figure is varied in such a manner that the

    power dissipated in the 3 Ω resistor is maximum. Under this condition, the value of r will be

    (a) 3 Ω (B)9Ω (3) 12 Ω (4)6 Ω

    9. Maximum power transfer theorem finds application in

    (a) Power circuits (b) distribution circuits

    (c) Communication circuits (d) both communication and power circuits

  • 15

    EXPERIMENT NO: 5 DATE:

    AIM : To verify Reciprocity Theorem.

    APPARATUS:

    (1) Regulated power supply (D.C) 0 - 30V 2

    (2) Board containing the network 1

    (3) Ammeters 0 - 250 mA 3

    (4) Voltmeter 0 - 30 V 1

    THEORY:

    The reciprocity theorem states that in a linear, bilateral, single source network the ratio of

    excitation to response is constant when the positions of excitation and response are

    interchanged.

    On the basis of mesh current analysis with a single voltage source acting in the network, the

    theorem may be demonstrated by considering the following equation for mesh current Ir.

    Ir = V1 (1r/z) + V2 (2r/z) +……….. + Vr (rr / z) + Vs (sr / z)

    Let the only source in the network be Vs then

    Ir = Vs (sr / z)

    The ratio of excitation to response is

    Vs / Ir = z / sr = Ztransfer sr ------------------------------(1)

    Now when the position excitation and response are interchanged the source becomes Vr and

    the current Is.

    Is = Vr(rs / z)

    The ratio of excitation to response is

    Vr / Is = z / rs = Ztransfer rs --------------------------(2)

    The two transfer impedances in (1) and (2) are equal in any linear, bilateral network since in

    such networks the impedance matrix [z] is symmetrical with respect to the principal diagonal,

    and the cofactors rs and sr are equal. Thus the current in mesh r which results from a voltage

    source in mesh s is the same as the current in mesh s when the voltage source is moves to mesh.

    It must be noted that currents in other parts of the network will not remain same.

  • 16

    The reciprocity theorem also applies to networks containing a single current source. Here the

    theorem states that the voltage which results at a pair of terminal m n due to a current source

    acting at terminals a b is the same as the voltage at terminals a b when the current source is

    moved at terminals m n. It should be noted that voltages at other points in the network would

    not remain the same.

    PROCEDURE:

    1. For the circuit shown in figure (4), calculate the values of current (I) for different values of

    source voltage and record them in the observation table.

    2. Connect the circuit as shown in figure (4), measure then values of current (I) (for source voltage

    of same values in step 1) and record them in the observation table.

    3. For the circuit shown in figure (5), calculate the values of current (I’) (for source voltage of

    same values as in step 1) and record them in the observation table.

    4. Connect the circuit as shown in figure (5), measure the values of current (I’)(for source voltage

    of same values as in step 1) and record them in the observation table.

    OBSERVATION TABLE:

    Sr No. Voltage

    (V)

    Current (I) A/mA Voltage

    (V)

    Current (I’) (A/mA)

    Exp. The. Exp. The.

    1.

    2.

    3.

    4.

    TELLEGEN`S THEOREM

    THEORY :

    Tellegen`s Theorem is one of the most general theorems in network theory. It applies to

    any network made up of lumped two terminal network elements, regardless of their nature,

    i.e., the elements may be linear or non-linear, passive or active, time invariant or time

    varying. The circuit may contains independent of dependent sources.

    If N1 and N2 are two different circuits, as shown in fig (4) and fig(5), having the same graph

    with the same reference directions assigned to the branches in the two circuits. Let vk and

    ik be the voltages and currents in N1 and vk` and ik` similarly be the voltages and currents

  • 17

    in N2, where all vk and vk` satisfy Kirchhoff`s Voltage Law (KVL) and all ik and ik` satisfy

    Kirchhoff`s Current Law (KCL). Then Tellegen`s theorem state that

    b b

    vk ik = 0 and vk` ik` = 0 k=1 k=1

    where b = No. of branches.

    PROCEDURE :

    (1) For the circuit shown in figure(4), measure the values of all the currents. ( i1, i2, i3, etc.)

    (2) Measure other voltages (v1, v2, v3 etc.) for the same value of v1.

    (3) Repeat steps 1 and 2 for the circuit shown in figure (5).

    OBSERVATION :

    i1 = v1 = i1` = v1` =

    i2 = v2 = i2` = v2` =

    i3 = v3 = i3` = v3` =

    i4 = v4 = i4` = v4` =

    CALCULATIONS:

    b

    vk ik = k=1

    b

    vk` ik` = k=1

    b

    vk ik` = k=1

    b

    vk` ik = k=1

  • 18

    CONCLUSION:

    QUIZ:

    1. Power delivered by the independent sources of the network must be equal the sum of power

    absorbed in all other branches of the network. True/False

    2. Verify Tellegen’s Thoerem considering two networks having identical graphs.

    3. Which theorem is a manifestation of Law of Conservation of Energy?

    4. To which networks is Reciprocity theorems applicable?

    5. Tellegen’s theorem can be applied to __________ networks.

    6. What is the use of Tellegen’s theorem?

    7. If the current in the 7 Ω resistor branch is 0.5A as shown in the figure and now if the source is

    connected in series with 7 Ω branch and the terminals AB are shorted, the current in the 5 Ω

    resistor is,

    (a) 1 A (b) 9.5 A (c) 9.75 A (d) none of the above

    8. Reciprocity theorem is applicable to

    (a) Any electric circuit (b) a linear network

    (c) a linear network and constant voltage source (d) a linear, bilateral network

  • 19

    EXPERIMENT NO: 6 DATE:

    AIM : (i) To determine z parameters of a given Two–Port Resistive

    Network.

