Fundamental Network Proporties for S-Parameters
Fundamental Network Theorems Applied to S-Parameters
Once a microwave network has been characterized using S-parameters, several important physical properties can be identified directly from the scattering matrix. These properties arise from fundamental electromagnetic principles such as reciprocity, symmetry, and conservation of energy.
For a general two-port microwave network:
\[ \begin{bmatrix} b_1\\ b_2 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix} \begin{bmatrix} a_1\\ a_2 \end{bmatrix} \]
The characteristics of the network can be determined directly from the values and relationships among the S-parameters.
1. Reciprocity Theorem
A microwave network is said to be reciprocal when electromagnetic energy transfers equally in both directions.
If a signal travels from Port 1 to Port 2 with a certain transmission coefficient, then the same transmission coefficient exists from Port 2 back to Port 1.
Reciprocity is a direct consequence of Maxwell's equations for linear passive networks constructed from isotropic materials.
Mathematical Condition for Reciprocity
For a reciprocal network:
\[ [S] = [S]^T \]
where:
\[ [S]^T \]
is the transpose of the scattering matrix.
For a two-port network:
\[ \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix} = \begin{bmatrix} S_{11} & S_{21}\\ S_{12} & S_{22} \end{bmatrix} \]
Comparing corresponding elements gives:
\[ S_{12}=S_{21} \]
Physical Interpretation
The power transmitted from Port 1 to Port 2 is identical to the power transmitted from Port 2 to Port 1.
If:
\[ S_{21}=0.8 \]
then:
\[ S_{12}=0.8 \]
must also hold for a reciprocal network.
Examples of Reciprocal Networks
- Transmission lines
- Waveguides
- Passive attenuators
- Passive filters
- Directional couplers
- Power dividers
Active devices such as transistors and amplifiers generally do not satisfy reciprocity.
2. Symmetry Property of Microwave Networks
A microwave network is said to be symmetric when its electrical characteristics remain unchanged after interchanging the input and output ports.
In other words, if Port 1 and Port 2 are swapped physically, the network behaves exactly the same.
Symmetry is a stronger condition than reciprocity. While reciprocity only requires equal transmission in both directions, symmetry additionally requires identical reflection characteristics at both ports.
Physical Meaning of Symmetry
Consider a two-port microwave network.
If a signal is applied at Port 1, the network exhibits a certain input reflection coefficient and transmission coefficient.
Now suppose the network is physically reversed and the signal is applied at Port 2 instead.
For a symmetric network:
- The reflection seen at Port 2 must be identical to the reflection originally seen at Port 1.
- The transmission characteristics must remain unchanged.
- The electrical response is independent of signal direction.
Consequently, a user cannot determine which side of the network is the input and which side is the output based solely on electrical measurements.
General Two-Port S-Matrix
The scattering matrix of a general two-port network is:
\[ [S] = \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix} \]
where:
- \(S_{11}\) = Input reflection coefficient
- \(S_{22}\) = Output reflection coefficient
- \(S_{21}\) = Forward transmission coefficient
- \(S_{12}\) = Reverse transmission coefficient
Derivation of the Symmetry Condition
Assume the network is physically reversed.
Because the network remains electrically unchanged after swapping ports:
The reflection coefficient originally measured at Port 1 must equal the reflection coefficient measured at Port 2.
Therefore:
\[ S_{11}=S_{22} \]
Similarly, the transmission coefficient measured from Port 1 to Port 2 must equal the transmission coefficient measured from Port 2 to Port 1.
Thus:
\[ S_{21}=S_{12} \]
Combining both conditions:
\[ S_{11}=S_{22} \]
and
\[ S_{12}=S_{21} \]
These equations define a symmetric two-port network.
Symmetric S-Matrix Form
Substituting the symmetry conditions into the general S-matrix:
\[ [S] = \begin{bmatrix} S_{11} & S_{21}\\ S_{21} & S_{11} \end{bmatrix} \]
Only two independent parameters remain because the remaining elements are determined automatically by symmetry.
Relationship Between Reciprocity and Symmetry
It is important to distinguish reciprocity from symmetry.
A reciprocal network satisfies:
\[ S_{12}=S_{21} \]
However, reciprocity alone does not require:
\[ S_{11}=S_{22} \]
Therefore:
- Every symmetric network is reciprocal.
- Not every reciprocal network is symmetric.
