Directional Coupler
Directional Coupler Parameters
A directional coupler is designed to split or sample microwave power between different ports, and when it is designed to divide power equally between two ports, it is known as a hybrid coupler. A directional coupler is a reciprocal, lossless, matched four port microwave network that allows a known portion of the power travelling through the main transmission path to be coupled to another port. It is widely used in microwave systems where it is necessary to sample the forward or reverse travelling wave without significantly disturbing the main transmission path. The operation of a directional coupler is based on controlled electromagnetic coupling between two transmission paths, so that power entering one port is mainly transferred to the through port while a specified fraction appears at the coupled port. The remaining port is ideally isolated from the input port.
2080 Bhadra (BEI) “Explain the working principle of a two-hole directional coupler with a neat diagram. Derive its scattering parameters.”
Primary and Secondary Waveguides in a Directional Coupler
A conventional directional coupler can be understood as two waveguides or transmission lines placed close enough to each other for electromagnetic energy to be coupled from one line to the other. The first transmission path is called the primary waveguide, while the second transmission path is called the secondary waveguide. In a four port directional coupler, ports 1 and 2 are associated with the primary waveguide, while ports 3 and 4 are associated with the secondary waveguide. The primary waveguide provides the main path for microwave power, whereas the secondary waveguide receives only a controlled portion of that power through electromagnetic coupling.
When power is applied at port 1, the main portion of the power travels along the primary waveguide toward port 2. Because the two waveguides are coupled, a portion of the electromagnetic energy is transferred to the secondary waveguide. Depending on the direction of coupling and the physical construction of the coupler, this coupled power appears at one of the secondary ports, while the other secondary port ideally receives no power. Thus, the four ports have distinct roles: the input port receives the incident power, the through port receives the major portion of the transmitted power, the coupled port receives a controlled sample of the input power, and the isolated port ideally receives no power.
Power Transmission and Coupling Between Ports
When all four ports of a directional coupler are terminated in their characteristic impedances, the network is properly matched and no reflection occurs at the ports. When power is applied to port 1, the main portion of the power is transmitted through the primary waveguide from port 1 to port 2. Since the ports are matched, this transmission takes place without reflection. However, there is ideally no power transmission from port 1 to port 3 because these ports form the isolated path and no effective coupling exists between them.
Similarly, there is no direct power transmission between port 2 and port 4 in the ideal directional coupler. The two ports belong to corresponding positions on the primary and secondary waveguides where the unwanted coupling is ideally cancelled. Therefore, 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 degree of coupling between port 1 and port 4, and correspondingly between port 2 and port 3, depends on the physical structure of the directional coupler. Factors such as the spacing between the coupled transmission lines or waveguides, the coupling length, the dimensions of the coupling region, and the operating frequency determine how much electromagnetic energy is transferred from the primary waveguide to the secondary waveguide. Therefore, the coupled-port power is not necessarily equal to the through-port power; it is determined by the required coupling factor of the particular coupler.
The directional behaviour of the coupler can also be understood from the phase relationship of the coupled waves. When power travels from port 1 toward port 2, electromagnetic energy is coupled into the secondary waveguide at different points along the coupling region. At the desired coupled port, these coupled components combine constructively and produce the required output. At the isolated port, the corresponding components ideally combine with equal magnitude and opposite phase, resulting in cancellation. Consequently, the isolated port receives no power in the ideal case.
Ideal Directivity and Isolated Port
In an ideal directional coupler, the isolated port receives zero power when the input is applied to the input port. For the port arrangement considered here, when port 1 is excited, port 3 is the isolated port. Therefore,
\[ P_3=0 \]
The directivity is defined as the ratio of the desired coupled power to the unwanted power appearing at the isolated port:
\[ D(\mathrm{dB})= 10\log_{10}\left(\frac{P_4}{P_3}\right) \]
Since \(P_3=0\) for the ideal case, the directivity approaches infinity:
\[ \boxed{D\rightarrow\infty} \]
This infinite directivity represents perfect directional behaviour. It means that the coupler can completely distinguish the desired power travelling in the coupling direction from the unwanted power travelling toward the isolated port. In practice, however, \(P_3\) cannot be made exactly zero because of imperfect amplitude and phase cancellation, impedance mismatch, manufacturing tolerances, and other practical limitations. Therefore, a practical directional coupler has a small but nonzero isolated-port power and consequently a finite directivity.
The statement that port 2 and port 4 are perfectly matched should be understood as part of the ideal matched-network condition. Proper matching prevents reflections from these ports, while the phase cancellation of the coupled waves is what produces the ideal zero power at the isolated port. Thus, matching and directional cancellation work together to obtain the ideal directional-coupler behaviour.
