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Reciprocity Theorem

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    RECIPROCITY THEOREM

    PREPARED BY:

    Namrata V. L.

    Assistant Professor

    Government Engineering College, Rajkot

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    RECIPROCITY THEOREM

    The reciprocity theorem is most powerful

    theorem in circuit and field theories both.

    The original theorem is due to Rayleigh

    Helmholtz which was generalized, to include

    continuous media, by J.R. Carson. Thatswhy

    it is also known as Rayleigh reciprocity

    theorem.

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    RECIPROCITY THEOREM cont.

    STATEMENT:

    If an emf is applied to the terminals of an antenna no. 1 andthe current measured at the terminals of another antenna no.2, then an equal current both in amplitude and phase will be

    obtained at the terminals of antenna no. 1 if the same emf isapplied to the terminals of antenna no. 2.

    OR

    If a current I, at the terminals of antenna no. 1 induces anemf E21at the open terminals of antenna no. 2 and a current

    I2at the terminals of antenna no. 2 induces an emf of E12atthe open terminals of antenna no. 1, then E12=E21providedI1=I2.

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    RECIPROCITY THEOREM cont.

    ASSUMPTIONS:

    Emf sare of same frequency.

    Medium between the two antennas are linear, passive

    and isotropic. Generator producing emf and the ammeter for measuring

    the current have zero impedance of if not, then both the

    generator and the ammeter impedances are equal.

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    RECIPROCITY THEOREM cont.

    1. A transmitter of frequency f and zero impedance be

    connected to the terminals of antenna no. 2 which is

    generating a current I2 and inducing an emf E12 at the open

    terminals of antenna no. 1 .

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    RECIPROCITY THEOREM cont.

    1. Now the same transmitter is transferred to antenna no. 1

    which is generating a current I1and inducing a voltage E21 at

    the open terminals of antenna no. 2

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    RECIPROCITY THEOREM cont.

    According to the statement of reciprocity theorem,

    The ratio of voltage of one circuit (E1) to the current (I2)in

    the second circuit is defined as the transfer impedance.

    Thus from reciprocity theorem two impedances are equal,

    or

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    RECIPROCITY THEOREM cont.

    Limitations:

    1. Although the Rayleigh-Carson theorem is applicable to radio

    communication but it fails to be true, only when the

    propagation of the radio wave is appreciably effected by the

    presence of the Earthsmagnetic field.

    2. It holds good for all practical radio work but for long

    distance communication through ionosphere. However, still it

    is expected to apply results averaged over a reasonable

    interval of time in which case it cannot be expected to beexactly correct at every given time.

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    APPLICATION OF RECIPROCITY THEOREM

    Reciprocity theorem may be used to derive the folllowing

    very important properties of transmitting and receiving

    antennas.

    1. Equality of directional patterns.

    2. Equality of directivities.

    3. Equality of effective lengths.

    4. Equality of antenna impedances.

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    APPLICATION OF RECIPROCITY THEOREM

    cont.Equality of effective lengths:

    For transmitting antenna:

    1. That length of an equivalent linear antenna

    that has a current I(c) at all points along its

    length and that radiates the same electric

    field strength as the actual antenna in thedirection perpendicular to its length. OR

    2. The moment of transmitting antenna current

    distribution divided by the input current

    where the moment of the current

    distribution is defined as the sum of the

    moments of its current elements.

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    APPLICATION OF RECIPROCITY THEOREM

    cont.Equality of effective lengths:

    Symbolically for transmitting antenna:

    If an emf E is applied at the centre point C, then currents I(c) at the centrepoint C, and I(z) at any point will be produce along the antenna. The

    value of current at the centre point C is given by

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    APPLICATION OF RECIPROCITY THEOREM

    cont.Equality of effective lengths:

    For Receiving antenna: Effective length of receiving antenna may be defined as the ration of open

    circuit voltage developed at the terminals of antenna and the received

    field strength i.e.