    (ii) To determine the z – parameters of series connection of two 2-port

    resistive networks and verify the result by direct calculation.

    APPARATUS :

    (1) Ammeter 0-50mA 2

    (2) Voltmeter 0-10V 1

    (3) Regulated power supply 0 - 30V. 1

    (4) Board containing two port network 1

    THEORY:

    In electrical network theory a port may be regarded as a pair of terminals in which current

    in to one terminal equals the current out of the other. A network may have one, two or n

    ports in general. A one port network is completely identified when voltage current

    relationship at the terminals of the port is given.

    A general two port network shown in fig (1) has two pairs of voltage - current relationships.

    The V1 and I1 are the variables at port 1 and V2 and I2 are the variables at port 2. Only

    two of the four variables are independent and specifications of any two of them

    determine the remaining two. The dependence of two of the four variables on the other

    two is described in a number of ways, depending on which of the variables are chosen to

    be independent variables. As such there are six possible sets of equations describing a two

    port network, six different types of parameters are defined as z parameters, y parameters,

    transmission parameters, inverse transmission parameters, hybrid parameters and inverse

    hybrid parameters.

    Z - parameters:

    In case of z parameters, V1 and V2 are expressed in terms of I1 and I2.

    i.e. V1 = z11 I1 + z12 I2 - (1)

    V2 = z21 I1 + z22 I2 - (2)

    These parameters may be defined in terms of a single voltage and current by letting either

    I1 = 0 or I2 = 0.

  • 20

    Thus,

    V1

    z11 = ___

    I1 I2 = 0

    V1

    z12 = ___

    I2 I1 = 0

    V2

    z21 = ___

    I1 I2 = 0

    V2

    z22 = ____

    I2 I1 = 0

    It may be observed that (i) all the z parameters have the dimensions of impedance

    and (ii) they are specified only when the current in one of ports is zero i.e open

    circuit at port 1 or port 2. Hence z parameters are designated as open circuit

    impedance parameters.

    Z – PARAMETERS OF SERIES CONNECTION OF TWO 2-PORT

    RESISTIVE NETWORK:

    Two port network analysis is useful for finding different parameters. The z

    Parameters are useful in characterizing series connected two port networks. They

    are found under open circuit conditions and hence they are referred as open circuit

    impedance functions. They are defined and found as under:

    The z parameters are useful in characterizing series connected two port networks.

    The overall z parameters from the individual z parameters can be found as under

    when the networks are connected in series.

    For network Na

    V1a

    =

    z11a z12a I1a

    V2a z21a z22a I2a ---------(1)

    For network Nb

    V1b

    =

    z11b z12b I1b

    V2b z21b z22b I2b ---------(2)

  • 21

    For overall network N

    V1

    =

    z11 z12 I1

    V2 z21 z22 I2 ---------(3)

    Note that

    I1 = I1a = I1b and V1 = V1a + V1b

    I2 = I2a = I2b and V2 = V2a + V2b ------(4)

    Combining equation (1), (2) and (4), we get

    V1

    =

    Z11a+z11b z12a +z12b I1

    V2 Z21a+z21b z22a +z22b I2 ---------(5)

    Comparing equation (5) with equation (3), we get

    z11 = z11a + z11b

    z12 = z12a + z12b

    z21 = z21a + z21b

    z22 = z22a + z22b ----------------(6)

    This result may be generalized for any number of networks connected i

    n series. The individual parameters are added to determine the overall Z

    parameters.

    PROCEDURE :

    1. Connect the circuit for Network Na as shown in fig (1).

    2. Apply voltage at port 1 keeping port 2 open circuited as shown in

    fig (4). Measure voltages and current at the port terminals. Keep levels of

    voltages and current such that meter readings are not exceeded their ranges

    and ratings of the resistances.

    3. Apply voltage at port 2 keeping port 1 open circuited as shown in

    fig (5). Measure voltages and current at the port terminals.

    4. Calculate z parameters using measured values of voltages and currents and

    verify the results theoretically.

    5. Connect the circuit as shown in fig (2) for network Nb only. Repeat steps 2 to 4 for

    Network Nb

    6. Connect the networks Na and Nb in series as shown in fig(3) to form the

    overall network N.

  • 22

    (7) Repeat steps 2 to 4 to find the z - parameters of network N

    and Verify the results theoretically.

    OBSERVATION TABLE:

    (1) Network : Na

    SR.

    NO.

    V1a

    Volts

    I1a

    mA

    V2a

    Volts

    I2a

    mA

    REMARK

    1 0 Port - 2 open circuited

    2 0 Port - 1 open circuited

    (2) Network : Nb

    SR.

    NO.

    V1b

    Volts

    I1b

    mA

    V2b

    Volts

    I2b

    mA

    REMARK

    1 0 Port - 2 open circuited

    2 0 Port - 1 open circuited

    (3) Network N :

    SR.

    NO.

