Two-Hole Directional Coupler
Construction and Working Principle of Two-Hole Directional Coupler
A two-hole directional coupler is a microwave device consisting of two parallel waveguides having the same general structure, with two small coupling holes provided in their common wall. These holes allow a fraction of the microwave energy travelling through the main waveguide to be transferred into the secondary waveguide. The two holes act approximately as small slot antennas and radiate a portion of the electromagnetic energy into the secondary waveguide. The spacing between the two holes is carefully selected so that the coupled waves travelling in the desired direction add constructively, while the waves travelling in the opposite direction cancel each other. This constructive and destructive interference of the coupled waves produces the directional property of the coupler.
2080 Bhadra (BEI) “Explain the working principle of a two-hole directional coupler with a neat diagram. Derive its scattering parameters.”
Construction of Two-Hole Directional Coupler
The basic structure consists of a main waveguide and a secondary waveguide placed parallel to each other. The two waveguides have a common wall through which electromagnetic energy can be coupled. Two small holes are made in this common wall at a fixed distance from each other. The holes are sufficiently small so that only a fraction of the energy travelling in the main waveguide is coupled into the secondary waveguide.
The main waveguide contains port 1 and port 2. When microwave power is applied at port 1, the main wave travels toward port 2. The secondary waveguide contains port 3 and port 4. The coupled energy travels in both directions inside the secondary waveguide. Due to the selected spacing between the two holes, the waves travelling toward port 4 reinforce each other, whereas the waves travelling toward port 3 cancel each other. Thus, port 4 acts as the coupled port and port 3 acts as the isolated port for excitation at port 1.
Here, \(L\) represents the distance between the centres of the two coupling holes and \(\lambda_g\) represents the guide wavelength in the waveguide. The holes are normally separated by a quarter guide wavelength for the minimum spacing required for directional operation.
\[ L=\frac{\lambda_g}{4} \]More generally, the hole spacing can be selected as an odd multiple of a quarter guide wavelength:
\[ L=\frac{(2n+1)\lambda_g}{4}, \qquad n=0,1,2,\ldots \]For the minimum spacing, \(n=0\), and therefore
\[ L=\frac{\lambda_g}{4} \]Function of the Coupling Holes
The two small holes in the common wall provide electromagnetic coupling between the main and secondary waveguides. When the microwave wave travelling through the main waveguide reaches the first hole, a small fraction of its energy passes through the hole and is radiated into the secondary waveguide. The same process occurs when the wave reaches the second hole. Therefore, each hole produces a coupled wave in the secondary waveguide.
The holes can be considered approximately as small slot antennas. The amount of energy transferred through each hole depends on the dimensions of the hole and the electromagnetic field distribution at its location. Only a fraction of the input power is coupled into the secondary waveguide, while the major portion continues through the main waveguide toward port 2.
Working Principle of Two-Hole Directional Coupler
When microwave power is applied to port 1, the electromagnetic wave travels through the main waveguide toward port 2. As the wave passes the first coupling hole, a fraction of the wave energy is coupled into the secondary waveguide. When the wave reaches the second coupling hole, another fraction of the energy is coupled into the secondary waveguide. Thus, two coupled waves are produced in the secondary waveguide, one from each coupling hole.
Each coupled wave travels in both directions along the secondary waveguide. Therefore, the two holes produce waves travelling toward port 3 and waves travelling toward port 4. The direction in which the coupled waves reinforce or cancel is determined by the phase difference between the waves. The hole spacing is selected so that the required phase relationship is obtained.
Forward Waves at Port 4
Consider the coupled waves travelling from the two holes toward port 4. The wave produced by the first hole travels a distance \(L\) before reaching the second hole location, while the wave produced at the second hole begins at that location. The phase relationship produced by the hole spacing is such that the two forward waves arrive at port 4 in the same phase. Therefore, the waves add constructively.
\[ \text{Forward waves} \longrightarrow \text{same phase} \longrightarrow \text{constructive addition at port 4} \]Consequently, the coupled power is reinforced in the direction of port 4. Port 4 therefore provides the required coupled output. The amount of power appearing at this port is determined by the coupling strength of the two holes.