Power Flow and Phase Relationship Between Ports
The directional behaviour of the coupler is obtained from the controlled electromagnetic interaction between the primary and secondary waveguides. When a signal is applied to port 1, power is transmitted from port 1 to port 2 through the primary waveguide, while a portion is coupled to the secondary waveguide. The coupled signals at different ports have specific amplitude and phase relationships. In an ideal directional coupler, the isolated port does not receive power from the input port because the coupled waves reaching that port cancel each other.
The cancellation can be understood in terms of the phase relationship between the coupled waves. At the isolated port, the waves arriving through the different coupling paths are ideally equal in magnitude and opposite in phase. Therefore, their vector sum becomes zero, and no power appears at the isolated port. At the coupled port, the corresponding coupled components combine constructively, allowing the required fraction of the input power to appear at that port. This phase relationship is one of the fundamental reasons why a directional coupler can distinguish between forward and reverse travelling power.
The phase relationship between ports also depends on the particular directional coupler structure and port numbering convention. In many coupled transmission line configurations, the through and coupled outputs have a defined phase difference, while the unwanted signal at the isolated port is suppressed by cancellation. For a hybrid coupler designed for equal power division, the two output signals have equal power and a specified phase relationship, commonly involving a quadrature phase difference of \(90^\circ\). Therefore, both the magnitude and phase of the coupled signals are important when describing the operation and performance of a directional coupler.
2077 Chaitra (BEX) “What is a directional coupler? Derive the $S$-matrix of a $4$-port directional coupler and define Coupling Factor, Directivity, and Isolation.”
Coupling Factor
The coupling factor is one of the most important parameters of a directional coupler. It indicates the amount of microwave power coupled from the primary transmission path to the secondary or coupled port. When the signal is applied to port 1, the input power is \(P_1\), while the power measured at the coupled port is \(P_4\) for the port numbering convention considered here. The coupling factor is defined as the ratio of input power to coupled power and is normally expressed in decibels.
\[ C(\mathrm{dB})=10\log_{10}\left(\frac{P_1}{P_4}\right) \]
Here, \(P_1\) is the power applied to port 1 and \(P_4\) is the power obtained at the coupled port. A smaller coupling factor in decibels represents stronger coupling because a larger fraction of the input power reaches the coupled port. Conversely, a larger coupling factor represents weaker coupling. For example, a coupler with a specified coupling factor is designed so that a known and predictable fraction of the main-line power can be sampled at the coupled port without significantly disturbing the main power flow.
The coupling factor is particularly useful in microwave measurements because the coupled-port power can be used to determine the power travelling in the main transmission line. If the coupling factor of the directional coupler is known accurately, measuring the power at the coupled port provides a convenient way to estimate the power present in the primary line. This makes the directional coupler useful for power monitoring, signal sampling, and measurement systems.
Directivity
Directivity describes the ability of a directional coupler to distinguish between power travelling in the desired direction and power travelling in the undesired direction. It is determined by comparing the power at the coupled port with the unwanted power appearing at the isolated port. For an input applied at port 1, the desired coupled power is \(P_4\), while the unwanted power appearing at the isolated port is \(P_3\).
The directivity is given by
\[ D(\mathrm{dB})=10\log_{10}\left(\frac{P_4}{P_3}\right) \]
A high directivity means that the desired coupled power is much greater than the unwanted power at the isolated port. Therefore, a high directivity indicates good directional performance. The directivity depends strongly on the accuracy of the coupling structure, phase relationships, matching conditions, and frequency of operation.
Isolation
Isolation indicates how effectively the isolated port is separated from the input port. Ideally, when power is applied at port 1, no power should appear at port 3. In a practical directional coupler, however, a small amount of power may reach the isolated port because of imperfect cancellation, impedance mismatch, fabrication tolerances, or other nonideal effects.
Isolation can be expressed as the ratio between the incident power and the unwanted power appearing at the isolated port. For a directional coupler, the isolation is directly related to both the coupling factor and directivity. A high isolation means that very little unwanted power reaches the isolated port.
\[ \text{Isolation in dB} = \text{Coupling Factor in dB} + \text{Directivity in dB} \]
Thus, isolation provides an overall indication of how well the input and isolated ports are separated. In an ideal directional coupler, the isolated port receives zero power, so the isolation becomes infinitely large.
Transmission Loss
The transmission loss describes the reduction in power along the main transmission path from the input port to the through port. If \(P_1\) is the input power at port 1 and \(P_2\) is the power delivered at port 2, the transmission loss is expressed as
\[ T(\mathrm{dB})=10\log_{10}\left(\frac{P_1}{P_2}\right) \]
In an ideal lossless directional coupler, the total available power is distributed between the through path and the coupled path, with no power dissipated inside the device. Therefore, the transmission loss is associated with the reduction in power available at the through port due to the power intentionally coupled to the secondary path. In a practical directional coupler, additional losses can also occur because of conductor loss, dielectric loss, and other imperfections in the physical structure.
