3-Port Directional Coupler
3-Port Directional Coupler: Structure and 3 dB Coupler Configuration
A 3-port directional coupler is a microwave network used to obtain a controlled portion of microwave power from a transmission path while allowing the remaining power to continue toward the output port. The structure is based on directional coupling between transmission paths, where the input power is divided into a through component and a coupled component. A 3 dB coupler can be used as the basic coupling arrangement when equal power division between the two output paths is required. In this arrangement, the directional behavior of the coupler determines which port receives the coupled power and which path is isolated from the input.
2082 Bhadra (BEI) Design a model of a three-port network and define its characteristic parameters. Prove that a lossless, reciprocal $3$-port junction cannot be matched simultaneously at all ports.”
Structure of 3-Port Directional Coupler
The structure of a 3-port directional coupler can be understood from the conventional 4-port directional coupler. A directional coupler normally consists of four ports arranged so that one port acts as the input port, one as the through port, one as the coupled port, and one as the isolated port. When the isolated port is terminated internally with a matched load, it is no longer available as an external port. The remaining three accessible ports form the 3-port directional coupler. Therefore, the three-port structure is obtained by using the directional coupling arrangement of a four-port network and terminating its isolated port in the characteristic impedance.
In the three-port configuration, the externally accessible ports can be identified as the input port, through port, and coupled port. When microwave power is applied at the input port, most or a controlled portion of the power travels toward the through port, while a specified fraction of the power is coupled to the coupled port. The internal termination absorbs the power that would otherwise appear at the isolated port. This arrangement allows the device to operate externally as a three-port network while retaining the directional coupling behavior of the original coupler.
Use of 3 dB Coupler in 3-Port Directional Coupler
A 3 dB coupler is particularly important when equal power division is required. A 3 dB coupling condition means that the input power is divided equally between two relevant output paths, apart from the phase relationship imposed by the coupler. Since power is proportional to the square of the magnitude of the wave amplitude, equal power division gives the magnitude relationship
\[ |S|=\frac{1}{\sqrt{2}} \]Thus, when an input signal is applied to a 3 dB coupler, the two corresponding output waves have equal magnitudes. The two waves also possess a definite phase relationship determined by the physical configuration of the coupler. For a quadrature type 3 dB coupler, the two output waves have a phase difference of \(90^\circ\). This equal-amplitude and controlled-phase behavior makes the 3 dB coupler suitable for forming directional microwave networks.
In a directional arrangement, the coupled transmission paths are designed so that the waves travelling toward one port reinforce each other, while the waves travelling toward the isolated direction cancel each other. This phase-dependent combination of coupled waves is the fundamental reason for directional operation. Therefore, the 3 dB coupler does not simply divide power equally; its structure and phase relationship determine how the divided power is directed toward the required ports.
Formation of the Three-Port Configuration
Consider the original four-port directional coupler with ports numbered as input, through, isolated, and coupled ports. The input port is connected to the microwave source, the through port carries the main transmitted power, and the coupled port provides a controlled sample of the input power. The fourth port is the isolated port. To obtain a three-port configuration, this isolated port is terminated by a matched load having the characteristic impedance of the network. Since the termination absorbs the wave incident on this port, there is no externally accessible connection at that location, leaving three external ports.
For the internal matched termination, the incident wave at the terminated port is taken as zero because there is no external source connected to that port. In scattering-parameter notation, if the internally terminated fourth port is numbered as port 4, the termination condition is written as
\[ a_4=0 \]The corresponding reflected wave from the matched termination is also zero because the load is matched to the characteristic impedance. Therefore, the internal termination does not introduce a reflected wave back into the directional coupler. The three remaining ports can then be treated as the externally accessible ports of the 3-port network.
Power Flow in the 3-Port Directional Coupler
When power is applied to the input port, the structure permits power to travel toward the through port and also provides a controlled coupled output at the coupled port. The relative amount of power appearing at these ports depends on the coupling arrangement. In the special case of a 3 dB coupler, the relevant output powers are equal because the input power is divided equally between the two output paths.
The directional nature of the structure is produced by the phase relationship between the waves coupled through the interaction region. Waves travelling toward one direction combine constructively, while waves travelling toward the unwanted direction combine destructively. As a result, the coupler can provide a useful output at the coupled port while suppressing the unwanted isolated path. This behavior is the basic structural principle behind directional power sampling.
