3.1 Scattering Parameters (S-Parameters)

As we move to higher frequencies, traditional methods of network analysis based on total voltages and currents (such as Z- or Y-parameters) become impractical. It is difficult to measure total voltage and current at microwave frequencies, and it is nearly impossible to maintain a true short or open circuit at a component’s ports for measurement. Therefore, a specialized framework is necessary. Scattering Parameters, or S-parameters, provide this framework. They are the preferred method for characterizing microwave junctions because they relate to traveling waves—incident, reflected, and transmitted—which are the quantities that can be readily and accurately measured at these frequencies.

A microwave junction or network with multiple ports can be completely described by its Scattering Matrix [S]. This is a square matrix whose elements, the scattering coefficients, quantify the power relationships between all ports. It describes how power “scatters” from a given input port to all other ports, including being reflected from the input port itself.

For a general n-port network, we can define an incident wave (a_n) and a reflected (or scattered) wave (b_n) at each port. The reflected wave at any given port is a linear superposition of the contributions from the incident waves at all ports. This relationship is defined by the scattering coefficients (S_{nm}), where S_{nm} represents the transmission from port m to port n and S_{nn} represents the reflection at port n.

This leads to a system of linear equations: b_1 = S_{11}a_1 + S_{12}a_2 + \dots + S_{1n}a_n b_2 = S_{21}a_1 + S_{22}a_2 + \dots + S_{2n}a_n \vdots b_n = S_{n1}a_1 + S_{n2}a_2 + \dots + S_{nn}a_n

This system can be expressed concisely in matrix form: \left [ b \right ] = \left [ S \right ]\left [ a \right ]

The [S] matrix has several key properties that are fundamental to its use in microwave analysis:

1. Square Matrix: For an n-port network, the [S] matrix is always a square matrix of dimension n x n, as it must relate the n incident waves to the n reflected waves.
2. Symmetric Matrix (): If the network is reciprocal—meaning it is made of passive materials and its transmission properties are the same in either direction between two ports—then its S-matrix will be symmetric. Most passive components fall into this category.
3. Unitary Matrix (): If the network is lossless, meaning it does not dissipate any power, then its S-matrix is unitary. This property implies that the total power exiting the network must equal the total power entering it. This is a crucial property used to derive the S-parameters of ideal components, as it provides a set of constraining equations.

These S-parameter principles will now be applied to analyze a family of fundamental microwave components: the waveguide Tee junctions.

3.2 Waveguide Tee Junctions

Waveguide Tees are fundamental three-port devices used for combining or splitting microwave power. Their name derives from their physical ‘T’ shape. They are essential building blocks in many microwave systems, including mixers, duplexers, and power dividers. This section will analyze the three primary types: the E-Plane Tee, the H-Plane Tee, and the combined E-H Plane (Magic) Tee. Our focus will be on deriving their respective S-matrices to gain a quantitative understanding of their unique operational behaviors.

E-Plane Tee (Series Tee)

The E-Plane Tee is constructed by attaching a third waveguide arm to the narrow wall of a rectangular waveguide. The axis of this side arm (Port 3) is parallel to the electric field (E-field) of the main guide, hence the name “E-Plane.”

