DIRECTIONAL COUPLERS
Ing. A. Ramón Vargas Patrón
rvargas@inictel.gob.pe
INICTEL
Abstract
This paper analyzes two types of Directional Couplers. First, magnetic coupling
between a transmission line and a secondary circuit is studied. It is then shown
that frequency independent samples of voltage and current on the transmission
line yield a feasible way of obtaining separate readings of the forward and reflected
waves.
The second type of coupler takes advantage of the electrostatic and magnetic
coupling existing between two parallel conductors. A simplified model of the coupler permits a straightforward analysis of the circuit. Expressions for the forward
and reflected voltages are then readily obtained. In both cases, a peak detector
circuit gives DC readings of the voltages.
1
Directional Couplers
A directional coupler (D.C.) is a device that detects and separates the incident and reflected
waves present in a transmission line, for instance, the one that links the radio transmitter with
the antenna system.
One type of D.C. that makes use of voltage and current coupling is shown in Fig. 1, where
it is suggested that the device be placed somewhere along the transmission line, between the
signal generator (radio transmitter, for example) and the load ZL (antenna). Usually, it is
more comfortable to make the connection at the transmitter output.
Consider an unbalanced line of length `, along with the circuit of Fig.1. If we call Ex the
transmission line voltage in the connection point to the secondary circuit and Ix the current
in the same point, we have:
Ex = Ef e−j ωx/υ + Eb e+j ωx/υ
Ix =
(1)
Ef −j ωx/υ Eb +j ωx/υ
e
−
e
Z0
Z0
where:
Ef
Eb
υ
Z0
x
ω
=
=
=
=
=
=
incident component of voltage
reflected component of voltage
velocity of propagation in the transmission line
characteristic impedance of the transmission line
position along the transmission line
angular frequency of the generator
We have for the secondary mesh, with I1 = Ix :
j ω M12 I1 = 2R2 + j ω L2 I2
(2)
Therefore:
I2
j ω M12
=
I1
2R2 + j ω L2
(3)
ω L2 2R2
(4)
If we make
then the expression (3) becomes:
Figure 1. Unbalanced line and secondary
circuit
I2
M12
=
I1
L2
(5)
Notice that for this condition I1 and I2 are in phase and the frequency dependent term vanishes.
Voltages in A and B will then be:
EA =
I2 R2 =
R2
M12
I1
L2
M12
EB = −I2 R2 = − R2
I1
L2
(6)
Now, a frequency independent voltage sample may be obtained with the help of a capacitive
divider, as is shown in Fig. 2. Then:
EC = E x
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C1
C 1 + C2
2
(7)
A. Ramón Vargas Patrón
Directional Couplers
or
EC ≈ E x
C1
C2
(8)
if C2 C1
Ec = E x
C1
C1 + C2
The circuit for our coupler would become the one in Fig. 3.
The following holds:
EAC = EA − EC = R2
Figure 2. Capacitive divider
M12
C1
I1 − Ex
L2
C2
Substituting for Ex and I1 the expressions labeled as (1), we have:
M12 Ef −j ωx/υ Eb +j ωx/υ
C1
−j ωx/υ
+j ωx/υ
EAC = R2
e
−
e
−
Ef e
+ Eb e
L2 Z0
Z0
C2
(9)
(10)
If the following holds:
C1
M12
= R2
C2
L2
1
Z0
(11)
the terms containing Ef cancel each other and
EAC = −2
C1
+j ωx/υ
Eb e
C2
Also:
EBD = EB − ED = −R2
(12)
M12
C1
I 1 − Ex
L2
C2
(13)
Substituting for Ex and I1 the expressions labeled as (1) and taking into account (11):
EBD = −2
C1
−j ωx/υ
Ef e
C2
(14)
We thus have a directional coupler with readings of the incident and reflected waves. Capacitor
C1 can be made adjustable for calibration purposes and to assure good directivity. If required,
DC voltages can be obtained to drive a galvanometer, rectifying and filtering voltages EAC for
the reflected wave and EBD for the incident component.
Figure 3. Basic coupler circuit topology
One possible implementation of this directional coupler is shown in Fig. 4 on following page.
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A. Ramón Vargas Patrón
Directional Couplers
D1 –D4 : Germanium diodes
1N60, 1N34, . . .
A practical implementation of Fig. 3
Figure 4.
In Fig. 5 is shown a second type of D.C. that utilizes two parallel conductors with magnetic
and electrostatic coupling. The main conductor is an extension of the transmission line that
links the instrument with the generator at one end and with the load (antenna) at the other.
The second conductor coupled to the previous one is end terminated with a resistive load and
a detector circuit, respectively.
In these circuits readings from the incident or reflected waves can be selected by the switch.
