3-Port S-Parameters

3-Port S-Parameters in Microwave Engineering

After mastering two-port S-parameters, the next step in microwave network analysis is understanding 3-port networks. Three-port devices are widely used in RF and microwave systems, including power dividers, power combiners, T-junctions, directional couplers, and ferrite circulators.

Like two-port networks, 3-port networks are characterized using Scattering Parameters (S-Parameters), which relate incident power waves to reflected or transmitted power waves at each port.

For students preparing for IOE, CSIT, BCA, BE Electronics, Communication Engineering, IIT, GATE, and competitive engineering examinations, understanding the physical meaning of all nine S-parameters is essential.

The 3-Port Scattering Matrix Framework

In a three-port microwave network, energy enters and leaves the device through three physical ports.

• a₁, a₂, a₃ → Incident power waves entering Ports 1, 2, and 3.
• b₁, b₂, b₃ → Reflected or scattered power waves leaving Ports 1, 2, and 3.

The relationship between incident and scattered waves is represented using the 3×3 scattering matrix:

\[ \begin{bmatrix} b_1\\ b_2\\ b_3 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} & S_{13}\\ S_{21} & S_{22} & S_{23}\\ S_{31} & S_{32} & S_{33} \end{bmatrix} \begin{bmatrix} a_1\\ a_2\\ a_3 \end{bmatrix} \]

Every parameter in the matrix is measured while all non-excited ports are terminated in matched loads.

For microwave systems, the standard reference impedance is:

\[ Z_0 = 50\Omega \]

Matched terminations ensure that no additional reflections are introduced during measurement.

Understanding the Nine S-Parameters

The nine entries of the scattering matrix can be divided into two major categories:

1. Reflection Coefficients (Diagonal Terms)
2. Transmission and Isolation Coefficients (Off-Diagonal Terms)

Reflection Coefficients (Diagonal Elements)

The diagonal terms describe how much energy is reflected back from the same port where the signal is applied.

These parameters indicate the impedance matching quality of each port.

1. Input Reflection Coefficient at Port 1 (S₁₁)

Mathematical Definition

\[ S_{11} = \left. \frac{b_1}{a_1} \right|_{a_2=0,\;a_3=0} \]

Physical Meaning

S₁₁ measures the fraction of incident power entering Port 1 that is reflected back toward Port 1 while Ports 2 and 3 are perfectly matched.

It is essentially the input reflection coefficient of Port 1.

A smaller value of \(|S_{11}|\) indicates better impedance matching.

Return Loss Formula

\[ \text{Return Loss} = -20\log_{10}|S_{11}| \]

Engineering Significance

• Antenna input matching
• RF amplifier input design
• Power divider input optimization
• Microwave filter input performance

2. Reflection Coefficient at Port 2 (S₂₂)

Mathematical Definition

\[ S_{22} = \left. \frac{b_2}{a_2} \right|_{a_1=0,\;a_3=0} \]

Physical Meaning

S₂₂ measures the amount of energy reflected back from Port 2 when a signal is applied to Port 2 and the remaining ports are terminated in matched loads.

It represents the output reflection coefficient of Port 2.

Magnitude in Decibels

\[ |S_{22}|_{dB} = 20\log_{10}|S_{22}| \]

Engineering Significance

• Output matching evaluation
• Wilkinson power divider output analysis
• Load impedance optimization
• Transmission efficiency verification

3. Reflection Coefficient at Port 3 (S₃₃)

Mathematical Definition

\[ S_{33} = \left. \frac{b_3}{a_3} \right|_{a_1=0,\;a_2=0} \]

Physical Meaning

S₃₃ measures the reflected power returning from Port 3 when Ports 1 and 2 are matched.

Like S₁₁ and S₂₂, it evaluates the matching quality of the corresponding port.

Magnitude in Decibels

\[ |S_{33}|_{dB} = 20\log_{10}|S_{33}| \]

Engineering Significance

• Power combiner output matching
• Circulator port verification
• Three-terminal RF device analysis
• Directional coupler design

Transmission and Isolation Coefficients (Off-Diagonal Elements)

While the diagonal S-parameters measure reflections at individual ports, the off-diagonal S-parameters describe how microwave energy travels between different ports.

These parameters are often called transmission coefficients because they quantify power transfer through the network. In many microwave components, they also represent isolation characteristics between ports.

For a 3-port network, there are six off-diagonal parameters:

• S₂₁
• S₃₁
• S₁₂
• S₃₂
• S₁₃
• S₂₃

These parameters describe signal propagation in both forward and reverse directions between ports.

4. Forward Transmission Coefficient: Port 1 to Port 2 (S₂₁)

Mathematical Definition

\[ S_{21} = \left. \frac{b_2}{a_1} \right|_{a_2=0,\;a_3=0} \]

Physical Meaning

S₂₁ measures how much of the incident signal entering Port 1 successfully reaches Port 2.