Return Loss
Return loss indicates the amount of incident power that is reflected back because of impedance mismatch. When a directional coupler is properly matched, very little power is reflected at its ports. If \(P_i\) represents the incident power and \(P_r\) represents the reflected power, the return loss is given by
\[ R(\mathrm{dB})=10\log_{10}\left(\frac{P_i}{P_r}\right) \]
A high return loss corresponds to a small reflected power and therefore indicates good impedance matching. Since directional couplers are generally designed as matched microwave networks, a high return loss is desirable at the operating frequency range. Poor matching can increase reflections and can also affect the directivity and isolation of the coupler.
Relationship Between Coupling Factor, Directivity and Isolation
The main performance parameters of a directional coupler are closely related. The coupling factor determines how much of the input power is intentionally sampled at the coupled port, while directivity determines how effectively the coupler separates the desired coupled signal from the unwanted signal at the isolated port. Isolation combines these effects and indicates the overall separation between the input and isolated ports.
The important relationship is
\[ \boxed{ \text{Isolation} = \text{Coupling Factor} + \text{Directivity} } \]
The three quantities can also be expressed directly using the port powers as
\[ C=10\log_{10}\left(\frac{P_1}{P_4}\right) \]
\[ D=10\log_{10}\left(\frac{P_4}{P_3}\right) \]
Therefore,
\[ \begin{aligned} C+D &= 10\log_{10}\left(\frac{P_1}{P_4}\right) + 10\log_{10}\left(\frac{P_4}{P_3}\right)\\ &= 10\log_{10}\left(\frac{P_1}{P_3}\right) \end{aligned} \]
Hence,
\[ \boxed{ \text{Isolation} = 10\log_{10}\left(\frac{P_1}{P_3}\right) = C+D } \]
This relationship is useful because the isolation of a directional coupler can be understood from its coupling factor and directivity. For a given coupling factor, improving the directivity reduces the unwanted power at the isolated port and consequently improves the isolation.
Ideal and Practical Directivity
For an ideal directional coupler, the isolated port receives no power when the input signal is applied to the input port. Thus, \(P_3=0\). Since directivity is defined as the ratio of desired coupled power to unwanted isolated-port power, the ideal directivity is infinite.
\[ D=10\log_{10}\left(\frac{P_4}{P_3}\right) \]
For \(P_3=0\), the ideal directivity approaches infinity. This represents perfect directional behaviour, where the coupler completely separates the desired coupled signal from the unwanted reverse or isolated signal.
In a practical directional coupler, the isolated-port power is not exactly zero. Imperfect amplitude and phase balance, impedance mismatch, manufacturing tolerances, conductor and dielectric losses, and frequency-dependent effects prevent perfect cancellation at the isolated port. Consequently, practical directivity is finite. A well-designed practical directional coupler can typically achieve directivity in the range of approximately 30 to 35 dB, although the actual value depends on the construction and operating frequency.
Therefore, the ideal and practical cases can be summarized by the fact that an ideal coupler has zero power at the isolated port and infinite directivity, whereas a practical coupler has a small amount of unwanted power at the isolated port and therefore a finite directivity. The closer the practical device approaches the ideal condition, the better its ability to measure or sample power in one direction without significant interference from power travelling in the opposite direction.
Directional Coupler Parameters
The important parameters used to describe the performance of a directional coupler are coupling factor, directivity, isolation, transmission loss, and return loss. The coupling factor specifies the fraction of power sampled from the primary waveguide, directivity specifies the ability to distinguish the desired coupled signal from the unwanted signal, isolation specifies the separation between the input and isolated ports, transmission loss describes the power reduction along the main transmission path, and return loss describes the power reflected because of impedance mismatch.
- Coupling factor: Measures the ratio of input power to coupled-port power.
- Directivity: Measures the ratio of desired coupled power to unwanted isolated-port power.
- Isolation: Measures the separation between the input and isolated ports.
- Transmission loss: Measures the reduction of power between the input and through ports.
- Return loss: Measures the amount of incident power that is not reflected because of good impedance matching.
- Important relationship: \(\text{Isolation}=\text{Coupling Factor}+\text{Directivity}\).
- Ideal directivity: Infinite, because the isolated-port power is zero.
- Practical directivity: Finite because perfect cancellation and matching cannot be achieved in practice.
S-Matrix of Directional Coupler
The S-matrix of a directional coupler is used to describe the relationship between the incident and reflected or outgoing waves at its four ports. Since a directional coupler is a four-port microwave network, four incident waves and four outgoing waves are considered. The scattering matrix provides a convenient way to represent how power entering any one port is distributed among the remaining ports. The properties of the directional coupler, such as matching, reciprocity, symmetry, coupling, isolation, and transmission, can be expressed directly in terms of its S-parameters.