Mathematically:
\[ \text{Symmetry} = \left\{ \begin{array}{l} S_{11}=S_{22}\\ S_{12}=S_{21} \end{array} \right. \]
whereas reciprocity only requires:
\[ S_{12}=S_{21} \]
Example 1: Uniform Transmission Line
A uniform transmission line having identical characteristic impedance and geometry at both ends is symmetric.
Its S-matrix satisfies:
\[ S_{11}=S_{22} \]
and
\[ S_{21}=S_{12} \]
Therefore the line behaves identically regardless of signal direction.
Example 2: Symmetric Bandpass Filter
Many microwave filters are designed using identical resonator structures on both sides of the network.
Because the input and output sections are mirror images:
\[ S_{11}=S_{22} \]
and
\[ S_{21}=S_{12} \]
making the filter both reciprocal and symmetric.
Example 3: Non-Symmetric Attenuator
Consider a resistive attenuator whose input and output impedances are different.
Even though it may still be reciprocal:
\[ S_{12}=S_{21} \]
the reflections at the two ports are unequal:
\[ S_{11}\neq S_{22} \]
Therefore the attenuator is reciprocal but not symmetric.
Importance of Symmetry in Microwave Design
Symmetric networks simplify design and analysis because identical performance is guaranteed regardless of signal direction.
Many practical microwave components are intentionally designed to be symmetric because:
- Input and output matching become identical.
- Manufacturing tolerances are easier to control.
- Performance is predictable in both directions.
- Analysis requires fewer independent parameters.
A microwave network is symmetric when its electrical behavior remains unchanged after interchanging its ports.
The mathematical conditions for symmetry are:
\[ S_{11}=S_{22} \]
and
\[ S_{12}=S_{21} \]
Symmetry automatically implies reciprocity, but reciprocity alone does not guarantee symmetry. Many transmission lines, passive filters, and microwave structures are intentionally designed to satisfy these conditions because they provide identical performance in both signal directions.
3. Lossless Networks
A microwave network is said to be lossless when no electromagnetic power is dissipated within the network.
In other words, the network contains no resistive losses and converts none of the incident RF power into heat.
All power entering the network must either be reflected back or transmitted to another port.
Therefore:
\[ P_{\text{incident}} = P_{\text{scattered}} \]
This principle is known as power conservation.
Incident and Scattered Power Waves
For an N-port microwave network:
\[ [a] = \begin{bmatrix} a_1\\ a_2\\ \vdots\\ a_N \end{bmatrix} \]
represents the incident power-wave vector and
\[ [b] = \begin{bmatrix} b_1\\ b_2\\ \vdots\\ b_N \end{bmatrix} \]
represents the scattered power-wave vector.
The S-parameter relationship is:
\[ [b]=[S][a] \]
Total Incident Power
The total incident power entering the network is:
\[ P_{\text{incident}} = [a]^{\dagger}[a] \]
where
\[ [a]^{\dagger} = ([a]^*)^T \]
is the Hermitian conjugate of the incident-wave vector.
Total Scattered Power
Similarly:
\[ P_{\text{scattered}} = [b]^{\dagger}[b] \]
Substituting:
\[ [b]=[S][a] \]
gives:
\[ P_{\text{scattered}} = ([S][a])^{\dagger} ([S][a]) \]
Applying the Hermitian operation:
\[ P_{\text{scattered}} = [a]^{\dagger} [S]^{\dagger} [S] [a] \]
Power Conservation Requirement
For a lossless network:
\[ P_{\text{incident}} = P_{\text{scattered}} \]
Therefore:
\[ [a]^{\dagger}[a] = [a]^{\dagger} [S]^{\dagger} [S] [a] \]
Rearranging:
\[ [a]^{\dagger} \Big( [I] - [S]^{\dagger}[S] \Big) [a] = 0 \]
This equation must be true for every possible incident-wave vector.
The only way this condition can always be satisfied is:
\[ [S]^{\dagger}[S] = [I] \]
Unitary Matrix Condition
Therefore, the scattering matrix of every lossless microwave network must satisfy:
\[ [S]^{\dagger}[S] = [I] \]
A matrix satisfying this condition is called a unitary matrix.
This is the fundamental mathematical condition for losslessness.
Physical Meaning of a Unitary S-Matrix
The unitary property guarantees that:
- No power is created inside the network.
- No power is destroyed inside the network.
- Total RF power remains constant.
- Only redistribution of power between ports occurs.
Consequently, the network behaves as a perfect energy-conserving structure.