It is the most important transmission parameter because it directly represents gain, attenuation, insertion loss, or power transfer through the network.

Power Interpretation

\[ |S_{21}|^2 = \text{Fraction of Input Power Delivered to Port 2} \]

Example: Wilkinson Power Divider

For an ideal equal-split Wilkinson power divider, half the input power reaches Port 2 and the remaining half reaches Port 3.

\[ |S_{21}| = \frac{1}{\sqrt{2}} \]

\[ 20\log_{10} \left( \frac{1}{\sqrt{2}} \right) = -3.01\text{ dB} \]

This is why an ideal two-way power divider has a transmission coefficient of approximately -3 dB.

5. Forward Transmission Coefficient: Port 1 to Port 3 (S₃₁)

Mathematical Definition

\[ S_{31} = \left. \frac{b_3}{a_1} \right|_{a_2=0,\;a_3=0} \]

Physical Meaning

S₃₁ represents the fraction of energy entering Port 1 that emerges at Port 3.

In symmetrical microwave power dividers, S₃₁ is typically identical to S₂₁.

Symmetrical Divider Condition

\[ S_{31} = S_{21} \]

Engineering Applications

• Wilkinson Power Dividers
• Power Splitters
• RF Distribution Networks
• Corporate Feed Antenna Arrays

6. Reverse Transmission Coefficient: Port 2 to Port 1 (S₁₂)

Mathematical Definition

\[ S_{12} = \left. \frac{b_1}{a_2} \right|_{a_1=0,\;a_3=0} \]

Physical Meaning

S₁₂ measures signal transmission from Port 2 back toward Port 1.

Unlike S₂₁, which measures forward transmission, S₁₂ measures reverse transmission.

This parameter becomes particularly important when evaluating amplifiers, isolators, and active microwave circuits.

Reciprocal Network Condition

For reciprocal passive microwave networks:

\[ S_{12} = S_{21} \]

Examples include:

• Transmission Lines
• Waveguides
• Passive Filters
• Power Dividers
• Directional Couplers

Active devices such as amplifiers generally do not satisfy this condition.

Transmission Gain in Decibels

Forward and reverse transmission parameters are commonly expressed in decibel form:

\[ \text{Transmission Gain (dB)} = 20\log_{10}|S_{ij}| \]

Positive values indicate gain, while negative values indicate attenuation.

For passive devices:

\[ |S_{ij}| < 1 \]

For amplifiers:

\[ |S_{21}| > 1 \]

which corresponds to positive gain in decibels.

Key Properties and Constraints of 3-Port Networks

The scattering matrix of a 3-port network is not completely arbitrary. Physical laws such as energy conservation and electromagnetic reciprocity impose strict mathematical constraints on the values of the nine S-parameters. These constraints are extremely important when designing microwave components such as power dividers, directional couplers, circulators, and RF distribution networks.

A. Reciprocity in 3-Port Networks

A microwave network is said to be reciprocal if the transmission characteristics remain identical when the source and load positions are interchanged. Reciprocity is a consequence of Maxwell's equations for linear passive structures built using isotropic materials.

Mathematically, a reciprocal network possesses a symmetric scattering matrix:

$[S] = [S]^T$

Therefore:

$S_{12}=S_{21}$

$S_{13}=S_{31}$

$S_{23}=S_{32}$

Originally, a 3-port network contains nine independent S-parameters. However, reciprocity reduces the number of independent variables because each transmission pair becomes equal.

Transmission Pair	Reciprocal Relationship
Port 1 ↔ Port 2	S12 = S21
Port 1 ↔ Port 3	S13 = S31
Port 2 ↔ Port 3	S23 = S32

As a result, the number of independent parameters decreases from nine to six:

• S₁₁
• S₂₂
• S₃₃
• S₂₁
• S₃₁
• S₃₂

Common examples of reciprocal microwave devices include:

• Transmission lines
• Waveguide sections
• Power dividers
• Passive filters
• T-junctions

Non-reciprocal devices such as ferrite circulators and isolators intentionally violate these relationships through magnetic biasing.

B. Losslessness in 3-Port Networks

A network is said to be lossless when no microwave power is converted into heat inside the device. All incident power must emerge from one or more output ports.

Therefore:

$P_{incident}=P_{scattered}$

In matrix form, power conservation requires the scattering matrix to be unitary:

$[S]^{\dagger}[S]=[I]$

where:

• [S]^† = Hermitian conjugate (complex conjugate transpose)
• [I] = Identity matrix

The unitary condition generates several important equations used in microwave design and network verification.

Power Conservation Equations

$|S_{11}|^2+|S_{21}|^2+|S_{31}|^2=1$

$|S_{12}|^2+|S_{22}|^2+|S_{32}|^2=1$

$|S_{13}|^2+|S_{23}|^2+|S_{33}|^2=1$

These equations state that the total power leaving a column of the scattering matrix must equal the total incident power entering that column.