    V1

    Volts

    I1

    mA

    V2

    Volts

    I2

    mA

    REMARK

    1 0 Port - 2 open circuited

    2 0 Port - 1 open circuited

    CALCULATION:

    For Network Na :

    V1a

    z11a = ___ = _________________________________

    I1a I2a = 0

    V1a

    z12a = ___ = _________________________________

    I2 a I1a = 0

    V2a

    z21a = ___ = ________________________________

    I1 a I2a= 0

  • 23

    V2a

    z22a = ___ = ________________________________

    I2a I1a = 0

    For Network Nb :

    V1b

    z11b = ___ = _________________________________

    I1b I2b = 0

    V1b

    z12b = ___ = _________________________________

    I2b I1b = 0

    V2b

    z21b = ___ = ________________________________

    I1b I2b = 0

    V2b

    z22b = ___ = ________________________________

    I2b I1b = 0

    For Network N:

    V1

    z11 = ___ = _________________________________

    I1 I2 = 0

    V1

    z12 = ___ = _________________________________

    I2 I1 = 0

    V2

    z21 = ___ = ________________________________

    I1 I2 = 0

  • 24

    V2

    z22 = ___ = ________________________________

    I2 I1 = 0

    Check:

    (1) z11 = z11a + z11b = ____________ = ____________

    (2) z12 = z12a + z12b = ____________ = ____________

    (3) z21 = z21a + z21b =____________ = ____________

    (4) z22 = z22a + z22b = ____________ = ____________

    RESULT TABLE:

    NETWORK Practical Theoretical

    Network Na

    z11a =______ z21a = ______

    z11a =______ z21a = ______

    z11a =______ z21a = ______

    z11a =______ z21a = ______

    Network Nb

    z11b =______ z21b = ______

    z11b =______ z21b = ______

    z11b =______ z21b = ______

    z11b =______ z21b = ______

    Network N

    z11 =______ z21 = ______

    z11 =______ z21 = ______

    z11 =______ z21 = ______

    z11 =______ z21 = ______

    CONCLUSION: -

  • 25

    QUIZ: -

    1. What do you mean by two port network?

    2. Z parameters are known as _________ circuit parameters.

    3. The Z11 and Z22 parameters of the given network are,

    4. Why two networks are connected in series to get overall z parameters? Discuss by

    taking detailed example.

    5. For two networks connected in series if z21 a = 4 Ω and z21b = 6 Ω, what will be the

    value of z21 ?

    6. The equivalent circuit of a two-port reciprocal network using z-parameters is shown here

    The z-parameters are (z11,z12,z21,z22)

    (a) 10,5,5,15 (b) 15,5,5,20 (c) 5,5,5,10 (d) 10,10,5,15

    7. When a number of 2-ports networks are connected in cascade, the individual

    (a) Zoc matrices are added (b) Ysc matrices are added

    (c) Chain matrices are multiplied (d) h-matrices are multiplied

  • 26

    EXPERIMENT NO: 7 DATE:

    AIM : (i) To determine y parameters of a given Two–Port Resistive

    Network.

    (ii) To determine the y – parameters of the parallel connection of two

    2-port resistive networks and verify the result by direct

    calculation.

    APPARATUS :

    (1) Ammeter 0-50mA 2

    (2) Voltmeter 0-10V 1

    (3) Regulated power supply 0 - 30V. 1

    (4) Board containing two port network 1

    THEORY:

    y parameters :

    In case of y parameters, I1 and I2 are expressed in terms of V1 and V2

    i.e I1 = y11 V1 + y12 V2

    I2 = y21 V1 + y22 V2

    The individual y parameters are defined by

    I1

    y11 = ___

    V1 V2 = 0

    I1

    y12 = ___

    V2 V1= 0

    I2

    y21 = ____

    V1 V2 = 0

    I2

    y22 = ___

    V2 V1 = 0

    It may be observed that

    (i) All the y-parameters have the dimensions of admittance.

    (ii) They are specified only when voltage at one of the ports is zero i.e. short circuit at port

    1 or port 2. Hence y parameters are known as short circuit admittance parameters.

  • 27

    Y – PARAMETERS OF PARALLEL CONNECTION OF TWO 2- PORT RESISTIVE

    NETWORK.

    The y - parameters (short - circuit admittance parameters) are useful in characterizing

    parallel connected two - port networks.

    They are found under short circuit conditions and hence they are referred as short circuit

    admittance parameters.

    The y-parameters are useful in characterizing parallel connected two port networks.

    The overall y parameters from the individually parameters can be found as under

    when the networks are connected in parallel.

    For network Na

    I1a

    y11a y12a V1a

    I2a

    =

    y21a y22a

    V2a

    ---------(1)

    For network Nb

    I1b

    y11b y12b V1b

    I2b

    =

    y21b y22b

    V2b

    ---------(2)

    For overall network N

    I1

    y11 y12 V1

    I2

    =

    y21 y22

    V2

    ---------(3)

    Note that

    V1 = V1a = V1b and I1 = I1a + I1b

    V2 = V2a = V2b and I2 = I2a + I2b ------(4)

    Combining equation (1), (2) and (4), we get

    I1

    y11a+y11b y12a +y12b V1

    I2

    =

    y21a+y21b y22a +y22b

    V2

    ---------(5)

  • 28

    Comparing equation (5) with equation (3), we get

    y11 = y11a + y11b

    y12 = y12a + y12b

    y21 = y21a + y21b

    y22 = y22a + y22b ----------------(6)

    This result may be generalized for any number of networks connected in parallel.