• Key Characteristic: The defining feature of the E-Plane Tee is that an input signal at the E-arm (Port 3) will be split equally between the collinear arms (Ports 1 and 2), but the two outputs will be 180° out of phase with each other.
• Scattering Matrix Derivation: We derive the 3×3 S-matrix for an ideal E-Plane Tee by applying its known physical properties to the S-matrix framework.
  1. The network is reciprocal, so the matrix is symmetric: S_{ij} = S_{ji}.
  2. The side arm is perfectly matched, so there is no reflection at Port 3: S_{33} = 0.
  3. The 180° phase relationship for an input at Port 3 means: S_{23} = -S_{13}.
  4. From the unitary property, [S][S]^* = I, which means the product of any row with the complex conjugate of another row is zero. Let’s examine the product of row 3 with the conjugate of row 1 (R_3 C_1^* = 0): S_{31}S_{11}^* + S_{32}S_{12}^* + S_{33}S_{33}^* = 0 Substituting our known properties (S_{31}=S_{13}, S_{32}=S_{23}=-S_{13}, and S_{33}=0): S_{13}S_{11}^* + (-S_{13})S_{12}^* + 0 = 0 \implies S_{13}(S_{11}^* – S_{12}^*) = 0 Since power is transmitted to port 1 from port 3 (S_{13} \ne 0), we must conclude that S_{11}^* = S_{12}^*, which means S_{11} = S_{12}.
  5. Due to the physical symmetry of the junction, S_{11} = S_{22}. Therefore, S_{11} = S_{22} = S_{12}.
  6. Now, applying the unitary property to row 3 multiplied by its own conjugate (R_3 C_3^* = 1): |S_{31}|^2 + |S_{32}|^2 + |S_{33}|^2 = 1 \implies |S_{13}|^2 + |-S_{13}|^2 + 0 = 1 \implies 2|S_{13}|^2 = 1 This gives |S_{13}| = 1/\sqrt{2}.
  7. Finally, using R_1 C_1^* = 1: |S_{11}|^2 + |S_{12}|^2 + |S_{13}|^2 = 1 \implies |S_{11}|^2 + |S_{11}|^2 + (1/\sqrt{2})^2 = 1 2|S_{11}|^2 + 1/2 = 1 \implies 2|S_{11}|^2 = 1/2 \implies |S_{11}| = 1/2 The final S-matrix for the ideal E-Plane Tee is: [S] = \begin{bmatrix} 1/2 & 1/2 & 1/\sqrt{2} \\ 1/2 & 1/2 & -1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} & 0 \end{bmatrix}

H-Plane Tee (Shunt Tee)

The H-Plane Tee is formed by attaching a third arm to the broad wall of a rectangular waveguide. In this case, the axis of the side arm (Port 3) is parallel to the magnetic field (H-field), hence the name “H-Plane.”

• Key Characteristic: The defining feature of the H-Plane Tee is that an input signal at the H-arm (Port 3) will be split equally between the collinear arms (Ports 1 and 2), and these two outputs will be in phase with each other.
• Scattering Matrix Derivation: The derivation follows a similar process. Key properties are: symmetry (S_{ij} = S_{ji}), a matched side arm (S_{33}=0), and the in-phase relationship (S_{13}=S_{23}). From physical symmetry, S_{11}=S_{22}. Applying the unitary property R_3 C_1^* = 0: S_{31}S_{11}^* + S_{32}S_{12}^* + S_{33}S_{33}^* = 0 \implies S_{13}S_{11}^* + S_{13}S_{12}^* = 0 This leads to S_{11} = -S_{12}. Applying the remaining unitary conditions yields |S_{13}| = 1/\sqrt{2} and |S_{11}| = 1/2. The final S-matrix for the ideal H-Plane Tee is: [S] = \begin{bmatrix} 1/2 & -1/2 & 1/\sqrt{2} \\ -1/2 & 1/2 & 1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} & 0 \end{bmatrix}

E-H Plane Tee (Magic Tee)

The E-H Plane Tee, commonly known as the Magic Tee or Hybrid Tee, is a four-port junction that combines the properties of both E-Plane and H-Plane Tees. It consists of two collinear ports (1 and 2), an H-arm (Port 3, also called the sum port), and an E-arm (Port 4, also called the difference port).

• Unique Characteristics: The magic tee possesses several “magical” properties:
  • An input at the H-arm (Port 3) splits equally and in phase between Ports 1 and 2, with no output at Port 4.
  • An input at the E-arm (Port 4) splits equally and 180° out of phase between Ports 1 and 2, with no output at Port 3.
  • This implies perfect isolation between the sum and difference ports (S_{34} = S_{43} = 0).
  • The two collinear ports are perfectly isolated from each other (S_{12} = S_{21} = 0).
  • A perfectly constructed Magic Tee is matched at all ports (S_{11}=S_{22}=S_{33}=S_{44}=0).
• Scattering Matrix: Applying these known characteristics along with the principles of symmetry and the unitary property allows for a methodical derivation of its 4×4 S-matrix. The result shows how power is divided when fed into any port. The final S-matrix for the ideal Magic Tee is: [S] = \begin{bmatrix} 0 & 0 & 1/\sqrt{2} & 1/\sqrt{2} \\ 0 & 0 & 1/\sqrt{2} & -1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} & 0 & 0 \\ 1/\sqrt{2} & -1/\sqrt{2} & 0 & 0 \end{bmatrix}
• Primary Applications:
  • Impedance Measurement: Used in microwave bridges to measure unknown impedances.
  • Duplexer: Allows a single antenna to be used for both transmitting and receiving by isolating the transmitter and receiver from each other.
  • Mixer: Combines a local oscillator signal with an incoming RF signal to produce an intermediate frequency (IF) signal.