General 4-Port S-Matrix
Consider a four-port directional coupler with incident waves \(a_1,a_2,a_3,a_4\) and outgoing waves \(b_1,b_2,b_3,b_4\). The relationship between the incident and outgoing waves is written as
2080 Bhadra (BEI) “Explain the working principle of a two-hole directional coupler with a neat diagram. Derive its scattering parameters.”
\[ \begin{bmatrix} b_1\\ b_2\\ b_3\\ b_4 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} & S_{13} & S_{14}\\ S_{21} & S_{22} & S_{23} & S_{24}\\ S_{31} & S_{32} & S_{33} & S_{34}\\ S_{41} & S_{42} & S_{43} & S_{44} \end{bmatrix} \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4 \end{bmatrix} \]
or, in compact form,
\[ \boxed{[b]=[S][a]} \]
Here, \(S_{ij}\) represents the ratio of the outgoing wave at port \(i\) to the incident wave at port \(j\), with all other ports terminated in matched loads. Thus, the column number identifies the excited port, while the row number identifies the port at which the resulting outgoing wave is observed.
Matched-Port Condition
A directional coupler is normally designed as a matched four-port network. When a particular port is perfectly matched, an incident wave entering that port produces no reflected wave at the same port. Therefore, the corresponding reflection coefficient is zero. For all four ports to be matched, the diagonal elements of the S-matrix must be zero.
Hence, the matched-port condition is
\[ \boxed{ S_{11}=S_{22}=S_{33}=S_{44}=0 } \]
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 means that the directional coupler has no reflection at any port under the ideal matched condition. The remaining off-diagonal S-parameters describe transmission between different ports.
Reciprocity
A directional coupler is a reciprocal network because it is constructed from passive reciprocal transmission-line or waveguide structures. Reciprocity means that the transmission characteristics between two ports are the same when the direction of excitation is reversed. Therefore, the S-matrix of a reciprocal network is symmetric.
The reciprocity condition is
\[ \boxed{S_{ij}=S_{ji}} \]
For the four-port directional coupler, this gives
\[ S_{12}=S_{21} \]
\[ S_{13}=S_{31} \]
\[ S_{14}=S_{41} \]
\[ S_{23}=S_{32} \]
\[ S_{24}=S_{42} \]
\[ S_{34}=S_{43} \]
Therefore, after applying both the matched-port and reciprocity conditions, the matrix can be written as
\[ [S]= \begin{bmatrix} 0 & S_{12} & S_{13} & S_{14}\\ S_{12} & 0 & S_{23} & S_{24}\\ S_{13} & S_{23} & 0 & S_{34}\\ S_{14} & S_{24} & S_{34} & 0 \end{bmatrix} \]
Symmetry of the Directional Coupler
The physical symmetry of a directional coupler introduces additional relationships between its S-parameters. A conventional directional coupler consists of a primary waveguide and a secondary waveguide. Ports 1 and 2 belong to the primary waveguide, while ports 3 and 4 belong to the secondary waveguide. Because of the symmetric construction of the coupled region, corresponding transmission paths have equal magnitudes.
The S-parameters can therefore be grouped according to the physical paths between the ports. The transmission between the two ports of the primary and secondary waveguides has one common magnitude, the coupling paths have another magnitude, and the unwanted isolated paths have a third magnitude.
Through, Coupled and Isolated Ports
To understand the S-matrix of a directional coupler, the four ports can be classified according to their function. Consider an excitation applied at port 1. Ports 1 and 2 form the primary waveguide, while ports 3 and 4 form the secondary waveguide.
The major portion of the input power travels through the primary waveguide from port 1 to port 2. Therefore, port 2 is called the through port. The portion of the input power intentionally transferred to the secondary waveguide appears at the coupled port. With the port numbering considered here, port 4 is the coupled port.
The remaining secondary port, port 3, is called the isolated port. Ideally, no power reaches this port when port 1 is excited. This occurs because the coupled waves arriving at the isolated port are equal in magnitude and opposite in phase, resulting in cancellation.
Thus, for excitation at port 1:
- Port 1: Input port.
- Port 2: Through port.
- Port 4: Coupled port.
- Port 3: Isolated port.
Under the ideal directional-coupler condition, the important transmission coefficients for excitation at port 1 are therefore \(S_{21}\), \(S_{41}\), and \(S_{31}\). Here, \(S_{21}\) represents transmission from port 1 to the through port, \(S_{41}\) represents coupling from port 1 to the coupled port, and \(S_{31}\) represents unwanted transmission from port 1 to the isolated port.
Relationships Between S-Parameters
For a symmetric directional coupler, the S-parameters can be grouped into three important categories. Let \(a\), \(b\), and \(g\) represent the magnitudes associated with the through, coupled, and isolated transmission paths, respectively. The symmetry and reciprocity of the network give the following relationships.
