RF Network Analysis and S-Parameters

Introduction

RF networks are difficult to characterize directly by open-circuit voltages and short-circuit currents because practical opens and shorts become parasitic, radiative and potentially unstable at microwave frequencies. Scattering parameters avoid these problems by relating incident and reflected travelling waves at specified reference planes and impedances. They are therefore the principal language of vector network analyzers (VNAs), amplifier design, filter synthesis, antenna interfaces and microwave system integration.

Nevertheless, impedance, admittance and transmission matrices remain indispensable. Z and Y matrices reveal terminal relationships; ABCD matrices simplify cascades; scattering matrices connect directly to measurable waves; and chain-scattering matrices permit stable algebraic cascading of measured two-ports. Correct analysis requires careful attention to wave convention, reference impedance, reference-plane position, stability and measurement uncertainty.

Learning Objectives

After completing this chapter, the reader should be able to:

• formulate one-port and multiport networks using Z, Y, ABCD and S matrices;
• convert between network representations under stated reference-impedance conditions;
• cascade two-ports using ABCD and chain-scattering matrices;
• calculate return loss, insertion loss, gain, reciprocity and isolation;
• renormalize S-parameters and shift measurement reference planes;
• evaluate conditional and unconditional stability, including stability circles;
• distinguish unilateral approximation error from Mason’s invariant gain;
• interpret RF measurements critically and apply sound de-embedding practice.

1. Ports, Phasors and Network Variables

An RF port is a terminal pair or transmission-line cross-section at which a single dominant propagation mode is assumed. Port voltage and current are complex phasors, with current conventionally defined as entering the network.

This chapter uses RMS phasors. Consequently, the time-average power entering a port is

\[

P=\operatorname{Re}\{VI^*\}.

\]

If peak phasors were used, the corresponding expression would contain a factor of one-half. Stating the convention is essential because otherwise travelling-wave power equations are ambiguous by a factor of two.

For a positive real reference impedance \(Z_0\), define power waves

\[

a=\frac{V+Z_0I}{2\sqrt{Z_0}},\qquad

b=\frac{V-Z_0I}{2\sqrt{Z_0}}.

\]

The inverse relations are

\[

V=\sqrt{Z_0}(a+b),\qquad

I=\frac{a-b}{\sqrt{Z_0}}.

\]

With RMS phasors, \(|a|^2\) and \(|b|^2\) have units of watts, and

\[

P=|a|^2-|b|^2.

\]

These simple power identities assume real, positive reference impedances. For complex reference impedances, power-wave and pseudo-wave definitions differ; the convention used by the measurement system must be stated explicitly.

2. Impedance and Admittance Representations

For an \(N\)-port linear network,

\[

\mathbf V=\mathbf Z\mathbf I,

\qquad

\mathbf I=\mathbf Y\mathbf V,

\]

where \(\mathbf Z\) has units of ohms and \(\mathbf Y\) has units of siemens. If both inverses exist,

\[

\mathbf Y=\mathbf Z^{-1}.

\]

For a two-port,

\[

\begin{bmatrix}V_1\\V_2\end{bmatrix}

=

\begin{bmatrix}Z_{11}&Z_{12}\\Z_{21}&Z_{22}\end{bmatrix}

\begin{bmatrix}I_1\\I_2\end{bmatrix}.

\]

Thus, for example, \(Z_{21}=V_2/I_1\) with \(I_2=0\), an open-circuit condition. Y-parameters are similarly defined using short-circuit conditions.

For a reciprocal network, \(Z_{12}=Z_{21}\) and \(Y_{12}=Y_{21}\), provided the same port-variable conventions are used. Reciprocity does not imply symmetry: a reciprocal network can still have \(Z_{11}\ne Z_{22}\).

2.1 Conversion between Z, Y and S

Let \(\mathbf R\) be a diagonal matrix containing positive real port reference impedances. Define normalized matrices

\[

\mathbf z=\mathbf R^{-1/2}\mathbf Z\mathbf R^{-1/2},

\qquad

\mathbf y=\mathbf R^{1/2}\mathbf Y\mathbf R^{1/2}.