Role of the Internal Matched Termination
The internal matched termination is an important part of the three-port structure because it removes the isolated port from external operation without introducing an unwanted reflection. In the original four-port directional coupler, the isolated port is designed to receive ideally no power from the input port. In practice, however, a small amount of power can reach this port because of imperfect cancellation, manufacturing tolerances, and other nonideal effects. The matched termination absorbs this power and prevents it from being reflected back into the network.
Consequently, the externally accessible device has only three ports, while the internal termination maintains the proper termination condition for the original isolated port. The structure can therefore be represented externally as a three-port directional coupler while its directional behavior originates from the underlying four-port coupling arrangement.
3 dB Coupler and Equal Power Division
For a 3 dB coupler, equal power division is the defining feature of the coupling arrangement. If the input power is \(P_{\mathrm{in}}\), the ideal equal division gives one half of the input power to each of the two relevant output paths. Thus, neglecting losses,
\[ P_{\mathrm{out1}}=\frac{P_{\mathrm{in}}}{2} \] \[ P_{\mathrm{out2}}=\frac{P_{\mathrm{in}}}{2} \]The corresponding amplitude ratio for each output is
\[ \left|\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}\right| = \frac{1}{\sqrt{2}} \]This is why a 3 dB coupler is commonly used when equal power splitting and a known phase relationship are required. In a directional configuration, the equal-amplitude coupled waves are combined according to their relative phase so that the desired output direction is reinforced and the unwanted direction is suppressed.
Physical Interpretation of the 3-Port Structure
The complete structure can therefore be viewed as a directional coupling section having four ports, with the isolated port terminated internally by a matched load. The remaining three ports provide the external input, through, and coupled connections. When a 3 dB coupling arrangement is used, the coupling section produces equal-amplitude waves with a defined phase relationship. The physical geometry of the coupling region determines the phase and direction in which these waves combine.
The important structural concept is that the 3-port directional coupler is not formed by simply removing one port from a four-port coupler. Instead, the fourth port is properly terminated with a matched load. This maintains the required boundary condition and prevents unwanted reflections. The 3 dB coupler provides the equal power division and controlled phase relationship, while the internal termination converts the externally accessible arrangement into a three-port network.
2078 Baishakh (BEX) Properties of 3-Port Directional Coupler“Prove using S-matrix properties why a $3$-port directional coupler cannot be lossless, reciprocal, and matched simultaneously. How do 4-port couplers overcome this restriction?”
A 3-port directional coupler has three externally accessible ports through which microwave signals can enter or leave the network. These ports are generally identified according to their function as the input port, through port, and coupled port. The input port receives the microwave signal, the through port carries the main transmitted power, and the coupled port provides a controlled sample of the power from the main transmission path. The fourth port of the original directional coupler is internally terminated with a matched load, so it does not appear as an external port. Thus, the three-port device retains the directional coupling behavior of the original four-port network while providing only three accessible connections.
Matched Port Condition
For a properly designed directional coupler, the accessible ports are normally designed to operate with the characteristic impedance of the transmission system. When a port is terminated in its characteristic impedance, the termination does not produce a reflection. In scattering parameter notation, a perfectly matched port has a reflection coefficient of zero. Therefore, the corresponding diagonal element of the S-matrix becomes zero. For a general 3-port network, if all three externally accessible ports are matched, the condition can be written as
\[ S_{11}=S_{22}=S_{33}=0 \]This condition is useful when developing the S-matrix because it establishes the absence of reflected waves at the corresponding ports. However, for a three-port network, matching all three ports simultaneously cannot be combined with ideal losslessness and reciprocity in the same way as for a four-port directional coupler. This limitation becomes important when the complete S-matrix is derived.
Directional Power Coupling
The main property of a directional coupler is its ability to transfer a controlled amount of microwave power from the main transmission path to another port while suppressing coupling in the opposite direction. When a signal is applied to the input port, power is primarily transmitted toward the through port, while a specified portion is coupled to the coupled port. The directional behavior is obtained from the phase relationship between the waves produced by the coupling structure. The waves combine constructively in the desired direction and destructively in the unwanted direction.
Because of this directional behavior, the coupled port can be used to sample the forward travelling power without significantly disturbing the main transmission path. This makes the directional coupler useful for power monitoring, measurement, and signal sampling in microwave systems. The amount of power transferred to the coupled port is determined by the coupling level of the coupler.