The through-path parameters are
\[ \boxed{ a=|S_{12}|=|S_{21}|=|S_{34}|=|S_{43}| } \]
The coupling-path parameters are
\[ \boxed{ b=|S_{14}|=|S_{41}|=|S_{23}|=|S_{32}| } \]
The isolated-path parameters are
\[ \boxed{ g=|S_{13}|=|S_{31}|=|S_{24}|=|S_{42}| } \]
These relationships show that the corresponding paths have equal S-parameter magnitudes because of the physical symmetry and reciprocal nature of the directional coupler. The three quantities \(a\), \(b\), and \(g\) therefore provide a convenient way of describing the through, coupled, and isolated transmission characteristics of the four-port network.
Definitions of \(a\), \(b\), and \(g\)
The parameter \(a\) represents the magnitude of the transmission coefficient along the through path. For excitation at port 1, the through transmission is represented by \(S_{21}\). Therefore, \(a\) describes the major portion of the power transmitted through the primary waveguide.
The parameter \(b\) represents the magnitude of the coupling coefficient. For excitation at port 1, the coupled transmission is represented by \(S_{41}\). Thus, \(b\) determines the amount of power sampled from the primary waveguide and transferred to the secondary waveguide.
The parameter \(g\) represents the magnitude of the unwanted transmission toward the isolated port. For excitation at port 1, this is represented by \(S_{31}\). In an ideal directional coupler, the isolated port receives no power, so
\[ \boxed{g=0} \]
In a practical directional coupler, \(g\) is small but not exactly zero because perfect cancellation cannot be achieved over a finite frequency range.
Coupling in Terms of S-Parameters
The coupling factor can be expressed directly in terms of the magnitude of the coupling coefficient. For excitation at port 1, the coupled-port transmission coefficient is \(S_{41}\). Since \(b=|S_{41}|\), the coupling factor is
\[ \boxed{ C=-20\log_{10}|S_{41}| } \]
Using \(b=|S_{41}|\), this can also be written as
\[ \boxed{ C=-20\log_{10}b } \]
The same coupling relationship applies to the corresponding reciprocal coupling path, so \(S_{14}\), \(S_{23}\), and \(S_{32}\) have the same magnitude under the symmetry conditions.
Transmission Through the Main Line
The transmission coefficient between the input port and the through port is \(S_{21}\). Since \(a=|S_{21}|\), the attenuation or transmission loss associated with the through path can be expressed as
\[ \boxed{ T=-20\log_{10}|S_{21}| } \]
or, using \(a\),
\[ \boxed{ T=-20\log_{10}a } \]
This parameter indicates how much the signal is reduced while travelling from the input port to the through port. For an ideal lossless coupler, the power not appearing at the through port is transferred to the coupled path rather than dissipated as loss.
Isolation and Directivity in Terms of S-Parameters
For excitation at port 1, \(S_{31}\) represents the transmission from the input port to the isolated port. Therefore, the magnitude of the unwanted isolated signal is \(g=|S_{31}|\). The isolation is consequently related to the ratio between the input power and the power reaching the isolated port.
\[ \boxed{ I=-20\log_{10}|S_{31}|^2 } \]
Since power is proportional to the square of the magnitude of the scattering parameter, this is equivalently written as
\[ \boxed{ I=-20\log_{10}|S_{31}| } \]
For the standard power-based definition, the isolation is obtained from
\[ \boxed{ I=10\log_{10}\left(\frac{P_1}{P_3}\right) } \]
The directivity compares the desired coupled signal with the unwanted signal at the isolated port. Therefore, for excitation at port 1, directivity is expressed as
\[ \boxed{ D=20\log_{10} \left( \frac{|S_{41}|}{|S_{31}|} \right) } \]
Using the parameters \(b\) and \(g\), the same expression becomes
\[ \boxed{ D=20\log_{10}\left(\frac{b}{g}\right) } \]
This expression clearly shows that directivity improves when the desired coupled signal becomes large compared with the unwanted signal at the isolated port. In an ideal directional coupler, \(g=0\), so the directivity becomes infinite.
Important S-Parameter Conditions
The important properties of an ideal directional coupler can therefore be represented through a small set of S-parameter conditions. Since the coupler is matched, reciprocal, and symmetric, the diagonal S-parameters are zero, corresponding transmission parameters occur in equal-magnitude pairs, and the coupling paths have corresponding equal magnitudes. The isolated path is ideally zero.
The main conditions are
\[ \boxed{ S_{11}=S_{22}=S_{33}=S_{44}=0 } \]
\[ \boxed{ S_{ij}=S_{ji} } \]
\[ \boxed{ |S_{12}|=|S_{21}|=|S_{34}|=|S_{43}|=a } \]
\[ \boxed{ |S_{14}|=|S_{41}|=|S_{23}|=|S_{32}|=b } \]
\[ \boxed{ |S_{13}|=|S_{31}|=|S_{24}|=|S_{42}|=g } \]
For an ideal directional coupler,
\[ \boxed{g=0} \]
and therefore the isolated-port transmission vanishes. These relationships form the basis for obtaining the complete scattering matrix of the directional coupler. The next stage is to apply the lossless condition and the required phase relationships between the through and coupled signals to determine the actual S-matrix elements.