\]

The scattering matrix is then

\[

\mathbf S=(\mathbf z-\mathbf I)(\mathbf z+\mathbf I)^{-1},

\]

or equivalently

\[

\mathbf S=(\mathbf I-\mathbf y)(\mathbf I+\mathbf y)^{-1}.

\]

The inverse conversions are

\[

\mathbf z=(\mathbf I+\mathbf S)(\mathbf I-\mathbf S)^{-1},

\]

\[

\mathbf y=(\mathbf I-\mathbf S)(\mathbf I+\mathbf S)^{-1}.

\]

Finally,

\[

\mathbf Z=\mathbf R^{1/2}\mathbf z\mathbf R^{1/2},

\qquad

\mathbf Y=\mathbf R^{-1/2}\mathbf y\mathbf R^{-1/2}.

\]

The required inverses must exist. Near singularities, direct matrix inversion may be numerically ill-conditioned, so pivoted factorization or alternative parameter representations should be used.

Worked Example 1: S-to-Y Conversion

A 50 Ω two-port has

\[

\mathbf S=

\begin{bmatrix}0.20&0.05\\0.80&0.10\end{bmatrix}.

\]

Using \(\mathbf y=(\mathbf I-\mathbf S)(\mathbf I+\mathbf S)^{-1}\),

\[

\mathbf y=

\begin{bmatrix}

0.71875&-0.078125\\

-1.25&0.875

\end{bmatrix}.

\]

Because \(\mathbf y=50\mathbf Y\),

\[

\mathbf Y=

\begin{bmatrix}

0.014375&-0.0015625\\

-0.025&0.0175

\end{bmatrix}\text{ S}.

\]

Since \(Y_{12}\ne Y_{21}\), the network is nonreciprocal.

3. ABCD Transmission Matrices

For cascading two-ports, this chapter adopts

\[

\begin{bmatrix}V_1\\I_1\end{bmatrix}

=

\begin{bmatrix}A&B\\C&D\end{bmatrix}

\begin{bmatrix}V_2\\-I_2\end{bmatrix},

\]

where both physical port currents enter the two-port. Under this convention, cascaded networks are represented by ordered matrix multiplication:

\[

\mathbf M_{\text{total}}=\mathbf M_1\mathbf M_2\cdots\mathbf M_n.

\]

Common building blocks are

\[

\mathbf M_{\text{series }Z}=

\begin{bmatrix}1&Z\\0&1\end{bmatrix},

\qquad

\mathbf M_{\text{shunt }Y}=

\begin{bmatrix}1&0\\Y&1\end{bmatrix}.

\]

A reciprocal two-port satisfies

\[

AD-BC=1.

\]

For equal real reference impedances \(Z_0\), define

\[

Q=A+\frac{B}{Z_0}+CZ_0+D.

\]

Then

\[

S_{11}=\frac{A+B/Z_0-CZ_0-D}{Q},

\]

\[

S_{21}=\frac{2}{Q},

\]

\[

S_{12}=\frac{2(AD-BC)}{Q},

\]

\[

S_{22}=\frac{-A+B/Z_0-CZ_0+D}{Q}.

\]

Conversely, when \(S_{21}\ne0\),

\[

A=\frac{(1+S_{11})(1-S_{22})+S_{12}S_{21}}{2S_{21}},

\]

\[

B=Z_0\frac{(1+S_{11})(1+S_{22})-S_{12}S_{21}}{2S_{21}},

\]

\[

C=\frac{1}{Z_0}\frac{(1-S_{11})(1-S_{22})-S_{12}S_{21}}{2S_{21}},

\]

\[

D=\frac{(1-S_{11})(1+S_{22})+S_{12}S_{21}}{2S_{21}}.

\]

Worked Example 2: Cascaded Reactive Network

A series impedance \(Z=j20\ \Omega\) is followed by a shunt admittance \(Y=j0.01\ \text{S}\). Their cascade is

\[

\mathbf M=

\begin{bmatrix}1&j20\\0&1\end{bmatrix}

\begin{bmatrix}1&0\\j0.01&1\end{bmatrix}

=

\begin{bmatrix}0.8&j20\\j0.01&1\end{bmatrix}.