Figure 5. A C-M type directional coupler
A relatively simple analysis of the coupler can be made utilizing the equivalent circuit of
Fig. 6. Here, C represents the distributed capacitance between conductors; M is the mutual
inductance of the system; R1 and R2 are the end terminations of the secondary conductor and
L is the self-inductance of this conductor. The transmission line current at the point where the
device connects is I (complex quantity) and E is the voltage on the line at that same point.
The next calculations assume that the following
inequality holds for R1 (and also for R2 ):
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4
ω L R1 1
ωC
(15)
A. Ramón Vargas Patrón
Directional Couplers
M1 = M/2
M2 = L/4
Figure 6. The directional coupler’s equivalent circuit
In Fig. 6:
e0 = −j ω M1 I − j ω M1 I + j ω
L
L
I1 − j ω M2 I2 + j ω M2 I1 − j ω I2 + I1 R1
4
4
(16)
Substituting for M1 and M2:
e0 = −j ω M I + j ω
L
L
L
L
I1 + j ω I1 − j ω I2 − j ω I2 + I1 R1
4
4
4
4
(17)
L
L
I1 − j ω I2 + I1 R1
2
2
(18)
e0 = −j ω M I + j ω
According to inequality (15), we can write:
e0 ≈ −j ω M I + I1 R1 − j ω
On the other hand:
L
I2
2
(19)
e1 = −j ω M1 I + j ω
L
I1 − j ω M2 I2 + I1 R1
4
(20)
e1 = +j ω M1 I + j ω
L
I2 − j ω M2 I1 + I2 R2
4
(21)
Also:
Equating (20) with (21):
L
L
R1 + j ω
I1 − j ω M1 I − j ω M2 I2 = R2 + j ω
I2 + j ω M1 I − j ω M 2 I1
4
4
∴
or
L
L
R1 + j ω + j ω M2 I1 = 2j ω M1 I + R2 + j ω + j ω M2 I2
4
4
L
R1 + j ω
2
L
I1 = j ω M I + R2 + j ω
2
(22)
(23)
I2
(24)
For the sake of inequality (15):
I1 R1 ≈ j ω M I + I2 R2
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(25)
A. Ramón Vargas Patrón
Directional Couplers
Also:
I1 + I2 = (E − e1 ) j ω C
(26)
L
M
L
I1 + I2 = E − R1 + j ω
I1 + j ω I + j ω I2 j ω C
4
2
4
(27)
with (20):
or
I 1 + I2 =
M
L
L
E + j ω I j ω C − I1 R1 + j ω
j ω C + j ω I2 j ω C
2
4
4
(28)
From inequality (15) we get:
ω 2 LC ω R1 C 1
(29)
Expression (26) becomes then:
I 1 + I2 ≈
M
E + jω I jωC
2
(30)
Solving for I1 from the last expression:
M
I1 = E + j ω I j ω C − I2
2
(31)
Substituting in (25):
M
E + j ω I j ω C − I2 R1 = j ω M I + I2 R2
2
j ω CR1
j ω CR1 E − j ω M I 1 −
= I2 (R1 + R2 )
2
(32)
(33)
From (33) and according to (15):
I2 ≈
j ω CR1 E − j ω M I
R1 + R2
(34)
On the other hand:
e0 = I2 R2
(35)
(34) in (35):
e0 =
R2
(−j ω M I + j ω CR1 E)
R2 + R1
(36)
If:
R1 = R2 = R and M = CR1 Z0
∴
e0 =
1
j ω CR (E − Z0 I)
2
(37)
(38)
From the equations for a transmission line
E = Ef e−j βx + Eb e+j βx
I = If e
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−j βx
(39)
− Ib e
6
+j βx
A. Ramón Vargas Patrón
Directional Couplers
With β = ω/υ, and letting x = 0 (generator side)
E = E f + Eb
I = If − I b =
(40)
Eb
Ef
−
Z0
Z0
Then:
E − Z0 I = 2 Eb
(41)
e0 = j ω CR Eb
(42)
Consequently:
From (25), (37) and (42):
1
j ω CR (E − Z0 I)
2
1
1
j ω CR Z0 I + j ω CR E
=
2
2
1
j ω CR (E + Z0 I)
=
2
(43)
I1 R = j ω CR Z0 I +
(44)
(45)
and according to (40):
I1 R =
1
j ω CR (2Ef )
2
(46)
= j ω CR Ef
(47)
Expressions (42) and (47) show us that we have separated the transmission line’s incident and
reflected waves.
Conclusions
Employing adequate circuit models two directional couplers have been studied. It has been
shown that it is possible to obtain separate readings of the voltages for the incident and
reflected components. This study helps the understanding of the principles of operation of
actual devices in use at frequencies in the HF to UHF range.
References
1
KUECKEN, JOHN A., Antennas and Transmission Lines, chapter 23,
Howard W. Sams & Co., Inc., 1969
2 VARGAS PATRON, R., Lab Notes
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A. Ramón Vargas Patrón