Through Power and Coupled Power
When microwave power is applied at the input port, the portion that continues along the main transmission path appears at the through port. This is referred to as through power. Another portion of the input power is transferred through the coupling mechanism and appears at the coupled port. The relative levels of through power and coupled power depend on the design of the directional coupler.
For a 3 dB coupler, the two relevant output paths have equal power under ideal conditions. If the input power is \(P_{\mathrm{in}}\), the equal power division is given by
\[ P_{\mathrm{through}}=\frac{P_{\mathrm{in}}}{2} \] \[ P_{\mathrm{coupled}}=\frac{P_{\mathrm{in}}}{2} \]The corresponding magnitude of the transmission coefficient for each equal-power path is
\[ |S|=\frac{1}{\sqrt{2}} \]Thus, a 3 dB coupler provides equal power division between the relevant output paths while maintaining the phase relationship required by the coupling structure.
Isolated Path and Internal Termination
In the original four-port directional coupler, one port is designated as the isolated port because ideally no power is coupled to that port from the input. In the three-port configuration, this isolated port is not externally accessible. Instead, it is terminated internally using a matched load. The internal termination absorbs any wave reaching that port and prevents the wave from being reflected back into the coupler.
The matched internal termination can be represented by the condition
\[ a_4=0 \]when the isolated port of the original four-port network is numbered as port 4. Since the termination is matched, the reflected wave from the load is also zero. This internal termination is what allows the original four-port directional coupling structure to be represented externally as a three-port network.
3 dB Equal Power Division
A 3 dB coupler is characterized by equal division of power between the two relevant output paths. A power ratio of 3 dB corresponds to one half of the input power in each output path for an ideal lossless device. Therefore, the amplitude of each output wave is \(1/\sqrt{2}\) times the input wave amplitude.
\[ \frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}=\frac{1}{2} \] \[ \left|\frac{b}{a}\right|=\frac{1}{\sqrt{2}} \]The 3 dB condition therefore provides the magnitude of the relevant S-parameters. The remaining information required to completely describe the network is the phase relationship between these waves. This phase relationship is essential for establishing the directional behavior of the coupler.
Phase Relationship
The output waves of a directional coupler are not determined only by their magnitudes. Their relative phase is also important because the directional operation results from constructive and destructive interference of the coupled waves. In a quadrature type 3 dB coupler, the two output waves have a phase difference of \(90^\circ\).
\[ \Delta\phi=90^\circ \]This phase difference allows the waves travelling toward one direction to combine appropriately while the corresponding waves travelling toward the unwanted direction are cancelled. Therefore, both the equal amplitude condition and the phase relationship must be considered when determining the S-parameters of a 3 dB directional coupler.
Reciprocity
A passive directional coupler constructed from reciprocal materials is generally a reciprocal network. Reciprocity means that the transmission characteristics between two ports are the same when the direction of excitation is reversed. In terms of scattering parameters, reciprocity gives
\[ S_{ij}=S_{ji} \]For a 3-port network, the reciprocity conditions are therefore
\[ S_{12}=S_{21} \] \[ S_{13}=S_{31} \] \[ S_{23}=S_{32} \]These relationships reduce the number of independent S-parameters that need to be determined during the matrix derivation.
Lossless Condition
An ideal directional coupler is assumed to be lossless, meaning that no microwave power is dissipated inside the coupling structure. The total output power must therefore equal the total input power. In terms of the scattering matrix, the lossless condition is expressed as
\[ [S][S]^\dagger=[I] \]where \([S]^\dagger\) represents the conjugate transpose of the scattering matrix and \([I]\) is the identity matrix. This condition means that the columns of the S-matrix must be orthonormal. Consequently, the magnitudes and phases of the S-parameters cannot be selected independently; they must satisfy the lossless-network condition.
Power Conservation
For a lossless network, the power entering the coupler must be completely accounted for by the power leaving the network. If a wave is incident at one port, the sum of the powers carried by all outgoing waves must equal the incident power. For an excitation at port \(i\), this condition can be expressed as
\[ \sum_{k=1}^{3}|S_{ki}|^2=1 \]For example, if port 1 is excited, the total normalized output power is
\[ |S_{11}|^2+|S_{21}|^2+|S_{31}|^2=1 \]This relation is particularly useful in the S-matrix derivation because once the matching and 3 dB conditions determine some of the S-parameters, power conservation can be used to determine the remaining magnitudes.