Two-Port Lossless Network
For a two-port network:
\[ [S] = \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix} \]
Applying the unitary condition:
\[ [S]^{\dagger}[S] = [I] \]
requires evaluating:
\[ \begin{bmatrix} S_{11}^{*} & S_{21}^{*}\\ S_{12}^{*} & S_{22}^{*} \end{bmatrix} \begin{bmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \]
Performing the matrix multiplication yields:
\[ \begin{bmatrix} |S_{11}|^2+|S_{21}|^2 & S_{11}^{*}S_{12}+S_{21}^{*}S_{22} \\ S_{12}^{*}S_{11}+S_{22}^{*}S_{21} & |S_{12}|^2+|S_{22}|^2 \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \]
Special Case: Reciprocal and Lossless Two-Port Networks
Many practical microwave components such as transmission lines, directional couplers, and passive filters are both reciprocal and lossless. In this case, the network simultaneously satisfies:
\[ S_{12}=S_{21} \]
and
\[ [S]^\dagger[S]=[I] \]
Substituting reciprocity into the lossless conditions greatly simplifies the scattering matrix relationships.
Starting from:
\[ |S_{11}|^2+|S_{21}|^2=1 \]
\[ |S_{22}|^2+|S_{12}|^2=1 \]
Since:
\[ S_{12}=S_{21} \]
we obtain:
\[ |S_{11}|^2+|S_{21}|^2=1 \]
\[ |S_{22}|^2+|S_{21}|^2=1 \]
Subtracting the two equations gives:
\[ |S_{11}|^2=|S_{22}|^2 \]
Therefore:
\[ |S_{11}|=|S_{22}| \]
This means the magnitudes of the input and output reflection coefficients must be identical for any reciprocal lossless two-port network.
Special Case: Symmetrical, Reciprocal, and Lossless Networks
If the network is also physically symmetrical, then:
\[ S_{11}=S_{22} \]
and
\[ S_{12}=S_{21} \]
The scattering matrix reduces to:
\[ [S] = \begin{bmatrix} S_{11} & S_{21}\\ S_{21} & S_{11} \end{bmatrix} \]
Applying the unitary condition:
\[ [S]^\dagger[S]=[I] \]
produces:
\[ |S_{11}|^2+|S_{21}|^2=1 \]
and
\[ S_{11}^{*}S_{21} + S_{21}^{*}S_{11} = 0 \]
Factoring:
\[ 2\,\text{Re} \left( S_{11}^{*}S_{21} \right) = 0 \]
Hence:
\[ \text{Re} \left( S_{11}^{*}S_{21} \right) = 0 \]
This result shows that the transmission coefficient and reflection coefficient must be in quadrature, meaning their phase difference is:
\[ 90^\circ \]
or
\[ 270^\circ \]
This property is frequently encountered in ideal hybrid couplers and lossless microwave junctions.
Physical Interpretation of the Lossless Conditions
The lossless equations provide direct insight into power flow inside microwave networks.
The equation:
\[ |S_{11}|^2+|S_{21}|^2=1 \]
states that all incident power must either be reflected or transmitted.
If a network has:
\[ |S_{11}|=0 \]
then:
\[ |S_{21}|=1 \]
meaning the network is perfectly matched and transmits all available power.
Conversely, if:
\[ |S_{11}|=1 \]
then:
\[ |S_{21}|=0 \]
which means the structure behaves as a perfect reflector and no power reaches the output.
For practical microwave components, designers attempt to minimize:
\[ |S_{11}| \]
while maximizing:
\[ |S_{21}| \]
to achieve efficient power transfer and low insertion loss.
Fundamental S-Parameter Properties
- Reciprocity requires:
\[ S_{12}=S_{21} \]
- Symmetry requires:
\[ S_{11}=S_{22} \]
- Losslessness requires:
\[ [S]^\dagger[S]=[I] \]
- For a lossless two-port:
\[ |S_{11}|^2+|S_{21}|^2=1 \]
\[ |S_{22}|^2+|S_{12}|^2=1 \]
- Orthogonality condition:
\[ S_{11}^{*}S_{12} + S_{21}^{*}S_{22} = 0 \]
- Reciprocal lossless networks satisfy:
\[ |S_{11}|=|S_{22}| \]
- Symmetrical reciprocal lossless networks satisfy:
\[ \text{Re} \left( S_{11}^{*}S_{21} \right) = 0 \]
These properties form the theoretical foundation for analyzing microwave filters, couplers, attenuators, matching networks, and RF communication systems using scattering parameters.