    The individual

    short circuit admittance parameters are added to determine the overall Y parameters.

    PROCEDURE:

    (1) Connect the circuit diagram of Network Na as shown in fig(1).

    (2) Apply voltage at port 1 short circuiting the port 2 through an ammeter as

    shone in fig (4). Measure voltage and currents at both the port terminals.

    (3) Apply voltage at port 2 short circuiting the port 1 through an ammeter as

    shown in fig (5). Measure voltage and currents at both the ports.

    (4) Calculate y parameters using measured values of voltage and currents and

    verify the results theoretically.

    (5) Connect the circuit as shown in fig (2) for network Nb only. Repeat steps 2 to

    4 for Network Nb.

    (6) Connect the networks Na and Nb in parallel as shown in fig(3) to for

    m network N and repeat steps 2 to 4 for Network N.

    find its y - parameters. Verify the results theoretically.

    OBSERVATION TABLE:

    (1) Network: Na

    SR.

    NO.

    V1a

    VOLTS

    I1a

    mA

    V2a

    VOLTS

    I2a

    mA

    REMARK

    1 0 Port - 2 short circuited

    2 0 Port - 1 short circuited

    (2) Network: Nb

    SR.

    NO.

    V1b

    VOLTS

    I1b

    mA

    V2b

    VOLTS

    I2b

    mA

    REMARK

    1 0 Port - 2 short circuited

    2 0 Port - 1 short circuited

  • 29

    (3) Network N:

    SR.

    NO.

    V1

    VOLTS

    I1

    mA

    V2

    VOLTS

    I2

    mA

    REMARK

    1 0 Port - 2 short circuited

    2 0 Port - 1 short circuited

    CALCULATION:

    For Network Na:

    I1a

    y11a = ___ = ________________________________

    V1a V2a = 0

    I1a

    y12a = ___ = ________________________________

    V2a V1a = 0

    I2a

    y21a = ___ = _________________________________

    V1a V2a = 0

    I2a

    y22a = ___ = ________________________________

    V2a V1a = 0

    For Network Nb :

    I1b

    y11b = __ = _________________________________

    V1b V2b = 0

    I1b

    y12b = ___ = _________________________________

    V2b V1b = 0

    I2b

    y21b = ___ = _________________________________

    V1b V2b = 0

  • 30

    I2b

    y22b = ___ = ________________________________

    V2b V1b = 0

    For Network N :

    I1

    y11 = ___ = ________________________________

    V1 V2 = 0

    I1

    y12 = ___ = ________________________________

    V2 V1 = 0

    I2

    y21 = ___ = _________________________________

    V1 V2 = 0

    I2

    y22 = ___ = ________________________________

    V2 V1 = 0

    Check :

    (1) y11 = y11a + y11b = ____________ = ____________

    (2) y12 = y12a + y12b = ____________ = ____________

    (3) y21 = y21a + y21b =____________ = ____________

    (4) y22 = y22a + y22b = ____________ = ____________

    RESULT TABLE:

    NETWORK Practical Theoretical

    Network Na

    y11a =______ y21a = ______

    y11a =______ y21a = ______

    y11a =______ y21a = ______

    y11a =______ y21a = ______

  • 31

    Network Nb

    y11b =______ y21b = ______

    y11b =______ y21b = ______

    y11b =______ y21b = ______

    y11b =______ y21b = ______

    Network N

    y11 =______ y21 = ______

    y11 =______ y21 = ______

    y11 =______ y21 = ______

    y11 =______ y21 = ______

    CONCLUSION: -

    QUIZ: -

    1. y parameters are also known as _______ circuit parameters.

    2. If for any two port passive network y12 is 0.4 mho, y21 = ______.

    3. If two networks Na and Nb are connected in parallel y11a = 3 mho and y11b = 4 mho

    what will be the value of y11 = ______.

    4. For the port network shown, select the correct statement

    (a)It does not have z-parameters (b)It has z-parameters

    (c)It does not have y-parameters (d)It does not have ABCD parameters

  • 32

    EXPERIMENT NO: 8 DATE:

    AIM : (i) To determine ABCD parameters of a given two–port resistive

    network.

    (ii) To determine the ABCD parameters of the cascade connection of two 2-

    port resistive networks and verify the result by direct calculation.

    APPARATUS:

    (1) Network board

    (2) Ammeters 0 - 50mA 2

    (3) Voltmeter 0 - 10V 1

    (4) Regulated power supply 0-30 V 1

    THEORY :

    The transmission parameters serve to relate the voltage and current at one port to voltage and

    current at the other port. In equation form,

    V1 = AV2 - BI2

    I1 = CV2 - DI2

    where A, B, C and D are the transmission parameters. They are also known as chain

    parameters, the ABCD parameters and general circuit parameters. Their first use is in the

    analysis of power transmission lines. From the circuit conditions, they can be found as

    follows,

    V1

    A = _____

    V2 I1=0

    V1

    -B = _____

    I2 V2=0

    I1

    C = _____

    V2 I2=0

    I1

    -D = _____

    I2 V2=0

  • 33

    ABCD PARAMETERS OF CASCADE CONNECTION OF TWO 2-PORT RESISTIVE

    NETWORK.

    The transmission parameters are useful in describing two port networks which are

    connected in cascade or in a chain arrangement. The overall parameters from the

    individual parameters can be found as under when the networks are connected in cascade.

