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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.