Limitation of an Ideal Matched Reciprocal Lossless 3-Port
An important property of a three-port network is that a network cannot simultaneously satisfy the conditions of being reciprocal, lossless, and perfectly matched at all three ports while also providing nonzero transmission between the ports. This is a fundamental limitation of a three-port scattering network and must be considered before attempting to construct its ideal S-matrix.
Suppose a reciprocal 3-port network is perfectly matched at all three ports. Its S-matrix would have zero diagonal elements, so it would have the form
\[ [S]= \begin{bmatrix} 0&S_{12}&S_{13}\\ S_{12}&0&S_{23}\\ S_{13}&S_{23}&0 \end{bmatrix} \]If the network is also lossless, the columns of this matrix must be mutually orthogonal. For example, the inner product of the first and second columns must be zero, giving
\[ S_{13}^{*}S_{23}=0 \]Similarly, orthogonality of the other pairs gives conditions requiring products of the transmission coefficients to vanish. This means that at least one of the corresponding transmission paths must be zero. Applying the same requirement to all three pairs makes it impossible for a nontrivial reciprocal, lossless, perfectly matched three-port network to have transmission between all ports.
This is why the three-port directional coupler should not be treated as an independent ideal matched reciprocal lossless network in the same manner as the original four-port directional coupler. Instead, the practical three-port arrangement is obtained by taking the corresponding four-port directional coupler and terminating its isolated port with a matched load. The internal termination supplies the fourth network condition while only three ports remain externally accessible.
Important Properties for S-Matrix Derivation
The properties discussed above provide the conditions required for deriving the scattering matrix. The three accessible ports establish the external network, while the internal matched termination represents the isolated port of the original four-port coupler. The 3 dB condition determines the equal power division and therefore the magnitude of the relevant transmission coefficients. The phase relationship determines the relative phase of the output waves, reciprocity relates \(S_{ij}\) to \(S_{ji}\), and the lossless condition imposes power conservation and orthogonality on the scattering matrix.
Therefore, the S-matrix derivation should not begin by assigning arbitrary values to the matrix elements. It should begin with the general \(3\times3\) S-matrix and then apply the physical properties of the network systematically. The apparent conflict between matching, reciprocity, and losslessness also explains why the internal termination of the original four-port directional coupler is essential when describing the device as a three-port network.
S-Matrix of 3-Port Directional Coupler and Its Derivation
The S-matrix of a 3-port directional coupler can be obtained by starting from the four-port directional coupler from which the three-port arrangement is formed. The fourth port, which is the isolated port of the original directional coupler, is terminated internally with a matched load. Therefore, the three-port network observed externally is not obtained simply by assuming an independent lossless three-port network. Instead, the internal termination must first be included in the four-port scattering equations, and the externally accessible three-port S-matrix can then be obtained by applying the termination condition. This approach is physically consistent because the internally terminated port can absorb power, meaning that the resulting external three-port network need not itself be lossless.
General S-Matrix of a 3-Port Network
For a three-port network, the incident and reflected travelling waves are related by the scattering matrix. Let \(a_1,a_2,a_3\) represent the incident waves at the three accessible ports and \(b_1,b_2,b_3\) represent the corresponding outgoing waves. The scattering relation is
\[ \begin{bmatrix} b_1\\ b_2\\ b_3 \end{bmatrix} = [S_3] \begin{bmatrix} a_1\\ a_2\\ a_3 \end{bmatrix} \]The general \(3\times3\) scattering matrix is
\[ [S_3]= \begin{bmatrix} S_{11}&S_{12}&S_{13}\\ S_{21}&S_{22}&S_{23}\\ S_{31}&S_{32}&S_{33} \end{bmatrix} \]Each element represents the response at one port due to excitation at another port. For example, \(S_{21}\) represents the outgoing wave at port 2 due to an incident wave at port 1 when all other external ports are terminated in their characteristic impedances.
Applying Reciprocity
A passive directional coupler constructed using reciprocal materials is a reciprocal network. Therefore, transmission from one port to another is the same when the direction of excitation is reversed. For the three-port network, reciprocity gives
\[ S_{12}=S_{21} \] \[ S_{13}=S_{31} \] \[ S_{23}=S_{32} \]Applying these relationships to the general matrix gives
\[ [S_3]= \begin{bmatrix} S_{11}&S_{12}&S_{13}\\ S_{12}&S_{22}&S_{23}\\ S_{13}&S_{23}&S_{33} \end{bmatrix} \]Thus, reciprocity reduces the number of independent S-parameters that must be determined during the derivation.