    For network Na

    V1a

    Aa Ba V2a

    I1a

    =

    Ca Da

    -I2a

    ---------(1)

    For network Nb

    V1b

    Ab Bb V2b

    I1b

    =

    Cb Db

    -I2b

    ---------(2)

    For overall network N

    V1

    A B V2

    I1

    =

    C D

    -I2

    ---------(3)

    Note that

    V1a = V1 V2a = V1b I2b = I2

    I1a = I1 I1b = - I2a V2b = V2 ------(4)

    Substituting these in equation (1) and equation (2), we get

    V1

    Aa Ba Ab Bb V2

    I1

    =

    Ca Da

    Cb Db

    -I2

    ---------(5)

    Comparing equation (5) with equation (3), we get

    A B

    Aa Ba

    Ab Bb

    AaAb+BaCb AaBb + BaDb

    C D

    =

    Ca Da

    Cb Db

    =

    CaAb + DaCb CaBb+ DaDb

    ------(6)

  • 34

    PROCEDURE:

    (1) Connect circuit diagram of Network Na as shown in fig (1).

    (2) Apply voltage at port 1 of network Na short circuiting the port 2 through an

    ammeter as shown in fig (4). Measure voltages and currents at both the ports.

    (3) Apply voltage at port 1 of network Na keeping port 2 open circuited as shown

    in fig (5). Measure voltages and currents at both the ports.

    (4) Calculate ABCD parameters using measured values of voltages and currents.

    (5) Connect the circuit as shown in fig (2) for network Nb only. Repeat steps 2 to 4

    for network Nb.

    (6) Connect both the networks in cascade as shown in fig (3). This forms network

    N.

    (7) To measure parameters of network N follow the steps 2 to 4.

    (8) Verify the parameters theoretically and tabulate the results.

    (9) For each network verify that AD - BC = 1.

    OBSERVATION TABLE:

    (1) Network: Na

    SR.

    NO.

    V1a

    Volts

    I1a

    mA

    V2a

    Volts

    I2a

    mA

    REMARK

    1 0 Port - 2 short circuited

    2 0 Port - 2 open circuited

    (2) Network : Nb

    SR.

    NO.

    V1b

    Volts

    I1b

    mA

    V2b

    Volts

    I2b

    mA

    REMARK

    1 0 Port - 2 short circuited

    2 0 Port - 2 open circuited

    (3) Network N :

    SR.

    NO.

    V1

    Volts

    I1

    mA

    V2

    Volts

    I2

    mA

    REMARK

    1 0 Port - 2 short circuited

    2 0 Port - 2 open circuited

  • 35

    CALCULATION:

    For Network Na:

    V1a

    Aa = ___ = _________________________________

    V2a I2a = 0

    V1a

    Ba = ___ = _________________________________

    -I2a V2a = 0

    I1a

    Ca = ___ = _______________________________

    V2a I2a = 0

    I1a

    Da = ___ = _______________________________

    -I2a V2a = 0

    For Network Nb:

    V1b

    Ab = ___ = _________________________________

    V2b I2b = 0

    V1b

    Bb = ___ = _________________________________

    -I2b V2b = 0

    I1b

    Cb = ___ = _______________________________

    V2b I2b = 0

    I1b

    Db = ___ = ________________________________

    -I2b V2b = 0

  • 36

    For Network N:

    V1

    A = ___ = _________________________________

    V2 I2 = 0

    V1

    B = ___ = _________________________________

    -I2 V2 = 0

    I1

    C = ___ = ________________________________

    V2 I2 = 0

    I1

    D = ___ = ________________________________

    -I2 V2 = 0

    Check:

    A = AaAb+BaCb =_________________

    B = AaBb + BaDb =_________________

    C = CaAb + DaCb =_________________

    D = CaBb+ DaDb =_________________

  • 37

    RESULT TABLE:

    NETWORK - Na NETWORK - Nb NETWORK - N

    Pract. Theo. Pract. Theo. Pract. Theo.

    Aa Ab A

    Ba Bb B

    Ca Cb C

    Da Db D

    CONCLUSION:

    QUIZ :

    1. ABCD parameters are also known as ___________ or _________ parameters.

    2. Why two networks are connected in cascade connection to get overall ABCD parameter?

    3. If A= 7 , B= 8 ohm and C = 2.5 mho , what will be the value of D?

    4. State the conditions for a network to be loss less in terms of ABCD parameters?

    5. State the condition for a network to be reciprocal and symmetrical.

    6. For _________ connection of two 2-port networks, ABCD parameters have to be

    multiplied.

    7. Are the ABCD parameters A(s), B(s),C(s) and D(s) the network functions?

    8. The relation AD – BC = 1 is valid for ________ and _________ networks.

    9. Why negative sign is introduced in the equations?

  • 38

    EXPERIMENT NO: 9 DATE:

    AIM : (i) To determine h - parameter of a given Two–Port Resistive

    Network.

    (ii) To determine g - parameter of a given Two–Port Resistive

    Network.

    APPARATUS:

    (1) Network board

    (2) Ammeters 0 - 50mA 2

    (3) Voltmeter 0 - 10V 1

    (4) Regulated power supply 0-30 V 1

    THEORY :

    Hybrid Parameters (h parameters)

    h parameters representation is widely used in modeling of electronic components

    and circuits, particularly transistors. As both short circuit and open circuit

    terminal conditions are utilized hence, this parameter representation is known as

    hybrid parameter representation. In this form of representation, the voltage of the

    input poet and the current of the output port are expressed in terms of the current

    of the input poet and the voltage of the output port.