Backward Waves at Port 3
Now consider the coupled waves travelling in the opposite direction, from right to left toward port 3. The two waves have to travel different distances before reaching port 3. The difference in propagation distance between the two coupled waves is \(2L\). Therefore, the phase difference between them is
\[ \Delta\phi = \frac{2\pi}{\lambda_g}(2L) \]Hence,
\[ \Delta\phi = \frac{4\pi L}{\lambda_g} \]For the minimum hole spacing
\[ L=\frac{\lambda_g}{4} \]the phase difference becomes
\[ \Delta\phi = \frac{4\pi}{\lambda_g} \left(\frac{\lambda_g}{4}\right) = \pi \]Therefore, the two backward waves arriving at port 3 are \(180^\circ\) out of phase. Since the two coupled waves have equal magnitude under the ideal symmetrical arrangement, they cancel each other.
\[ \Delta\phi=\pi=180^\circ \] \[ \boxed{\text{Backward waves cancel at port 3}} \]As a result, ideally no coupled power appears at port 3 when port 1 is excited. Port 3 is therefore called the isolated port for excitation at port 1.
Directional Operation of Two-Hole Coupler
The directional operation of the two-hole directional coupler is therefore obtained from the interference of the waves generated by the two coupling holes. The waves travelling toward port 4 are arranged to combine constructively, while the waves travelling toward port 3 combine destructively. This means that the energy coupled from the main waveguide is preferentially directed toward port 4.
The power flow for excitation at port 1 can be represented as
\[ \boxed{ \text{Port 1} \rightarrow \text{Port 2} \quad\text{through power} } \] \[ \boxed{ \text{Port 1} \rightarrow \text{Port 4} \quad\text{coupled power} } \] \[ \boxed{ \text{Port 1} \rightarrow \text{Port 3} \quad\text{ideally isolated} } \]Thus, the two-hole arrangement provides a controlled sample of the input power at port 4 without allowing the same coupled energy to appear at port 3. The main wave continues toward port 2, while the small fraction of energy coupled through the holes is directed toward port 4.
Importance of Hole Spacing
The distance between the two coupling holes is the most important structural parameter responsible for directional operation. If the holes are separated by the appropriate distance, the phase difference between the coupled waves produces constructive interference in one direction and destructive interference in the opposite direction. For the minimum spacing, the distance between the centres of the holes is one quarter of the guide wavelength.
\[ L=\frac{\lambda_g}{4} \]More generally, the same phase condition can be obtained when the hole spacing is an odd multiple of a quarter wavelength:
\[ L=\frac{(2n+1)\lambda_g}{4} \]Therefore, the hole spacing determines the phase relationship between the coupled waves and consequently determines the direction in which the coupled energy is reinforced. This is the fundamental operating principle of the two-hole directional coupler.
Working Principle in Summary
In a two-hole directional coupler, two parallel waveguides are coupled through two small holes made in their common wall. When a microwave signal is applied at port 1, a fraction of its energy passes through each hole and is radiated into the secondary waveguide. The coupled waves travel in both directions. With the holes separated by \(L=\lambda_g/4\), the forward waves travelling toward port 4 arrive in the required phase and add constructively. The backward waves travelling toward port 3 have a phase difference of \(180^\circ\) and cancel each other. Hence, port 4 receives the coupled power while port 3 remains ideally isolated. This constructive and destructive interference is what gives the two-hole directional coupler its directional characteristic.
Phase Relationship and Directional Operation of Two-Hole Directional Coupler
The directional operation of a two-hole directional coupler is based on the phase relationship between the electromagnetic waves coupled through the two holes. When a wave travels through the main waveguide, each coupling hole transfers a small portion of the microwave energy into the secondary waveguide. The waves produced by the two holes then travel in both directions along the secondary waveguide. The spacing between the holes is selected so that the waves travelling toward port 4 reinforce each other, while the waves travelling toward port 3 cancel each other. Therefore, the directionality of the coupler is obtained from the controlled constructive and destructive interference of the two coupled waves.