    For receiving case tml no. 1 is short circuited.

    Applying the reciprocity theorem notations, the voltage E21 is applied at

    the tml no.2 by shorting the tml no.1, then

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    APPLICATION OF RECIPROCITY THEOREM

    cont.Equality of effective lengths:

    For Receiving antenna: By reciprocity theorem,

    Integrating both sides,

    By Theveninstheorem, open circuit voltage is given by,

    For constant incident field along the entire length of antenna,

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    APPLICATION OF RECIPROCITY THEOREM

    cont.

    Equality of effective lengths:

    For Receiving antenna:

    This proves that maximum effective lenght of an antenna is same whethertransmitting or receiving

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    APPLICATION OF RECIPROCITY THEOREM

    cont.Equality of Antenna Impedances:

    It is to be proved here that impedance ofantenna, away from ground and other objects is

    same whether it is transmitting or receiving.

    PROOF:

    Let two antennas with a wide separation in between. Ant no.2 is quite away from ant no. 1 so the mutual

    impedance between the two is neglected.

    Ant no. 1 is transmitting => self impedance is

    obtained.

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    APPLICATION OF RECIPROCITY THEOREM

    cont.Equality of Antenna Impedances: But this assumption is not true when ant no. 1 is receiving. During reception it is the mutual impedance (Z12) between the two

    antennas which provides coupling.

    If an equivalent ant no.1 is drawn under load (ZL), then little

    consideration would show that Z12I2acts as voltage generator. See figure.

    Now assume that ant no. 2 is quite

    away from 1change in ZLdoes not

    cause change in I2 of voltage source

    Z12I2acts as an ideal zero impedance,

    constant voltage generator.

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    APPLICATION OF RECIPROCITY THEOREM

    cont.Equality of Antenna Impedances: The equivalent circuit of receiving antenna under open and short circuit

    conditions are shown below in figure.

    In such case receiving impedance and transmitting impedance are equal

    as ant no.1 have the terminal behavior of voltage generator with internal

    impedance Z11.

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    APPLICATION OF RECIPROCITY THEOREM

    cont.Equality of Antenna Impedances:

    It is seen that (Z12I2) is a voltage source and Z11is the internal impedance

    and the ratio of the two gives short circuit current.

    Further, this is not only true for one distant antenna but for any number of

    antennas if they are away from the antenna whose impedance is being

    considered.

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    ANTENNA FEEDING METHODS

    Depending upon where the feed line connected to the dipolethere are different methods as,

    1. Centre fed

    2. Off centre fed

    3. Delta matching4. Inverted L-type

    The current and voltage distribution on a half-wave dipole is

    shown in figure below.

    NOTE:

    At centre where current is maximum the impedance

    is minimum of 73. But at ends where the current is

    minimum, the impedance is about 2500.

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    ANTENNA FEEDING METHODS

    1. Centre fed The 75 transmission line can be connected to exactly at the

    centre of the dipole.

    This matches the 73 impedance of half-wave dipole with the

    feed line impedance.

    This is the simplest feed arrangement provided the feed line 75 is available.

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    ANTENNA FEEDING METHODS

    Off - Centre fed When 75 cable is not available, dipole is fed off the centre.

    The position of the feed connection must be done such that

    antenna impedance at that point matches with feed line

    impedance.

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    ANTENNA FEEDING METHODS

    Delta matching Commonly used for connection.

    Here two wire line approaches the antenna, it is fanned gradually.

    The radiation resistance of the dipole increases symmetrically as

    the distance from the centre. Therefore fanning of the

    transmission line is done to such an extent that impedance of theline becomes equal to the impedance of the section of antenna

    wire between the points where the line is transmitted.

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    ANTENNA FEEDING METHODS

    Inverted L-type feeding Another simple method.

    Vertical and horizontal section of the wire is connected to form

    inverted L shape.

    It uses the narrowest range of frequencies.


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