Applying the Matching Conditions
If an externally accessible port is perfectly matched, an incident wave at that port does not produce a reflected wave at the same port. The corresponding reflection coefficient is therefore zero. For a three-port network in which the external ports are matched, the diagonal elements satisfy
\[ S_{11}=S_{22}=S_{33}=0 \]The matrix would then have the form
\[ [S_3]= \begin{bmatrix} 0&S_{12}&S_{13}\\ S_{12}&0&S_{23}\\ S_{13}&S_{23}&0 \end{bmatrix} \]However, this condition must be interpreted carefully. A reciprocal three-port network cannot simultaneously be perfectly matched at all ports and be both lossless and nontrivially coupled. Therefore, the three-port directional coupler should be derived from its original four-port directional-coupler structure rather than by imposing all ideal four-port properties directly on this \(3\times3\) matrix.
Starting from the 4-Port Directional Coupler
Consider the ideal four-port 3 dB directional coupler from which the three-port configuration is obtained. For a quadrature type coupler, one possible port numbering and reference-plane convention gives the scattering matrix
\[ [S_4]= \frac{1}{\sqrt{2}} \begin{bmatrix} 0&-j&0&-1\\ -j&0&-1&0\\ 0&-1&0&-j\\ -1&0&-j&0 \end{bmatrix} \]The exact signs of the matrix elements can change with port numbering and reference-plane conventions, but the important physical properties remain the same: the relevant output paths have equal magnitude, the required phase relationship is maintained, and the isolated path is ideally suppressed.
For the four-port network, the scattering relation is
\[ \begin{bmatrix} b_1\\ b_2\\ b_3\\ b_4 \end{bmatrix} = [S_4] \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_4 \end{bmatrix} \]Internal Termination of the Isolated Port
To convert the four-port directional coupler into the three-port configuration, port 4 is terminated internally by a matched load. Since there is no external source connected to this matched termination, the incident wave entering the coupler from the termination is zero. Therefore, the termination condition is
\[ a_4=0 \]Because the termination is matched, the wave reaching the load is absorbed without reflection. The four-port equations can therefore be written as
\[ \begin{bmatrix} b_1\\ b_2\\ b_3 \end{bmatrix} = \begin{bmatrix} S_{11}&S_{12}&S_{13}\\ S_{21}&S_{22}&S_{23}\\ S_{31}&S_{32}&S_{33} \end{bmatrix} \begin{bmatrix} a_1\\ a_2\\ a_3 \end{bmatrix} \]The fourth incident wave has disappeared because \(a_4=0\). Therefore, the externally observed three-port S-matrix is the submatrix obtained by retaining the rows and columns corresponding to the three accessible ports.
Deriving the 3-Port S-Matrix
Using ports 1, 2, and 3 as the externally accessible ports and terminating port 4 internally, the required \(3\times3\) submatrix is obtained from the upper-left portion of the four-port matrix. Hence,
\[ [S_3]= \frac{1}{\sqrt{2}} \begin{bmatrix} 0&-j&0\\ -j&0&-1\\ 0&-1&0 \end{bmatrix} \]This is the externally observed scattering matrix for this particular port configuration and reference-plane convention. It shows that port 1 is coupled to port 2 with magnitude \(1/\sqrt{2}\), while there is no direct coupling between port 1 and port 3 in this configuration.
For excitation at port 1, we have
\[ a_1\neq0,\qquad a_2=0,\qquad a_3=0 \]The resulting output waves are
\[ b_1=0 \] \[ b_2=-\frac{j}{\sqrt{2}}a_1 \] \[ b_3=0 \]The remaining power associated with the original four-port coupling structure is absorbed by the internally terminated fourth port. Therefore, the externally observed three-port network does not satisfy the lossless condition by itself.