Orthogonality Conditions for a Lossless 3-Port Network

The unitary condition

$[S]^\dagger[S]=[I]$

generates not only the power conservation equations but also a second set of equations known as the orthogonality conditions. These conditions require every pair of columns within the scattering matrix to be mutually orthogonal.

For a 3-port network:

$S_{11}^{*}S_{12}+S_{21}^{*}S_{22}+S_{31}^{*}S_{32}=0$

$S_{11}^{*}S_{13}+S_{21}^{*}S_{23}+S_{31}^{*}S_{33}=0$

$S_{12}^{*}S_{13}+S_{22}^{*}S_{23}+S_{32}^{*}S_{33}=0$

These equations ensure that power waves associated with different excitation ports remain mathematically independent and satisfy electromagnetic energy conservation.

The Fundamental 3-Port Theorem (Pozar's Rule)

A 3-port network cannot be simultaneously reciprocal, lossless, and perfectly matched at all ports.

This is one of the most important theoretical results in microwave engineering. The theorem applies to all passive 3-port structures regardless of physical implementation.

Assume we attempt to design a 3-port network that satisfies all three conditions:

• Perfectly matched at all ports
• Reciprocal
• Lossless

Step 1: Assume Perfect Matching

Perfect matching requires:

$S_{11}=S_{22}=S_{33}=0$

Therefore the scattering matrix becomes:

$[S]= \begin{bmatrix} 0 & S_{12} & S_{13}\\ S_{21} & 0 & S_{23}\\ S_{31} & S_{32} & 0 \end{bmatrix}$

Step 2: Apply Reciprocity

Reciprocity requires:

$S_{12}=S_{21}$

$S_{13}=S_{31}$

$S_{23}=S_{32}$

The matrix becomes:

$[S]= \begin{bmatrix} 0 & A & B\\ A & 0 & C\\ B & C & 0 \end{bmatrix}$

Step 3: Apply Lossless Conditions

Using the power conservation equations:

$|A|^2+|B|^2=1$

$|A|^2+|C|^2=1$

$|B|^2+|C|^2=1$

Solving these equations gives:

$|A|=|B|=|C|=\frac{1}{\sqrt{2}}$

Step 4: Apply Orthogonality Condition

Consider:

$S_{11}^{*}S_{12}+S_{21}^{*}S_{22}+S_{31}^{*}S_{32}=0$

Since:

$S_{11}=S_{22}=0$

the equation reduces to:

$S_{31}^{*}S_{32}=0$

Using reciprocity:

$B^{*}C=0$

This requires:

$B=0 \quad \text{or} \quad C=0$

which directly contradicts:

$|B|=|C|=\frac{1}{\sqrt{2}}$

Therefore the original assumptions cannot all be true simultaneously.

Conclusion: A 3-port network cannot be simultaneously reciprocal, lossless, and perfectly matched at every port.

Practical Microwave Devices and the 3-Port Theorem

1. Wilkinson Power Divider

The Wilkinson Power Divider is a popular three-port microwave device used for equal power splitting.

Characteristics

• Reciprocal
• Matched at all ports
• Uses an isolation resistor
• Not perfectly lossless under all operating conditions

By introducing a resistor between output ports, the Wilkinson divider sacrifices the strict lossless requirement and therefore avoids violating the 3-port theorem.

2. Ferrite Circulator

A circulator uses magnetically biased ferrite materials to force energy to travel in only one direction around the ports.

Characteristics

• Lossless (approximately)
• Matched at all ports
• Non-reciprocal

Typical power flow:

 

$Port\ 1 \rightarrow Port\ 2 \rightarrow Port\ 3 \rightarrow Port\ 1$

Since reciprocity is intentionally broken, the circulator satisfies the theorem.

3. E-Plane and H-Plane T-Junctions

Waveguide T-junctions are simple passive 3-port structures commonly used in microwave systems.

Characteristics

• Reciprocal
• Lossless
• Not matched at all ports

Reflections naturally occur at one or more ports, preventing simultaneous matching.

Comparison of Practical 3-Port Devices

Device	Reciprocal	Lossless	Matched at All Ports	How Theorem Is Satisfied
Wilkinson Power Divider	Yes	No (Isolation Resistor)	Yes	Sacrifices Losslessness
Ferrite Circulator	No	Yes	Yes	Breaks Reciprocity
E/H Plane T-Junction	Yes	Yes	No	Sacrifices Matching

Exam-Oriented Summary for IOE Students

• A 3-port network contains 9 S-parameters.
• Reciprocity reduces the independent parameters from 9 to 6.
• Lossless networks require a unitary scattering matrix.
• Perfect matching requires S11 = S22 = S33 = 0.
• No passive 3-port device can be reciprocal, lossless, and perfectly matched simultaneously.
• This result is known as the 3-Port Theorem or Pozar's Rule.
• Wilkinson dividers, circulators, and T-junctions are practical examples that demonstrate this theorem.