Coupled Waves from the Two Holes
Consider a microwave signal applied at port 1 of the main waveguide. The signal travels toward port 2 and encounters the first coupling hole. A small fraction of the electromagnetic energy is coupled through the first hole into the secondary waveguide. As the main wave continues along the main waveguide, it reaches the second coupling hole, where another small fraction of the energy is coupled into the secondary waveguide.
Thus, the two holes behave as two coupling points that generate electromagnetic waves in the secondary waveguide. Each coupled wave propagates in both directions. One component travels toward port 4, while another component travels toward port 3. The final amount of power reaching either port depends on the relative phase of the waves generated at the two holes.
Let the distance between the centres of the two holes be \(L\). The coupled waves from the two holes therefore experience different propagation distances before reaching the output ports. This difference in propagation distance produces a phase difference between the waves.
Constructive Interference at Port 4
Consider the waves travelling toward port 4. The wave generated at the first hole has to travel an additional distance before it reaches port 4 compared with the wave generated at the second hole. The hole spacing is selected so that the phase relationship of these two coupled waves causes them to reinforce each other at port 4.
Therefore, the two coupled waves travelling toward port 4 combine constructively. Their amplitudes add in the required phase relationship, resulting in a measurable coupled signal at port 4. This port is consequently called the coupled port for excitation at port 1.
\[ \boxed{ \text{Constructive interference at port 4} } \]The constructive addition means that the energy coupled from the two holes is concentrated in the direction of port 4. The coupling level at this port depends on the strength of the individual coupling through the holes.
Destructive Interference at Port 3
Now consider the coupled waves travelling in the opposite direction toward port 3. The two waves do not travel equal distances before reaching port 3. The wave generated at the second hole has to travel an additional distance of \(L\) to reach the first hole position and then continues toward port 3. Consequently, the difference in propagation distance between the two waves reaching port 3 is \(2L\).
The additional propagation distance produces a phase difference between the two waves. Since the phase change of a wave travelling through a distance \(d\) in a waveguide is
\[ \phi=\frac{2\pi}{\lambda_g}d \]the phase difference produced by the additional propagation distance \(2L\) is
\[ \Delta\phi = \frac{2\pi}{\lambda_g}(2L) \]Therefore,
\[ \boxed{ \Delta\phi=\frac{4\pi L}{\lambda_g} } \]This phase difference is the key mathematical relationship responsible for the isolation of port 3.
Phase Difference for Minimum Hole Spacing
For the minimum hole spacing, the two holes are separated by one quarter of the guide wavelength:
\[ L=\frac{\lambda_g}{4} \]Substituting this value into the phase difference equation gives
\[ \Delta\phi = \frac{2\pi}{\lambda_g} \left( 2\frac{\lambda_g}{4} \right) \] \[ \Delta\phi = \frac{2\pi}{\lambda_g} \left( \frac{\lambda_g}{2} \right) \] \[ \boxed{ \Delta\phi=\pi } \]Since
\[ \pi=180^\circ \]the two coupled waves arriving at port 3 have a phase difference of \(180^\circ\). For equal magnitude waves, one wave therefore opposes the other, causing destructive interference.
\[ \boxed{ \text{Destructive interference at port 3} } \]Ideally, the two waves cancel completely at port 3. Hence, no coupled power appears at port 3 when port 1 is excited. Port 3 is therefore known as the isolated port for this excitation condition.
Why Port 4 is Coupled and Port 3 is Isolated
The two-hole directional coupler can now be understood directly from the phase relationship of the coupled waves. The two holes generate waves that travel in both directions in the secondary waveguide. The hole spacing introduces the required propagation phase difference between these waves. In one direction, the phase relationship causes constructive interference, so the waves reinforce each other. In the opposite direction, the phase relationship causes destructive interference, so the waves cancel each other.