Branch-Line Coupler S-Matrix and Important Design Formulae
The branch-line coupler is a four-port microwave network designed to divide the input power between two output ports with a specific amplitude and phase relationship. For an ideal equal power dividing coupler, the two output ports receive equal power and are separated by a \(90^\circ\) phase difference. The scattering matrix provides a convenient way to represent this behaviour because it directly relates the incident waves at the four ports to the outgoing waves. The S-matrix of the ideal coupler can be obtained by applying the conditions of matching, reciprocity, symmetry, equal power division, isolation, and the required \(90^\circ\) phase relationship.
General S-Matrix of a Branch-Line Coupler
Consider a four-port branch-line coupler with incident waves \(a_1,a_2,a_3,a_4\) and outgoing waves \(b_1,b_2,b_3,b_4\). The general scattering 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 four-port 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 at port \(i\) to the incident wave at port \(j\), with all other ports terminated in matched loads. The individual elements therefore describe reflection, transmission, coupling, and isolation between the four ports.
How the S-Matrix Is Obtained
The final S-matrix is obtained by progressively applying the physical properties of the branch-line coupler. Instead of assigning values to all the S-parameters immediately, the general matrix is first simplified using the matched condition, reciprocity, and symmetry. The equal power division condition then determines the magnitudes of the output coefficients, while the isolated-port condition eliminates one of the transmission paths. Finally, the \(90^\circ\) phase relationship determines the relative phase between the two output signals.
Matched Condition
An ideal branch-line coupler is matched at all four ports. Therefore, when a wave enters any port, no wave is reflected back to that same port. The diagonal elements of the scattering matrix, which represent the reflection coefficients, are consequently zero.
\[ \boxed{ S_{11}=S_{22}=S_{33}=S_{44}=0 } \]
The general matrix therefore becomes
\[ [S]= \begin{bmatrix} 0&S_{12}&S_{13}&S_{14}\\ S_{21}&0&S_{23}&S_{24}\\ S_{31}&S_{32}&0&S_{34}\\ S_{41}&S_{42}&S_{43}&0 \end{bmatrix} \]
Reciprocity
The branch-line coupler is a passive reciprocal network. For a reciprocal microwave network, transmission from port \(i\) to port \(j\) is equal to transmission from port \(j\) to port \(i\). Therefore, the S-matrix is symmetric and
\[ \boxed{S_{ij}=S_{ji}} \]
Thus,
\[ S_{12}=S_{21},\qquad S_{13}=S_{31},\qquad S_{14}=S_{41} \]
\[ S_{23}=S_{32},\qquad S_{24}=S_{42},\qquad S_{34}=S_{43} \]
Applying reciprocity gives
\[ [S]= \begin{bmatrix} 0&S_{12}&S_{13}&S_{14}\\ S_{12}&0&S_{23}&S_{24}\\ S_{13}&S_{23}&0&S_{34}\\ S_{14}&S_{24}&S_{34}&0 \end{bmatrix} \]
Symmetry
The physical symmetry of the coupler produces equal transmission characteristics for corresponding paths. Therefore, the magnitudes of the S-parameters associated with equivalent paths are equal. This symmetry allows the number of independent S-parameters to be reduced considerably and ensures that the same coupling behaviour is obtained when the coupler is operated from the corresponding ports.
For the ideal equal power dividing configuration, the two output paths corresponding to the through and coupled ports have equal magnitudes. The symmetry of the structure also ensures that the same relationships apply when the direction of excitation is reversed.
Equal Power Division
Suppose port 1 is excited while all other ports are terminated in matched loads. In an equal power dividing coupler, the input power is divided equally between the two output ports. If the output waves are represented by \(S_{21}\) and \(S_{41}\), equal power division requires
\[ |S_{21}|^2=|S_{41}|^2 \]
For a lossless coupler, the total output power must equal the input power. Therefore,
\[ |S_{21}|^2+|S_{41}|^2=1 \]
Since the two output powers are equal,
\[ |S_{21}|=|S_{41}|=\frac{1}{\sqrt{2}} \]
Thus, each output receives one-half of the input power.
\[ \boxed{ P_2=P_4=\frac{P_1}{2} } \]
Isolated Port Condition
A fundamental property of the directional or hybrid coupling action is that one port is isolated from the excited port. For excitation at port 1, let port 3 be the isolated port. Ideally, no power appears at port 3. Therefore,
\[ \boxed{S_{31}=0} \]
By reciprocity,
\[ \boxed{S_{13}=0} \]
The corresponding isolated paths are also zero according to the symmetry of the ideal coupler. This isolation is produced by cancellation of the unwanted coupled waves at the isolated port.