Why the 3-Port S-Matrix Is Not Lossless
The original four-port directional coupler is lossless, so its complete scattering matrix satisfies
\[ [S_4][S_4]^\dagger=[I] \]However, once one port is internally terminated by a matched load, power entering that internal port is absorbed by the termination. From the viewpoint of the three externally accessible ports, this absorbed power appears as a loss. Therefore, the reduced three-port matrix generally satisfies
\[ [S_3][S_3]^\dagger\neq[I] \]This does not mean that the original directional coupler is lossy. The coupling structure itself may be lossless; the apparent loss occurs because one of its ports has been connected to an internal matched load. The complete four-port network including the termination must be considered when applying the lossless power-conservation condition.
3 dB Condition in the S-Matrix
The 3 dB characteristic determines the magnitude of the relevant coupling coefficients. Equal power division requires one half of the available input power to appear in each corresponding output path. Since power is proportional to the square of the wave magnitude, the magnitude of each corresponding S-parameter is
\[ |S_{ij}|=\frac{1}{\sqrt{2}} \]For the matrix above, the nonzero transmission terms have magnitude
\[ |S_{12}|=|S_{21}|=\frac{1}{\sqrt{2}} \] \[ |S_{23}|=|S_{32}|=\frac{1}{\sqrt{2}} \]The factor \(1/\sqrt{2}\) corresponds to a power ratio of
\[ |S_{ij}|^2=\frac{1}{2} \]which represents equal power division in the corresponding 3 dB coupling paths.
Phase Relationship in the Matrix
The complex terms in the S-matrix also contain the phase information of the coupler. For example, the term \(-j/\sqrt{2}\) has a phase of \(-90^\circ\), while the term \(-1/\sqrt{2}\) has a phase of \(180^\circ\) relative to the chosen reference. These phase terms originate from the physical geometry and reference-plane definition of the 3 dB coupler.
Therefore, the S-matrix contains both the power division information and the phase information. The magnitude of an S-parameter determines the power transferred between the corresponding ports, while its phase determines the phase of the resulting travelling wave.
Physical Meaning of the S-Matrix Elements
The diagonal elements \(S_{11}\), \(S_{22}\), and \(S_{33}\) represent reflections at the three external ports. When the corresponding ports are matched, these terms are zero. The off-diagonal elements represent transmission or coupling between different ports. For example, \(S_{21}\) represents transmission from port 1 to port 2, while \(S_{12}\) represents transmission from port 2 to port 1. Reciprocity makes these two terms equal.
Similarly, \(S_{23}\) and \(S_{32}\) describe the reciprocal coupling between ports 2 and 3. The zero terms \(S_{13}\) and \(S_{31}\) indicate that there is no direct transmission between ports 1 and 3 for the selected port arrangement. The exact location of the zero and nonzero elements depends on how the external ports are selected from the original four-port directional coupler.
Final S-Matrix of the 3-Port Configuration
For the selected port numbering, with port 4 of the original 3 dB directional coupler internally terminated in a matched load, the externally observed three-port scattering matrix is
\[ \boxed{ [S_3]= \frac{1}{\sqrt{2}} \begin{bmatrix} 0&-j&0\\ -j&0&-1\\ 0&-1&0 \end{bmatrix} } \]This matrix demonstrates the main characteristics of the three-port arrangement: the external ports are represented by a \(3\times3\) matrix, reciprocity is maintained, the relevant coupling paths have equal magnitude corresponding to the 3 dB condition, and the phase relationship is represented by the complex coefficients. The internally terminated fourth port is not included as an external port, but its effect is essential because it absorbs the power directed toward that port.
S-Matrix Derivation
The derivation of the 3-port directional coupler S-matrix therefore follows a specific sequence. First, the general \(3\times3\) scattering relation is established. Reciprocity is then used to relate the forward and reverse transmission coefficients. The 3 dB condition establishes the magnitude of the relevant coupling coefficients as \(1/\sqrt{2}\), while the phase relationship determines the complex signs and phase factors. Most importantly, the three-port network is obtained from the corresponding four-port directional coupler by internally terminating the isolated port with a matched load, giving the condition \(a_4=0\). The externally observable S-matrix is then obtained from the four-port equations.
The complete four-port directional coupler remains the appropriate network for applying the ideal lossless condition. Once one port is terminated internally, the three-port network observed from outside can absorb power in that termination and therefore cannot generally be treated as an independently lossless reciprocal three-port network. This distinction is essential for obtaining a physically meaningful S-matrix and for understanding the relationship between the original 4-port directional coupler, the 3 dB coupler, and the resulting 3-port configuration.