\]

For \(Z_0=50\ \Omega\),

\[

Q=1.8+j0.9.

\]

Therefore,

\[

S_{21}=S_{12}=0.8889-j0.4444,

\]

\[

S_{11}=-0.1111,

\qquad

S_{22}=0.0667-j0.0889.

\]

Equality of \(S_{12}\) and \(S_{21}\) confirms reciprocity. The computed values also satisfy lossless power conservation within rounding error.

4. Scattering Matrices

An \(N\)-port scattering relation is

\[

\mathbf b=\mathbf S\mathbf a.

\]

For a two-port,

\[

\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 parameters have direct interpretations:

• \(S_{11}\): input reflection coefficient with port 2 matched;
• \(S_{22}\): output reflection coefficient with port 1 matched;
• \(S_{21}\): forward transmission coefficient;
• \(S_{12}\): reverse transmission coefficient.

For equal real references, a reciprocal network satisfies \(\mathbf S=\mathbf S^T\). A passive network satisfies

\[

\mathbf I-\mathbf S^{\dagger}\mathbf S\succeq0,

\]

meaning that the matrix is positive semidefinite. Equivalently, no singular value of \(\mathbf S\) exceeds unity. A lossless network satisfies

\[

\mathbf S^{\dagger}\mathbf S=\mathbf I.

\]

5. Chain-Scattering Matrices

Direct multiplication of S matrices does not represent cascade connection. A chain-scattering matrix, often called a T matrix, resolves this problem. Define

\[

\begin{bmatrix}a_1\\b_1\end{bmatrix}

=

\mathbf T

\begin{bmatrix}b_2\\a_2\end{bmatrix}.

\]

When \(S_{21}\ne0\),

\[

\mathbf T=

\begin{bmatrix}

1/S_{21}&-S_{22}/S_{21}\\

S_{11}/S_{21}&-\det(\mathbf S)/S_{21}

\end{bmatrix}.

\]

For cascaded stages,

\[

\mathbf T_{\text{total}}=\mathbf T_1\mathbf T_2\cdots\mathbf T_n.

\]

The S-parameters are recovered from

\[

S_{21}=\frac{1}{T_{11}},\quad

S_{22}=-\frac{T_{12}}{T_{11}},\quad

S_{11}=\frac{T_{21}}{T_{11}},\quad

S_{12}=\frac{\det(\mathbf T)}{T_{11}}.

\]

Chain matrices can become poorly conditioned for long, highly attenuating cascades because exponentially large and small quantities coexist. Scattering-based connection algorithms are often preferable for very large networks.

6. Gain, Loss, Matching and Isolation

For a one-port with reflection coefficient \(\Gamma\),

\[

\text{return loss}=-20\log_{10}|\Gamma|\ \text{dB},

\]

and

\[

\text{VSWR}=\frac{1+|\Gamma|}{1-|\Gamma|},\qquad |\Gamma|<1.

\]

For a two-port terminated in its reference impedance, the forward power ratio is \(|S_{21}|^2\). Thus,

\[

G_{\text{matched}}=20\log_{10}|S_{21}|\ \text{dB}.

\]

For a passive device, insertion loss is commonly

\[

IL=-20\log_{10}|S_{21}|\ \text{dB}.

\]

For an active device, this quantity may be negative and is better reported as gain. Reverse isolation is often

\[

I_{\text{rev}}=-20\log_{10}|S_{12}|\ \text{dB}.

\]

With source and load reflection coefficients \(\Gamma_S\) and \(\Gamma_L\), the transducer gain is

\[

G_T=

\frac{(1-|\Gamma_S|^2)|S_{21}|^2(1-|\Gamma_L|^2)}

{|(1-S_{11}\Gamma_S)(1-S_{22}\Gamma_L)-S_{12}S_{21}\Gamma_S\Gamma_L|^2}.