\(90^\circ\) Phase Relationship
Equal power division determines the magnitudes of the two output signals, but it does not by itself determine their relative phase. For an ideal quadrature hybrid, the two output signals have a phase difference of \(90^\circ\). Therefore, if one output coefficient is chosen as a real quantity, the other output coefficient contains a factor of \(j\).
For excitation at port 1, the two output coefficients can therefore be represented as
\[ S_{21}=-\frac{j}{\sqrt{2}} \]
and
\[ S_{41}=-\frac{1}{\sqrt{2}} \]
The particular signs depend on the selected port numbering and reference-plane convention. What is physically important is that the two output signals have equal magnitude and a \(90^\circ\) phase difference.
\[ \boxed{ \left|\frac{S_{21}}{S_{41}}\right|=1 } \]
and
\[ \boxed{ \angle S_{21}-\angle S_{41}=\pm90^\circ } \]
Final S-Matrix of the Ideal Branch-Line Coupler
Applying the matched condition, reciprocity, symmetry, equal power division, isolated-port condition, and \(90^\circ\) phase relationship gives the ideal four-port scattering matrix. For one common port numbering and reference-plane convention, the matrix is
\[ \boxed{ [S]=\frac{1}{\sqrt{2}} \begin{bmatrix} 0&-j&0&-1\\ -j&0&-1&0\\ 0&-1&0&-j\\ -1&0&-j&0 \end{bmatrix} } \]
The signs and phase factors can appear differently in other textbooks because they depend on the selected port numbering and reference-plane convention. However, the important physical properties remain unchanged: all ports are matched, the network is reciprocal, the required port is isolated, the two output ports receive equal power, and the two output signals have a \(90^\circ\) phase difference.
Physical Meaning of the S-Matrix Elements
The physical meaning of the matrix can be understood by exciting one port at a time. Consider an excitation at port 1. The first column of the S-matrix describes all outgoing waves produced by this excitation.
\[ \begin{bmatrix} b_1\\ b_2\\ b_3\\ b_4 \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 0\\ -j\\ 0\\ -1 \end{bmatrix} a_1 \]
The element \(S_{11}=0\) means that port 1 is matched and no power is reflected back to port 1. The element \(S_{21}=-j/\sqrt{2}\) represents transmission from port 1 to port 2, with one-half of the input power appearing at port 2. The element \(S_{31}=0\) indicates that port 3 is isolated from port 1. Finally, \(S_{41}=-1/\sqrt{2}\) represents coupling from port 1 to port 4, with one-half of the input power appearing at port 4.
Thus, the first column clearly demonstrates the basic operation of the equal power dividing coupler: the input power is divided equally between two output ports, one port is isolated, and the two output signals have a quadrature phase relationship.
Even and Odd Mode Results Used for Design
The even and odd mode method is used to analyse and design coupled transmission-line sections without repeating the complete derivation of the field distributions. A pair of coupled lines can be separated into two independent modes: the even mode, in which the two conductors have equal voltages with the same phase, and the odd mode, in which the conductors have equal magnitudes with opposite phase.
The corresponding characteristic impedances are denoted by \(Z_0^e\) and \(Z_0^o\). For a system having characteristic impedance \(Z_0\), the required even and odd mode impedances are related to the midband coupling coefficient \(C_0\) by
\[ \boxed{ Z_0^e = Z_0 \sqrt{\frac{1+C_0}{1-C_0}} } \]
and
\[ \boxed{ Z_0^o = Z_0 \sqrt{\frac{1-C_0}{1+C_0}} } \]
These equations provide the required even and odd mode impedances once the desired coupling and system impedance are known.
The product of the even and odd mode impedances satisfies
\[ \boxed{ Z_0^e Z_0^o=Z_0^2 } \]
This relationship is particularly useful in the design of a symmetric coupled-line structure because the required physical dimensions can be selected to obtain the required even and odd mode characteristic impedances.
Coupling Coefficient \(C_0\)
The coupling coefficient specifies the amount of signal coupled from one transmission line to the other at the centre or midband frequency. If \(a_1\) is the incident wave at the input and \(b_4\) is the coupled output wave, the midband coupling coefficient can be written as
\[ \boxed{ C_0=\frac{b_4}{a_1} } \]
For the coupled-line structure, the same coefficient can be expressed in terms of the even and odd mode characteristic impedances as
\[ \boxed{ C_0= \frac{Z_0^e-Z_0^o} {Z_0^e+Z_0^o} } \]
The magnitude of \(C_0\) determines the strength of coupling. A larger difference between the even and odd mode impedances produces stronger coupling, whereas a smaller difference produces weaker coupling.