    We know that

    V1 = h11I1 + h12V2

    I2 = h21I1 + h22V2

    In matrix form

    V1

    h11 h12 I1

    I2

    =

    h21 h22

    V2

    ---------(1)

    Where

    V1

    h11 = ___ = Input impedance when output is short circuited

    I1 V2 = 0

    V1

    h12 = ___ = Reverse voltage ratio when input open circuited

    V2 I1 = 0

    I2

  • 39

    h21 = ___ = Forward current ratio when output short circuited

    I1 V2 = 0

    I2

    h22 = ___ = Output admittance when input is open circuited

    V2 I1 = 0

    Inverse Hybrid Parameters (g parameters)

    Hybrid parameters (h parameters) and Inverse hybrid parameters (g parameters) are dual of

    each other. For g parameters both short circuit and open circuit terminal conditions are utilized.

    In this form of representation, the current of the input port and the voltage of the output port

    are expressed in terms of the voltage of the input port and the current of the output port.

    In case of g parameters, I1 and V2 are expressed in terms of V1 and I2.

    i.e. I1 = g11 V1 + g12 I2 - (1)

    V2 = g21 V1 + g22 I2 - (2)

    I1

    g11 g12 V1

    V2

    =

    g21 g22

    I2

    ---------(1)

    Where

    I1

    g11 = ___ = Input admittance when output is open circuited

    V1 I2 = 0

    I1

    g12 = ___ = Reverse current ratio when input short circuited

    I2 V1 = 0

    V2

    g21 = ___ = Forward voltage ratio when output open circuited

    V1 I2 = 0

    V2

  • 40

    g22 = ___ = Output impedance when input is short circuited

    I2 V1 = 0

    PROCEDURE :

    Hybrid Parameters

    (1) Connect the circuit diagram of network Na as shown in fig (1).

    (2) Apply voltage at port 1 keeping port 2 short-circuited. Measure voltages and

    current at the port terminals as shown in fig (4). Keep levels of voltages and

    current such that meter readings are not exceeded their ranges and ratings of

    the resistances.

    (3) Apply voltage at port 2 keeping port 1 open circuited as shown in fig (5).

    Measure voltages and current at the port terminals.

    (4) Calculate h parameters using measured values of voltages

    and currents and verify the results theoretically

    Inverse Hybrid Parameters

    (1) Connect the circuit diagram of network Na as shown in fig (1).

    (2) Open the output port and excite the input port with a known voltage source Vs

    as shown in fig (4) so that V1 = Vs and I2 = 0.

    (3) Determine I1 and V2 to obtain g11 and g21.

    (4) Then the input port is short circuited and output port is excited with the same

    voltage source Vs as shown in fig (5) so that V2 = Vs and V1 = 0.

    (5) Determine I1 and I2 to obtain g12 and g22.

    OBSERVATION TABLE:

    Hybrid parameters

    SR.

    NO.

    V1a

    Volts

    I1a

    mA

    V2a

    Volts

    I2a

    mA

    REMARK

    1 0 Port - 2 short circuited

    2 0 Port - 1 open circuited

    Inverse Hybrid Parameters

    SR.

    NO.

    V1a

    Volts

    I1a

    mA

    V2a

    Volts

    I2a

    mA

    REMARK

    1 0 Port - 2 open circuited

    2 0 Port - 1 short circuited

  • 41

    CALCULATION:

    Hybrid Parameters :

    V1a

    h11a = ___ =

    I1a V2a = 0

    V1a

    h12a = ___ =

    V2a I1 = 0

    I2a

    h21a = ___ =

    I1a V2a = 0

    I2a

    h22a = ___ =

    V2a I1a= 0

    Inverse Hybrid Parameters:

    I1a

    g11a = ___ =

    V1a I2a = 0

    I1a

    g12a = ___ =

    I2a V1a = 0

    V2a

    g21a = ___ =

    V1a I2a = 0

  • 42

    V2a

    g22a = ___ =

    I2a V1 = 0

    CONCLUSION: -

    QUIZ: -

    1. For _________ connection of 2 networks h-parameters have to be added.

    2. Will the h parameter matrix of a passive network be a symmetrical?

    3. If two networks Na and Nb are connected in series parallel, h11a =3 and h11b = 4 what will

    be the value of h11?

    4. For network shown, the parameters h11 and h21 are

    5. In a two-port network. the condition for reciprocal in terms of ‘h’ parameters is,

    (a) h12=h21 (b) h11= h22 (c) h11= -h22(d) h12=-h21

    6. The ideal transformer cannot be described by

    (a) h parameter (b) ABCD parameter

    (c) g parameter (d) z parameter

    7. For a symmetrical network

    (a)h11 = h22 (b)h12 = h21

    (c)h11 h22- h12 h21 = 0 (d)h11 h22- h12 h21 = 1

    8. For a single element two port network of the given figure , h21 is

    9. If a two port is reciprocal, which of the following is not true?

    (a) z21 =z12 (b) y21 = y12

    (c) h21 =h12 (d) AD = BC +1

  • 43

    EXPERIMENT NO: 10 DATE:

    AIM : To study the response in R-L-C series circuit and determine various time

    response specifications.