\]

This expression assumes passive source and load terminations, so \(|\Gamma_S|\le1\) and \(|\Gamma_L|\le1\).

7. Reference-Impedance Renormalization

S-parameters have meaning only relative to their reference impedances. Renormalization changes the wave basis without changing the physical network. A reliable procedure for positive real references is:

1. convert the original \(\mathbf S\) to \(\mathbf Z\) using the old reference matrix;

2. normalize \(\mathbf Z\) using the new reference matrix;

3. convert the normalized impedance matrix back to \(\mathbf S\).

For a one-port,

\[

Z=Z_{01}\frac{1+\Gamma_1}{1-\Gamma_1},

\qquad

\Gamma_2=\frac{Z-Z_{02}}{Z+Z_{02}}.

\]

Worked Example 3: One-Port Renormalization

A load has \(\Gamma=1/3\) in a 50 Ω system. Its impedance is

\[

Z=50\frac{1+1/3}{1-1/3}=100\ \Omega.

\]

Renormalizing to 75 Ω gives

\[

\Gamma_{75}=\frac{100-75}{100+75}=0.1429.

\]

The device has not changed; only its numerical reflection coefficient has changed.

8. Reference-Plane Transformations and De-embedding

If matched transmission-line lengths \(l_i\) are added between a device and new external reference planes, then

\[

S'_{ij}=S_{ij}e^{-(\gamma_i l_i+\gamma_j l_j)},

\]

where \(\gamma_i=\alpha_i+j\beta_i\). Therefore,

\[

S'_{ii}=S_{ii}e^{-2\gamma_i l_i}.

\]

The inverse transformation de-embeds the line sections. For lossless lines, magnitudes are unchanged and only phases rotate. With loss, both phase and magnitude change.

Reference-plane movement should not be confused with renormalization: the former changes the included electrical path, whereas the latter changes the wave reference impedance.

Practical de-embedding methods include port extension, fixture subtraction, TRL calibration and two-tier calibration. Their validity depends on fixture repeatability, modal purity, connector quality and accurate propagation constants.

9. Stability of Active Two-Ports

An active two-port may oscillate if source or load reflections produce a net regenerative loop. The input and output reflection coefficients are

\[

\Gamma_{in}=S_{11}+\frac{S_{12}S_{21}\Gamma_L}{1-S_{22}\Gamma_L},

\]

\[

\Gamma_{out}=S_{22}+\frac{S_{12}S_{21}\Gamma_S}{1-S_{11}\Gamma_S}.

\]

Define

\[

\Delta=S_{11}S_{22}-S_{12}S_{21}

\]

and, when \(S_{12}S_{21}\ne0\),

\[

K=\frac{1-|S_{11}|^2-|S_{22}|^2+|\Delta|^2}

{2|S_{12}S_{21}|}.

\]

A two-port is unconditionally stable for all passive source and load terminations if

\[

K>1\quad\text{and}\quad |\Delta|<1.

\]

An alternative input-oriented stability measure is

\[

\mu=\frac{1-|S_{11}|^2}

{|S_{22}-\Delta S_{11}^*|+|S_{12}S_{21}|}.

\]

Together with the corresponding output-oriented measure, values greater than unity establish unconditional stability.

9.1 Stability Circles

The load stability circle, defined by \(|\Gamma_{in}|=1\), has center and radius

\[

C_L=\frac{(S_{22}-\Delta S_{11}^)^}{|S_{22}|^2-|\Delta|^2},

\]

\[

r_L=\frac{|S_{12}S_{21}|}

{\left||S_{22}|^2-|\Delta|^2\right|}.

\]

The source stability circle, defined by \(|\Gamma_{out}|=1\), has

\[

C_S=\frac{(S_{11}-\Delta S_{22}^)^}{|S_{11}|^2-|\Delta|^2},

\]

\[

r_S=\frac{|S_{12}S_{21}|}

{\left||S_{11}|^2-|\Delta|^2\right|}.

