Reciprocity Theorem

RECIPROCITY THEOREM

PREPARED BY: Namrata V. L. Assistant Professor Government Engineering College, Rajk Rajkot ot

RECIPROCITY THEOREM •

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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. That’s why it is also known as Rayleigh reciprocity theorem.

RECIPROCITY THEOREM

cont….

STATEMENT: If an emf is applied to the terminals of an antenna no. 1 and the 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 is applied to the terminals of antenna no. 2. OR If a current I, at the terminals of antenna no. 1 induces an emf E21 at the open terminals of antenna no. 2 and a current I2 at the terminals of antenna no. 2 induces an emf of E 12 at the open terminals of antenna no. 1, then E12=E21 provided I1=I2. •

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cont….

ASSUMPTIONS: –

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Emf ‘s are 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.

RECIPROCITY THEOREM

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1. A transmitter of frequency f and zero impedance be connected to the terminals of antenna no. 2 which is generating a current I 2 and inducing an emf E 12 at the open terminals of antenna no. 1 .

RECIPROCITY THEOREM 1.

cont….

Now the same transmitter is transferred to antenna no. 1 which is generating a current I 1 and inducing a voltage E 21 at the open terminals of antenna no. 2


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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.

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Thus from reciprocity theorem two impedances are equal,

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or


1.

cont….

Limitations:

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 Earth’s magnetic 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 be exactly correct at every given time.

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.

APPLICATION OF RECIPROCITY THEOREM 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 the direction 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.

cont….


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cont….

Symbolically for transmitting antenna:

If an emf E is applied at the centre point C, then currents I(c) at the centre point 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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cont….

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


For Receiving antenna:

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By reciprocity theorem,

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Integrating both sides,

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By Thevenin’s theorem, open circuit voltage is given by,

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For constant incident field along the entire length of antenna,

cont….

APPLICATION OF RECIPROCITY THEOREM cont…. Equality of effective lengths: •

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For Receiving antenna:

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

APPLICATION OF RECIPROCITY THEOREM Equality of Antenna Impedances: •

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It is to be proved here that impedance of antenna, away from ground and other objects is same whether it is transmitting or receiving. PROOF: –

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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.

cont….


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cont….

But this assumption is not true when ant no. 1 is receiving. During reception it is the mutual impedance (Z 12) between the two antennas which provides coupling. If an equivalent ant no.1 is drawn under load (Z L), then little consideration would show that Z 12I2 acts as voltage generator. See figure. Now assume that ant no. 2 is quite away from 1  change in ZL does not cause change in I2 of voltage source Z12I2 acts as an ideal zero impedance, constant voltage generator.


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cont….

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.

APPLICATION OF RECIPROCITY THEOREM Equality of Antenna Impedances:

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cont….

It is seen that (Z 12I2) is a voltage source and Z 11 is 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.

ANTENNA FEEDING METHODS •

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

1.

Centre fed

2.

Off centre fed

3.

Delta matching

4.

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 Ω. But at ends where the current is minimum, the impedance is about 2500 Ω.

ANTENNA FEEDING METHODS 1.

Centre fed •

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The 75 Ω transmission line can be connected to exactly at the centre of the dipole. This matches the 73 feed line impedance.

Ω impedance

of half-wave dipole with the

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

Ω

ANTENNA FEEDING METHODS Off - Centre fed •

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When 75 Ω 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.

ANTENNA FEEDING METHODS Delta matching •

Commonly used for connection.

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

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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 the line becomes equal to the impedance of the section of antenna wire between the points where the line is transmitted.

ANTENNA FEEDING METHODS Inverted L-type feeding •

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Another simple method. Vertical and horizontal section of the wire is connected to form inverted L shape. It uses the narrowest range of frequencies.

Reciprocity Theorem

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