For excitation at port 1, the constructive interference occurs toward port 4 and the destructive interference occurs toward port 3. Therefore, port 4 receives the coupled signal, while port 3 ideally receives no signal.
The directional operation can therefore be summarized as
\[ \boxed{ \text{Port 4: constructive interference} } \] \[ \boxed{ \text{Port 3: destructive interference} } \]This is the fundamental reason why the device is called a directional coupler. The coupling holes do not simply transfer power into the secondary waveguide; their spacing and the resulting phase relationship determine the direction in which the coupled power is reinforced.
General Hole Spacing and Phase Condition
The minimum spacing \(L=\lambda_g/4\) is not the only spacing that can produce the required phase relationship. In general, the hole spacing is selected as an odd multiple of a quarter guide wavelength:
\[ L=\frac{(2n+1)\lambda_g}{4} \]Using this spacing in the phase difference equation gives
\[ \Delta\phi = \frac{4\pi}{\lambda_g} \left( \frac{(2n+1)\lambda_g}{4} \right) \] \[ \Delta\phi = (2n+1)\pi \]Thus,
\[ \boxed{ \Delta\phi=(2n+1)\pi } \]This corresponds to an odd multiple of \(180^\circ\), which maintains the required opposite phase relationship for cancellation in the isolated direction.
Power Flow in the Two-Hole Directional Coupler
When power is applied at port 1, most of the input microwave power continues through the main waveguide toward port 2. A smaller fraction of the power is coupled through the two holes into the secondary waveguide. Due to constructive interference, this coupled energy appears primarily at port 4. Ideally, the coupled waves cancel at port 3, so no power is delivered to the isolated port.
\[ \boxed{ P_1 \rightarrow P_2 + P_4 } \]For an ideal lossless coupler, the input power is divided between the through path and the coupled path. Therefore, in terms of power fractions,
\[ |S_{21}|^2+|S_{41}|^2=1 \]The exact values of the through and coupled powers depend on the coupling level of the particular directional coupler. For a 3 dB coupler, the through and coupled powers are equal, so each receives one half of the input power.
Ideal Directionality and Practical Limitations
In an ideal two-hole directional coupler, the destructive interference at the isolated port is complete. This means that the coupled signal at port 3 is zero when port 1 is excited. The ideal condition can therefore be represented by
\[ S_{31}=0 \]In a practical coupler, complete cancellation is difficult to achieve. Manufacturing tolerances, differences in hole dimensions, imperfect spacing, conductor losses, dielectric effects, frequency variation, and other imperfections can disturb the required amplitude and phase relationship between the coupled waves. Consequently, a small amount of power may appear at the isolated port.
Thus, practical directional couplers have finite directivity rather than infinite directivity. The directivity is related to the ratio between the desired coupled signal and the undesired signal appearing at the isolated port. For excitation at port 1, it can be expressed in terms of scattering parameters as
\[ D=20\log_{10} \left( \frac{|S_{41}|}{|S_{31}|} \right) \]For an ideal coupler, \(S_{31}=0\), and therefore the ideal directivity approaches infinity. In an actual device, \(S_{31}\) is very small but nonzero, giving a finite directivity.
Connection Between Phase Relationship and Directionality
The complete working mechanism can therefore be traced from the physical spacing of the holes to the directional behavior of the coupler. The microwave signal entering port 1 produces two coupled waves through the two holes. These waves propagate in both directions in the secondary waveguide. The difference in propagation distance produces a phase difference given by
\[ \Delta\phi = \frac{2\pi}{\lambda_g}(2L) \]For \(L=\lambda_g/4\), this phase difference becomes \(180^\circ\). The resulting phase relationship causes the waves to combine constructively in the coupled direction toward port 4 and destructively in the opposite direction toward port 3. Therefore, port 4 becomes the coupled port and port 3 becomes the isolated port.