\]

The formulas require nonzero denominators. The stable side cannot be inferred from the circle alone. Test a known point, usually the Smith-chart origin: for the load circle, the origin is stable if \(|S_{11}|<1\); for the source circle, it is stable if \(|S_{22}|<1\).

Worked Example 4: Stability-Circle Calculation

Consider

\[

S_{11}=0.5,\quad S_{22}=0.4,\quad S_{21}=2,

\quad S_{12}=0.05.

\]

All quantities are dimensionless and evaluated at one frequency. Then

\[

\Delta=(0.5)(0.4)-(0.05)(2)=0.1,

\]

\[

K=\frac{1-0.25-0.16+0.01}{2(0.1)}=3.

\]

Because \(K>1\) and \(|\Delta|=0.1<1\), the network is unconditionally stable.

The load stability circle is

\[

C_L=\frac{0.4-(0.1)(0.5)}{0.16-0.01}=2.333,

\qquad

r_L=\frac{0.1}{0.15}=0.667.

\]

The source circle is

\[

C_S=\frac{0.5-(0.1)(0.4)}{0.25-0.01}=1.917,

\qquad

r_S=\frac{0.1}{0.24}=0.417.

\]

Both circles lie outside the unit Smith chart, consistent with unconditional stability.

10. Maximum Gain and the Unilateral Approximation

For an unconditionally stable two-port, the maximum available gain is

\[

G_{MAG}=\left|\frac{S_{21}}{S_{12}}\right|

\left(K-\sqrt{K^2-1}\right).

\]

When \(K\le1\), the commonly quoted maximum stable gain

\[

G_{MSG}=\left|\frac{S_{21}}{S_{12}}\right|

\]

is a limiting design indicator, not a guarantee that a realizable broadband design will remain stable.

If reverse transmission is neglected, \(S_{12}\approx0\), the unilateral transducer gain becomes

\[

G_{TU}=

\frac{1-|\Gamma_S|^2}{|1-S_{11}\Gamma_S|^2}

|S_{21}|^2

\frac{1-|\Gamma_L|^2}{|1-S_{22}\Gamma_L|^2}.

\]

For \(|S_{11}|<1\) and \(|S_{22}|<1\), conjugate matching gives

\[

G_{TU,max}=rac{|S_{21}|^2}

{(1-|S_{11}|^2)(1-|S_{22}|^2)}.

\]

A standard unilateral approximation error measure is

\[

U_e=rac{|S_{11}S_{12}S_{21}S_{22}|}

{(1-|S_{11}|^2)(1-|S_{22}|^2)}.

\]

For \(U_e<1\), the gain obtained with specified unilateral-design terminations is bounded relative to the unilateral prediction by

\[

\frac{1}{(1+U_e)^2}

\le \frac{G_T}{G_{TU}}

\le \frac{1}{(1-U_e)^2}.

\]

This \(U_e\) is an approximation-error factor; it is not Mason’s invariant gain. Mason’s invariant gain may be written in Y-parameters as

\[

U_M=rac{|Y_{21}-Y_{12}|^2}

{4[\operatorname{Re}(Y_{11})\operatorname{Re}(Y_{22})-

\operatorname{Re}(Y_{12})\operatorname{Re}(Y_{21})]},

\]

when its denominator is positive. It is invariant under lossless reciprocal embedding networks and has a different purpose from \(U_e\).

11. Illustrative Case: Selecting an LNA Matching Strategy

Background

An illustrative receiver project requires a narrowband low-noise amplifier. At the design frequency, the transistor has

\[

S_{11}=0.5,

\quad S_{22}=0.4,

\quad S_{21}=2,

\quad S_{12}=0.05.

\]

Problem

The engineer must determine whether simultaneous matching is safe, estimate achievable gain and assess whether a unilateral design is defensible.

Analysis

The matched forward gain is

\[

|S_{21}|^2=4=6.02\text{ dB}.

\]

Input return loss is 6.02 dB, output return loss is 7.96 dB, and reverse isolation is

\[

-20\log_{10}(0.05)=26.02\text{ dB}.