    APPARATUS:

    ( i ) Decade resistance

    (ii ) Decade capacitance

    (iii ) Decade inductance

    (iv) C.R.O.

    ( v ) Square wave generator.

    THEORY : The behaviour of a circuit or system which contains two independent energy

    storing elements is completely determined by a second order differential

    equation.

    For the circuit in fig (1) writing equation by applying KVL we get,

    L di/dt + Ri + 1/c I dt = V

    On differentiation,

    L d2i / dt2 + R di/dt + 1/c i = 0

    d2i / dt2 + R/L di / dt + 1/LC i = 0

    Roots of characteristics equation are given by

    s1 , s2 = - R/2L + (R/2L)2 -(1/LC)

    On this basis, we can predict the nature of response for three conditions.

    (i ) (R/2L)2 > 1/LC , the response is over damped. Roots are real negative and

    distinct.

    (ii ) (R/2L)2 = 1/LC , roots are equal and negative repeated roots.

    The response is critically damped, the values of resistance to achieve this is

    called critical resistance, Rcr.

    Hence

    (Rcr/ 2L )2 = 1/LC

    which give Rcr = 2 L / C

    (iii) (R/2L)2 < 1/LC , we get under damped response. Roots are complex

    conjugate. Resistance R is less than RCR We define damping ratio = R /

    RCR = actual damping / critical damping

  • 44

    (iv) Natural frequency of oscillation is given by wn = 1 / LC

    It yields information about settling time ( ts ) i.e. the time to reach steady state

    value. The larger the wn the smaller the value will be the settling time.

    For under-damped system damping frequency of oscillation is given

    wd = wn (1- 2 ) rad / sec

    Percentage overshoot is determined the ratio of maximum overshoot above final

    steady level of square wave to find steady wave to final steady state level.

    PROCEDURE:

    (1) Connect the given circuit as shown in fig(1) and supply a square voltage of

    any magnitudes.

    (2) Starting from zero, increase the value of R to adjust the response to a

    critical damped case. At critical resistance, C.R.O will show a good square

    wave. Note down the value of RCR and compare it with theoretical values

    (given RCR =2 L/C )

    (3) Adjust the resistance R < RCR and get under-damped response on C.R.O.

    = R / RCR = is known as damping ratio.

    (4) Find the damped frequency of oscillation Wd = 2 / Td is obtained from

    C.R.O. shown in fig. where Td is on obtained from C.R.O as shown in

    fig(2) compare with the theoretical

    (5) Find percentage peak overshoot from C.R.O as shown in fig (2)

    Peak overshoot :- a/b X 100

    Compare it with theoretical value given by

    peak overshoot = e-- / (1 -- 2) x 100

    (6) Repeat above steps for another value of R in order to get under-damped

    response.

    L = _________

    C = _________

  • 45

    CALCULATIONS :

    RCR = 2 L/C

    wd = 2 / Td

    where Td is obtaind from C.R.O. as shown in fig (2).

    peak overshoot = a/b x 100 %

    where a and b is obtaind from C.R.O. as shown in fig (2).

    RESULT TABLE:

    QUANTITY THEORITICAL PRACTICAL

    RCR

    wd

    peak

    overshoot

    CONCLUSION: -

    QUIZ: -

    1. A load is modeled as a 250 mH inductor in parallel with a 12 resistor. A capacitor is

    needed to be connected to the load so that the network is critically damped at 60 Hz.

    Calculate the size of capacitor.

    2. Due to which factor, transient current is produced?

    3. A two-terminal black box contains one of the R,L,C elements. The black box is connected to a 220V a.c. supply. The current through the source is I. When a capacitance

    of 0.1 F is inserted in series between the source and box, the current through the source is

    2I. the element is

    (a) a resistance (b) an inductance

    (c) a capacitance of 0.5F (d) not readily identifiable from the given data

  • 46

    4. The open-circuit voltage ratio V2(s)/V1(s) of the network shown in the given figure is

    (a) 1+ 2s2 (b) 1/1+ 2s2 (c) 1+ 2s (d)1/1+ 2s

    5. Refer to the RLC circuit in the figure below, what kind of response will it produce?

    6. Consider parallel RLC circuit in the figure below, what type of response will it produce?

    7. In a series RLC circuit, setting R = 0 will produce

    (a) an overdamped response (b) a critically damped response

    (c) an underdamped response (d) an undamped response

    8. Why does capacitor act open circuit with DC?

    9. A RL circuit has R= 2 Ω and L = 4 H. The time needed for the inductor current to reach 40

    % of its steady state value is ___________.

  • 47

    EXPERIMENT NO: 11 DATE:

    AIM : To study the step response of first order R-C circuit and cascaded R-C

    sections

    APPARATUS :

    ( i ) Function generator

    (ii ) C.R.O.

    (iii ) 680 Ω resistor

    THEORY : The simplest RC circuit is a capacitor and a resistor in series. When a circuit

    consists of only a charged capacitor and a resistor, the capacitor will discharge its

    stored energy through the resistor. The voltage across the capacitor, which is time

    dependent, can be found by using Kirchhoff's current law, where the current

    through the capacitor must equal the current through the resistor. This results in the

    linear differential equation.

    CdV/dt + V/R = 0

    Solving this equation for V yields the formula for exponential decay:

    V (t) = Vo e -t/RC

    Where Vo is the capacitor voltage at time t = 0.