This phase-based explanation establishes the physical basis for the scattering parameters that will be derived next. In particular, the constructive coupling toward port 4 corresponds to a nonzero transmission coefficient \(S_{41}\), while the ideal cancellation toward port 3 corresponds to the isolation condition \(S_{31}=0\). The next part can therefore use these physical properties to construct and derive the complete scattering matrix of the two-hole directional coupler.
Two-Hole Directional Coupler S-Matrix and Scattering Parameter Derivation
The two-hole directional coupler is fundamentally a four-port microwave network, so its scattering behavior is described using a \(4 \times 4\) S-matrix. The physical operation discussed in the previous parts provides the conditions required to determine this matrix. When port 1 is excited, port 2 acts as the through port, port 4 acts as the coupled port, and port 3 acts as the isolated port. The scattering matrix can therefore be derived by starting with the general four-port S-matrix and then applying the matched-port, reciprocity, directional, lossless, and phase conditions.
General S-Matrix of the Two-Hole Directional Coupler
For a four-port network, the incident and reflected travelling-wave amplitudes are related by the scattering matrix. Let \(a_i\) represent the incident wave at port \(i\), and let \(b_i\) represent the outgoing wave from port \(i\). The general relationship is
\[ \begin{bmatrix} b_1\\ b_2\\ b_3\\ b_4 \end{bmatrix} = [S] \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4 \end{bmatrix} \]where the general scattering matrix is
\[ [S]= \begin{bmatrix} S_{11}&S_{12}&S_{13}&S_{14}\\ S_{21}&S_{22}&S_{23}&S_{24}\\ S_{31}&S_{32}&S_{33}&S_{34}\\ S_{41}&S_{42}&S_{43}&S_{44} \end{bmatrix} \]Each element \(S_{ij}\) represents the ratio of the outgoing wave from port \(i\) to the incident wave at port \(j\), with all other ports terminated in matched loads. Thus, the S-matrix completely describes the transmission, coupling, reflection, and isolation characteristics of the directional coupler.
Matched-Port Condition
An ideal two-hole directional coupler is designed with all four ports matched to their characteristic impedances. A matched port does not produce a reflected wave when excited, so the corresponding reflection coefficient is zero. Therefore, all diagonal elements of the S-matrix are zero.
\[ \boxed{ S_{11}=S_{22}=S_{33}=S_{44}=0 } \]Applying this condition to the general matrix gives
\[ [S]= \begin{bmatrix} 0&S_{12}&S_{13}&S_{14}\\ S_{21}&0&S_{23}&S_{24}\\ S_{31}&S_{32}&0&S_{34}\\ S_{41}&S_{42}&S_{43}&0 \end{bmatrix} \]This condition shows that the ideal coupler has no input reflection at any of its four ports.
Reciprocity Condition
A passive two-hole directional coupler is a reciprocal microwave network. For a reciprocal network, transmission between any two ports is the same in either direction when the reference conditions are properly defined. Mathematically, the scattering matrix satisfies
\[ \boxed{ S_{ij}=S_{ji} } \]Therefore, the corresponding pairs of scattering parameters are related as
\[ S_{12}=S_{21} \] \[ S_{13}=S_{31} \] \[ S_{14}=S_{41} \] \[ S_{23}=S_{32} \] \[ S_{24}=S_{42} \] \[ S_{34}=S_{43} \]Using reciprocity, the S-matrix can be written in the symmetric form
\[ [S]= \begin{bmatrix} 0&S_{21}&S_{31}&S_{41}\\ S_{21}&0&S_{32}&S_{24}\\ S_{31}&S_{32}&0&S_{34}\\ S_{41}&S_{24}&S_{34}&0 \end{bmatrix} \]Directional Isolation Condition
The defining feature of the two-hole directional coupler is its directional operation. When port 1 is excited, the wave coupled through the two holes is reinforced in the direction of port 4, while it is cancelled in the direction of port 3. Therefore, port 4 is the coupled port and port 3 is the isolated port.