Having analyzed Tee junctions, we will now turn our attention to other important components used for sampling, splitting, and combining microwave power.

3.3 Couplers and Junctions

Beyond the family of Tee junctions, several other passive components are essential for manipulating microwave power in specific and often highly precise ways. These devices are used to sample signals for measurement, combine signals from different sources, or divide a signal among multiple paths with specific phase relationships.

Rat-race Junction

The Rat-race junction is a four-port device, often constructed as a circular ring or “race” with a mean circumference of 1.5 wavelengths (1.5λ). The four ports are spaced around the ring at specific intervals to control the phase relationships of signals traveling between them.

• Operational Principles: The device’s behavior is dictated by the path lengths signals must travel between ports.
  • If power is input at Port 1, it splits and travels in both clockwise and counter-clockwise directions. The signals arrive in phase at Ports 2 and 4, which receive equal power. However, the paths to Port 3 differ by half a wavelength (λ/2), causing the signals to arrive 180° out of phase and cancel completely. Thus, Port 3 is isolated from Port 1.
  • Similarly, if power is input at Port 3, it splits equally to Ports 2 and 4, while Port 1 remains isolated.
• Scattering Matrix: The S-matrix for an ideal Rat-race junction reflects this isolation and power division: [S] = \begin{bmatrix} 0 & S_{12} & 0 & S_{14} \\ S_{21} & 0 & S_{23} & 0 \\ 0 & S_{32} & 0 & S_{34} \\ S_{41} & 0 & S_{43} & 0 \end{bmatrix}

Directional Couplers

A directional coupler is a four-port device designed to sample a small, known amount of microwave power flowing in a specific direction. It consists of a primary or main waveguide and a secondary or auxiliary waveguide that are coupled together.

• Ideal Properties: In an ideal directional coupler, power flowing from Port 1 to Port 2 will result in a fraction of that power appearing at the coupled Port 4, but no power appearing at the isolated Port 3. It is “directional” because it selectively couples power traveling in one direction.
• Key Performance Parameters:
  • Coupling Factor (C): The ratio of the incident power to the forward-coupled power, measured in dB. It indicates how much of the main-line power is sampled. C = 10 \: log_{10}\frac{P_i}{P_f}dB
  • Directivity (D): The ratio of the forward-coupled power to the back power (power at the isolated port), measured in dB. It is a measure of the coupler’s ability to distinguish between forward and reverse traveling waves. D = 10 \: log_{10}\frac{P_f}{P_b}dB
  • Isolation (I): The ratio of the incident power to the back power, measured in dB. It indicates the total separation between the input and isolated ports. I = 10 \: log_{10}\frac{P_i}{P_b}dB
• Two-Hole Directional Coupler: A common implementation involves two waveguides sharing a common wall with two small holes drilled in it, spaced a quarter of a guide wavelength ({\lambda_g}/{4}) apart. Power traveling from Port 1 leaks through both holes into the secondary guide. The signals traveling towards the coupled port (Port 4) are in phase and add constructively, while the signals traveling towards the isolated port (Port 3) have traveled different path lengths, resulting in a 180° phase difference, causing them to add destructively. This phase cancellation at the isolated port is what gives the device its high directivity.

While passive devices are essential for manipulating microwave energy, active devices are required to generate and amplify it. The next two modules will explore the two major classes of active devices: solid-state components and their vacuum tube counterparts.