\]

The previous stability calculation gives \(K=3\) and \(|\Delta|=0.1\), establishing unconditional stability at the evaluated frequency. The exact maximum available gain is

\[

G_{MAG}=40(3-\sqrt8)=6.863=8.36\text{ dB}.

\]

The unilateral maximum estimate is

\[

G_{TU,max}=\frac{4}{(0.75)(0.84)}=6.349=8.03\text{ dB}.

\]

The unilateral error factor is

\[

U_e=\frac{0.5(0.05)(2)(0.4)}{(0.75)(0.84)}=0.03175.

\]

This indicates modest reverse-coupling error, but it does not eliminate the need for full bilateral analysis.

Approach and Outcome

A simultaneous-conjugate-match design may target the exact available gain, while electromagnetic simulation must include package, bias and layout parasitics. Stability must be checked across the full operating and out-of-band frequency range, not merely at the design frequency. A small gain sacrifice may be justified to improve noise figure, bandwidth, manufacturing tolerance or robustness.

Learning Points

• Stability is frequency dependent.
• Matched gain is not maximum available gain.
• High reverse isolation does not justify automatically setting \(S_{12}=0\).
• Nominal S-parameters should be supplemented by tolerance and fixture analysis.

12. Measurement Limitations, Governance and Best Practice

S-parameter quality is limited by calibration residuals, connector repeatability, cable motion, fixture radiation, temperature drift, compression and unintended higher-order modes. Active devices must be measured below compression unless large-signal behavior is intended. Uncertainty should accompany critical gain, isolation and stability claims.

Professional practice includes preserving calibration metadata, reference impedance, power level, bias, temperature, frequency grid and reference-plane definition. Selective smoothing or removal of inconvenient measurements can conceal instability or noncompliance and is technically and ethically unacceptable. RF products must also satisfy applicable spectrum, electromagnetic compatibility, exposure and safety regulations.

Modern developments include mixed-mode S-parameters for differential circuits, multiport interconnect characterization, nonlinear X-parameters and load-pull measurements. Machine-learning methods can assist anomaly detection and surrogate modeling, but physically constrained models are preferable because unconstrained predictions may violate passivity, causality or reciprocity.

Chapter Summary

RF network representations provide complementary views of linear circuits. Z and Y matrices describe terminal relationships, ABCD and chain-scattering matrices support cascading, and S-parameters connect naturally to measurable travelling waves. Correct power interpretation requires an explicit RMS-phasor convention and positive real reference impedances for the basic power-wave equations.

Matrix conversion, renormalization and reference-plane transformation must be performed with consistent conventions. Gain and loss depend on source and load matching, while reciprocity, passivity and isolation follow from distinct matrix properties. Active-network design additionally requires bilateral stability analysis using \(K\), \(\Delta\), stability measures and stability circles. Unilateral approximation error must not be confused with Mason’s invariant gain. Reliable engineering ultimately depends on calibrated measurements, uncertainty awareness and verification across frequency, power, bias and environmental conditions.

Key Terms

ABCD matrix: Two-port transmission representation suited to cascaded networks.

Admittance matrix: Matrix relating port currents to port voltages.

Chain-scattering matrix: Wave matrix whose products represent cascaded two-ports.

Conditional stability: Stability for only a subset of passive source and load terminations.

De-embedding: Removal of fixture or transmission-line effects from measured data.

Insertion loss: Reduction in transmitted power caused by insertion of a passive network.

Isolation: Attenuation between ports in an undesired transmission direction.

Mason’s invariant gain: Gain invariant under lossless reciprocal embeddings.

Power wave: Normalized incident or reflected wave associated with power under a specified convention.

Reciprocity: Property producing equal forward and reverse transfer parameters under consistent references.

Reference impedance: Impedance used to define incident and reflected waves.

Reference plane: Physical or electrical location at which port variables are defined.

Renormalization: Re-expression of S-parameters using different reference impedances.

Return loss: Logarithmic measure of reflected-wave magnitude.

Scattering matrix: Matrix relating reflected waves to incident waves.