For ideal isolation, no power is transferred from port 1 to port 3. Hence, the corresponding scattering parameter is zero:
\[ \boxed{ S_{31}=0 } \]By reciprocity,
\[ \boxed{ S_{13}=S_{31}=0 } \]The same directional relationship applies to excitation from the opposite main-waveguide port. Therefore, port 2 and port 4 are isolated from each other in the ideal case:
\[ \boxed{ S_{24}=S_{42}=0 } \]After applying the matched-port, reciprocal, and directional conditions, the matrix becomes
\[ [S]= \begin{bmatrix} 0&S_{21}&0&S_{41}\\ S_{21}&0&S_{32}&0\\ 0&S_{32}&0&S_{34}\\ S_{41}&0&S_{34}&0 \end{bmatrix} \]Through and Coupling Coefficients
For excitation at port 1, \(S_{21}\) represents transmission from port 1 to port 2, while \(S_{41}\) represents coupling from port 1 to port 4. Let the magnitudes of these two coefficients be represented by \(a\) and \(b\), respectively:
\[ |S_{21}|=a \] \[ |S_{41}|=b \]Here, \(a\) is called the through coefficient, because it represents the fraction of the input wave that continues through the main waveguide. The quantity \(b\) is called the coupling coefficient, because it represents the magnitude of the wave coupled into the secondary waveguide.
The corresponding coefficients in the other transmission directions have the same magnitudes because of the symmetry and reciprocity of the ideal directional coupler.
Lossless Condition
For an ideal directional coupler, the network is assumed to be lossless. This means that the total output power is equal to the input power. The scattering matrix of a lossless network satisfies
\[ \boxed{ [S][S]^\dagger=[I] } \]where \([S]^\dagger\) is the Hermitian transpose of the scattering matrix and \([I]\) is the identity matrix.
For excitation at port 1, the ideal isolated port does not receive any power. Therefore, the input power is divided between the through port and the coupled port. The power conservation condition is
\[ |S_{21}|^2+|S_{41}|^2=1 \]Using the definitions of \(a\) and \(b\), this becomes
\[ a^2+b^2=1 \] \[ \boxed{ a^2+b^2=1 } \]This equation establishes the relationship between the through coefficient and the coupling coefficient of an ideal lossless directional coupler.
Phase Relationship and Final S-Matrix
The two-hole structure does not only determine the magnitudes of the through and coupled signals; it also establishes a specific phase relationship between the waves. Depending on the selected port numbering and reference-plane convention, the coupled terms are commonly represented with a \(90^\circ\) phase shift relative to the through terms. A \(90^\circ\) phase shift is represented mathematically by the factor \(j\).
Using the commonly adopted reference-plane convention, the scattering matrix of the ideal two-hole directional coupler can therefore be written as
\[ \boxed{ [S]= \begin{bmatrix} 0&a&0&jb\\ a&0&jb&0\\ 0&jb&0&a\\ jb&0&a&0 \end{bmatrix} } \]where
\[ \boxed{ a^2+b^2=1 } \]The matrix satisfies the important physical properties of the ideal directional coupler. Its diagonal elements are zero because the ports are matched. The matrix is symmetric because the network is reciprocal. The elements \(S_{31}\) and \(S_{13}\) are zero because port 3 is isolated from port 1. Similarly, \(S_{24}\) and \(S_{42}\) are zero because port 4 is isolated from port 2.
The \(j\) factor represents the relative \(90^\circ\) phase relationship between the through and coupled paths for this particular reference-plane convention. Different port numbering or reference-plane choices can change the signs or phase representation of individual S-parameters without changing the fundamental physical behavior of the coupler.
S-Matrix of a 3 dB Two-Hole Directional Coupler
A special and important case occurs when the directional coupler is designed to divide the input power equally between the through and coupled ports. Such a coupler is commonly referred to as a 3 dB directional coupler.