    The time required for the voltage to fall to is called the RC time constant and is

    given by

    τ = RC

    The time required for the voltage to fall to is called the RC time constant and is

    given by RC circuits are frequently used to model the timing characteristics of

    computer systems. When one logic gate drives another gate, the input circuit of the

    second gate can be modeled as an RC load. The propagation delay through the first

    gate can then be calculated assuming ideal square wave input and the RC load. The

    longer the delay time, the slower the circuit can be switched and the slower the

    computer is. Conversely, the shorter the delay time, the faster the computer is. This

    delay time is called “gate delay” since it relates to driving characteristics of a logic

    gate. Another use of RC circuits is to model wiring characteristics of bus lines on

  • 48

    integrated circuits (IC) or on printer-circuit boards (PCB). A wire can be modeled

    as many cascaded sections of simple RC circuits as shown in Figure 3 using 2

    sections. When a square wave is applied to one end of the bus, it takes time for the

    signal to propagate to the other end. This delay time due to the wire can be

    calculated based on the values of R and C in each section and the number of sections

    used to model the wire. The longer the wire, the more sections are needed for

    accurate model. A wire is also referred to as “interconnect” and the delay due to a

    wire is also called “interconnect delay.” In high-frequency systems, the

    interconnect delay tends to dominate the gate delay and is a fundamental constraint

    on how fast a computer can operate.

    PROCEDURE :

    (1) Build the circuit in Figure 2 using R = 10 Kand C = 0.01 F. Set the function

    generator to provide a square wave input as follows:

    a) Period T 3 ms (to ensure that T >> RC). This value of T guarantees that the

    output signal has sufficient time to reach a final value before the next input

    transition. Record the value of T. b) Set amplitude from 0 V to 5 V. Note that you

    need to set the offset to achieve this waveform. Use the oscilloscope to display this

    waveform on Channel 1 to make sure the amplitude is correct. We use this

    amplitude since it is common in computer systems.

    (2) Use Channel 2 of the oscilloscope to display the output signal waveform.

    Adjust the timebase to display 2 complete cycles of the signals. Record the

    maximum and the minimum values of the output signal.

    (3) Use the measurement capability of the scope to measure the period T of the

    input signal, the time value of the 10%-point of Vout, the time value of the 90%-

    point of Vout, and the time value of the 50%-point of Vout.

    (4) Clear all the measurements. Use the paired measurement capability of the

    scope to measure the voltage and time values at 10 points on the Vout waveform

    during one interval when Vout rises or falls with time (pick one interval only). Note

    that the time values should be referred to time t = 0 at the point where the input

    signal rises from 0 V to 5 V or falls from 5 V to 0 V. Record these 10 measurements.

  • 49

    For R-C cascaded sections:-

    (1) Build the circuit in Figure 3, using 2 identical resistors R = 10 Kand two

    identical capacitors C = 0.01 F. Use the same square input as above and display it

    on Channel 1.

    (2) Display Vout on Channel 2 and adjust the timebase to display 2 complete cycles

    of the signals.

    (3) Use the scope measurement capability to measure the two delay times tPHL and

    tPLH between the input and output signals.

    CALCULATIONS:

    From the equation for Vout and using the amplitude of Vs as 500 mV, compute the

    amplitude of Vout for both cases R1 = 50 and R1 = 27 K.

    RESULT TABLE:

    QUANTITY THEORITICAL PRACTICAL

    t(rise)

    t(fall)

    Vout for R = 50 Ω

    Vout for R = 27 kΩ

  • 50

    CONCLUSION: -

    QUIZ: -

    1. If you use a different input signal (e.g. a ramp waveform from the function generator) as

    source input to a R-C circuit, draw the response waveform look like on the scope.

    2. List out the applications of R-C cascaded networks.

    3. R-C model for a wire is good below which frequency?

    4. What difference is observed in the response a single R-C circuit and cascaded R-C

    sections?

    5. What would be the step response of R-L series circuit?

  • 51

    EXPERIMENT NO: 12 DATE:

    AIM : To design and test a passive constant-k High Pass Filter and measure its cut-off

    frequency

    APPARATUS :

    ( i ) Function generator

    (ii ) C.R.O.

    (iii) R = 680 Ω

    THEORY : The simple first-order electronic high-pass filter is implemented by placing an

    input voltage across the series combination of a capacitor and a resistor and using

    the voltage across the resistor as an output. The product of the resistance and

    capacitance (R×C) is the time constant (τ); it is inversely proportional to the cutoff

    frequency fc, at which the output power is half the input power. That is,

    fc = 1/2πRC

    PROCEDURE :

    (1) Connect the circuit of high pass filter using R and C components.

    (2) Set the input voltage, Vi = 5V using signal generator and vary the frequency

    from (0-1 MHz) in regular steps.

    (3) Note down the corresponding output voltage.

    (4) Plot the graph of output voltage v/s frequency.

    CALCULATIONS :

    Calculate cutoff frequency of this HPF.

  • 52

    OBSERVATION TABLE:

    Frequency (Hz) Output voltage (volts)

    RESULT TABLE :

    Cutoff frequency

    (Hz)

    THEORITICAL

    PRACTICAL

    CONCLUSION: -

    QUIZ: -

    1. What would be the response of low pass filter?

    2. How band pass filters differ from low pass and high pass filters?

    3. State the difference between active and passive filters.

    4. What are the demerits of constant k-filters?

    5. Define a time constant of a circuit.