Stability circle: Boundary separating stable and unstable source or load terminations.

Transducer gain: Power delivered to a load divided by power available from the source.

Unconditional stability: Stability for every passive source and load termination.

Review Questions

Short Answer Questions

1. Why are RMS and peak-phasor conventions not interchangeable in wave-power calculations?

2. What physical conditions are represented by \(S_{11}\) and \(S_{21}\)?

3. Distinguish reference-impedance renormalization from reference-plane movement.

4. State the two Rollett conditions for unconditional stability.

5. Why cannot ordinary S matrices be directly multiplied to represent cascades?

Long Answer Questions

1. Derive the relationship between normalized impedance and scattering matrices for an N-port.

2. Explain how ABCD and chain-scattering matrices are used to analyze cascaded RF systems.

3. Discuss reciprocity, symmetry, passivity and losslessness as distinct network properties.

4. Develop the transducer-gain expression and explain the roles of source and load mismatch.

5. Compare unilateral approximation error, maximum available gain and Mason’s invariant gain.

Analytical / Application Questions

1. A 50 Ω two-port has specified complex S-parameters. Convert them to Y-parameters, test reciprocity and determine whether the network is passive.

2. For a potentially unstable transistor, calculate its source and load stability circles, identify the stable regions and propose stabilizing terminations.

3. Design a de-embedding and renormalization workflow for an amplifier measured through nonidentical input and output fixtures.

Multiple Choice Questions with Answers

Question 1

Under the RMS-phasor and real-reference-impedance convention, what is the net average power entering a port?

A. |a|² + |b|²

B. (|a|² − |b|²)/2

C. |a|² − |b|²

D. 2(|a|² − |b|²)

Correct Answer: C. |a|² − |b|²

Question 2

A 100 Ω load has what reflection coefficient when renormalized to 75 Ω?

A. 0.1429

B. 0.2500

C. 0.3333

D. −0.1429

Correct Answer: A. 0.1429

Question 3

Which operation correctly combines the ABCD matrices of two cascaded two-ports?

A. Element-by-element addition

B. Ordered matrix multiplication

C. Matrix transposition

D. Element-by-element multiplication

Correct Answer: B. Ordered matrix multiplication

Question 4

For a reciprocal two-port with equal and consistently defined reference impedances, which relation holds?

A. S11 = S22

B. S11 = 0

C. S12 = 0

D. S12 = S21

Correct Answer: D. S12 = S21

Question 5

A two-port has K = 1.4 and |Δ| = 0.6. What is established?

A. It is unconditionally stable for passive terminations

B. It is necessarily lossless

C. It is unilateral

D. It is necessarily unstable with matched terminations

Correct Answer: A. It is unconditionally stable for passive terminations

Question 6

For the chain-scattering definition used in the chapter, which expression recovers S21?

A. S21 = T11

B. S21 = 1/T11

C. S21 = T21/T11

D. S21 = det(T)/T11

Correct Answer: B. S21 = 1/T11

Question 7

A matched passive two-port has |S21| = 0.5. What is its insertion loss?

A. 3.01 dB

B. −6.02 dB

C. 6.02 dB

D. 12.04 dB

Correct Answer: C. 6.02 dB

Question 8

A lossless matched line of length l is added at a one-port reference plane. What factor multiplies its reflection coefficient?

A. e^(−jβl)

B. e^(jβl)

C. e^(j2βl)

D. e^(−j2βl)

Correct Answer: D. e^(−j2βl)

Question 9

How is the stable side of a stability circle determined reliably?

A. Test a known point such as the Smith-chart origin

B. Always select the circle interior

C. Always select the region nearest Γ = 1

D. Select the region with the largest phase angle

Correct Answer: A. Test a known point such as the Smith-chart origin

Question 10

Which condition is necessary and sufficient for passivity of an S matrix at a frequency under real power-wave references?

A. Every element of S is real

B. I − S†S is positive semidefinite

C. det(S) = 1

D. S11 = S22

Correct Answer: B. I − S†S is positive semidefinite