For a 3 dB coupler, the power delivered to the through port and the coupled port is equal. Therefore, for excitation at port 1,
\[ |S_{21}|^2=|S_{41}|^2=\frac{1}{2} \]Hence,
\[ a=b=\frac{1}{\sqrt{2}} \]Substituting these values into the general S-matrix gives
\[ \boxed{ [S]= \frac{1}{\sqrt{2}} \begin{bmatrix} 0&1&0&j\\ 1&0&j&0\\ 0&j&0&1\\ j&0&1&0 \end{bmatrix} } \]This matrix represents an ideal 3 dB directional coupler in the selected reference-plane convention. The magnitude of each nonzero transmission or coupling coefficient is \(1/\sqrt{2}\), while the isolated-port coefficients remain zero.
Interpretation of Important Scattering Parameters
The scattering parameters can be interpreted directly from the physical operation of the two-hole directional coupler. The parameter \(S_{21}\) represents the transmission from port 1 to port 2. Since port 2 is the through port, \(S_{21}\) determines the amount of input signal that continues through the main waveguide.
\[ \boxed{ S_{21}=\text{transmission from port 1 to port 2} } \]The parameter \(S_{41}\) represents the coupling from port 1 to port 4. Since port 4 is the coupled port, \(S_{41}\) determines the amount of signal extracted from the main waveguide through the two coupling holes.
\[ \boxed{ S_{41}=\text{coupling from port 1 to port 4} } \]The parameter \(S_{31}\) represents transmission from port 1 to port 3. Since port 3 is the isolated port, ideal directional operation requires
\[ \boxed{ S_{31}=0 } \]Similarly, \(S_{24}\) represents transmission from port 4 to port 2. The ideal directional condition gives
\[ \boxed{ S_{24}=0 } \]The important scattering parameters can therefore be summarized as follows:
- \(S_{21}\): through transmission from port 1 to port 2.
- \(S_{41}\): coupling from port 1 to port 4.
- \(S_{31}\): isolation from port 1 to port 3.
- \(S_{24}\): isolation from port 4 to port 2.
Coupling Factor, Transmission and Directivity
The scattering parameters can also be used to express the main performance parameters of the directional coupler. The coupling factor describes the amount of power coupled from the input port to the coupled port. In terms of the coupling scattering parameter, it is given by
\[ \boxed{ C=-20\log_{10}|S_{41}| } \]The transmission or through loss describes the signal transmitted from port 1 to port 2. It can be expressed as
\[ \boxed{ T=-20\log_{10}|S_{21}| } \]The directivity compares the desired coupled signal at port 4 with the undesired signal appearing at the isolated port 3. It is given by
\[ \boxed{ D= 20\log_{10} \left( \frac{|S_{41}|}{|S_{31}|} \right) } \]For an ideal directional coupler, \(S_{31}=0\). Therefore, the ideal directivity approaches infinity:
\[ \boxed{ D\rightarrow\infty } \]In a practical two-hole directional coupler, the cancellation at the isolated port is not perfectly complete because of manufacturing tolerances, frequency dependence, unequal coupling through the holes, conductor losses, and other imperfections. Consequently, \(S_{31}\) is small but nonzero, resulting in finite directivity.
Final S-Matrix of the Ideal Two-Hole Directional Coupler
Combining the matched-port condition, reciprocity, directional isolation, lossless operation, and the phase relationship established by the two-hole structure gives the final ideal scattering matrix:
\[ \boxed{ [S]= \begin{bmatrix} 0&a&0&jb\\ a&0&jb&0\\ 0&jb&0&a\\ jb&0&a&0 \end{bmatrix}, \qquad a^2+b^2=1 } \]For the 3 dB case,
\[ \boxed{ [S]= \frac{1}{\sqrt{2}} \begin{bmatrix} 0&1&0&j\\ 1&0&j&0\\ 0&j&0&1\\ j&0&1&0 \end{bmatrix} } \]Thus, the physical operation of the two-hole directional coupler is directly reflected in its S-matrix. The nonzero through coefficient represents transmission through the main waveguide, the nonzero coupling coefficient represents power coupled through the two holes, and the zero isolation terms represent the cancellation produced by the phase relationship between the coupled waves. The S-matrix therefore provides the mathematical representation of the constructive and destructive interference mechanism established in the previous part.