Transmission and Coupling Coefficients
For a coupled transmission-line directional coupler, the through coefficient is associated with the transmission from the input port to the through port, while the coupling coefficient is associated with transmission from the input port to the coupled port. Using the S-parameters, these quantities can be represented as
\[ \boxed{ T=S_{21} } \]
and
\[ \boxed{ C=S_{41} } \]
The corresponding magnitudes determine the power delivered to the through and coupled ports. The coupling factor in decibels is therefore
\[ \boxed{ C(\mathrm{dB}) = -20\log_{10}|S_{41}| } \]
Similarly, the transmission loss of the main path can be written as
\[ \boxed{ T(\mathrm{dB}) = -20\log_{10}|S_{21}| } \]
Directivity of the Coupler
Directivity measures the ability of the coupler to distinguish the desired coupled signal from the unwanted signal appearing at the isolated port. For excitation at port 1, \(S_{41}\) represents the desired coupled signal and \(S_{31}\) represents the unwanted isolated signal. Therefore, directivity is
\[ \boxed{ D(\mathrm{dB}) = -20\log_{10} \left( \frac{|S_{31}|}{|S_{41}|} \right) } \]
or equivalently
\[ \boxed{ D(\mathrm{dB}) = 20\log_{10} \left( \frac{|S_{41}|}{|S_{31}|} \right) } \]
For an ideal directional coupler, the isolated-port coefficient is zero:
\[ S_{31}=0 \]
Therefore, the ideal directivity is infinite. In a practical coupler, \(S_{31}\) is small but nonzero because of imperfect amplitude and phase balance, impedance mismatch, fabrication tolerances, and frequency-dependent effects.
Frequency Dependence of the Coupler
The coupling behaviour of a coupled-line directional coupler depends on the electrical length of the coupled section. The phase shift along a coupled section of physical length \(l\) is
\[ \boxed{ \theta=\beta l } \]
where \(\beta\) is the phase constant. Since
\[ \beta=\frac{2\pi}{\lambda_g} \]
the phase shift can also be written as
\[ \boxed{ \theta=\frac{2\pi l}{\lambda_g} } \]
or, using the phase velocity \(u\),
\[ \boxed{ \theta=\frac{2\pi f l}{u} } \]
Thus, for a fixed physical coupling length, the electrical length changes with frequency. Consequently, the amount of power coupled to the secondary line and the phase relationship between the output signals also vary with frequency. This is why a directional coupler is normally designed around a specified centre or midband frequency.
For maximum coupling in the commonly used quarter-wavelength coupled section, the required electrical length at the centre frequency is
\[ \boxed{ l=\frac{\lambda_{g0}}{4} } \]
where \(\lambda_{g0}\) is the guided wavelength at the centre frequency. The corresponding electrical length is
\[ \boxed{ \theta=\frac{\pi}{2} } \]
More generally, the coupled voltage reaches a maximum when the electrical length is an odd multiple of \(\pi/2\), so that
\[ \boxed{ \theta=\frac{(2n+1)\pi}{2} } \]
where \(n=0,1,2,\ldots\). The minimum practical coupling length corresponding to the centre-frequency condition is therefore \(\lambda_{g0}/4\).
Final Design Relationships
The important design relationships for the coupled-line directional coupler can be summarized using the desired system impedance \(Z_0\), midband coupling coefficient \(C_0\), and the even and odd mode characteristic impedances. These relationships provide the connection between the required electrical performance and the physical coupled-line structure.
The midband coupling coefficient is
\[ \boxed{ C_0= \frac{Z_0^e-Z_0^o} {Z_0^e+Z_0^o} } \]
The even mode characteristic impedance is
\[ \boxed{ Z_0^e= Z_0\sqrt{\frac{1+C_0}{1-C_0}} } \]
The odd mode characteristic impedance is
\[ \boxed{ Z_0^o= Z_0\sqrt{\frac{1-C_0}{1+C_0}} } \]
and their product is
\[ \boxed{ Z_0^eZ_0^o=Z_0^2 } \]
The coupling factor is
\[ \boxed{ C(\mathrm{dB}) = -20\log_{10}|S_{41}| } \]
The through transmission loss is
\[ \boxed{ T(\mathrm{dB}) = -20\log_{10}|S_{21}| } \]
The directivity is
\[ \boxed{ D(\mathrm{dB}) = 20\log_{10} \left( \frac{|S_{41}|}{|S_{31}|} \right) } \]
Finally, the centre-frequency coupling section is commonly selected as a quarter guided wavelength:
\[ \boxed{ l=\frac{\lambda_{g0}}{4} } \]
These equations connect the required coupling performance with the even and odd mode properties of the coupled transmission lines. Together with the ideal S-matrix, they provide the fundamental analytical relationships required for understanding and designing a branch-line or coupled-line microwave coupler.