Sources and References

• D. M. Pozar, Microwave Engineering, 4th ed., John Wiley & Sons, 2012. (Primary textbook — the canonical reference for virtually all topics in this course.)
• G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, 2nd ed., Prentice Hall, 1997. (Amplifier stability, noise circles, and transistor oscillator design.)
• R. Ludwig and P. Bogdanov, RF Circuit Design: Theory and Applications, 2nd ed., Pearson/Prentice Hall, 2009. (Lumped-element matching, resonators, and system perspectives.)
• K. C. Gupta, R. Garg, I. Bahl, and P. Trivedi, Microstrip Lines and Slotlines, 2nd ed., Artech House, 1996. (Microstrip characteristic impedance and dispersion formulas.)
• K. Kurokawa, “Some Basic Characteristics of Broadband Negative Resistance Oscillator Circuits,” Bell System Technical Journal, vol. 48, pp. 1937–1955, 1969. (Kurokawa oscillation stability conditions.)
• D. B. Leeson, “A Simple Model of Feedback Oscillator Noise Spectrum,” Proceedings of the IEEE, vol. 54, pp. 329–330, 1966. (Phase noise model.)
• B. Razavi, RF Microelectronics, 2nd ed., Prentice Hall, 2012. (RF system context, LNA and mixer design from a circuit perspective.)
• MIT OpenCourseWare 6.013/6.014 — Electromagnetics and Applications. Public lecture notes, freely available at ocw.mit.edu. (Transmission line theory, waveguides, supplementary derivations.)

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Chapter 1: Foundations of Transmission Line Theory

1.1 The Telegrapher’s Equations

At microwave frequencies, the wavelength of electromagnetic signals becomes comparable to or smaller than the physical dimensions of circuit elements. A wire that is negligibly short at 60 Hz may span many wavelengths at 10 GHz, fundamentally changing how it behaves. The lumped-element approximation of conventional circuit theory — where voltage and current are assumed uniform throughout a component — breaks down completely. We must instead treat interconnects as distributed-parameter transmission lines and solve for wave propagation along them.

Consider a uniform two-conductor transmission line oriented along the \(z\)-axis. A short segment of length \(\Delta z\) can be modeled by four distributed parameters:

• \(R\) — resistance per unit length \([\Omega/\text{m}]\), representing conductor ohmic loss
• \(L\) — inductance per unit length \([\text{H/m}]\), representing the stored magnetic energy
• \(G\) — conductance per unit length \([\text{S/m}]\), representing dielectric leakage loss
• \(C\) — capacitance per unit length \([\text{F/m}]\), representing the stored electric energy

Applying Kirchhoff’s voltage and current laws to the incremental segment, taking the limit \(\Delta z \to 0\), and working in the frequency domain (phasor representation with time dependence \(e^{j\omega t}\)), we arrive at the telegrapher’s equations:

\[ \frac{dV(z)}{dz} = -(R + j\omega L)\, I(z) \]\[ \frac{dI(z)}{dz} = -(G + j\omega C)\, V(z) \]

These are two coupled first-order ordinary differential equations. Differentiating the first equation with respect to \(z\) and substituting the second gives a decoupled second-order equation for voltage:

\[ \frac{d^2 V(z)}{dz^2} = \gamma^2 V(z) \]

where the complex propagation constant \(\gamma\) is defined as

\[ \gamma \equiv \alpha + j\beta = \sqrt{(R + j\omega L)(G + j\omega C)} \]

Here \(\alpha\) is the attenuation constant in Np/m and \(\beta\) is the phase constant in rad/m. A completely analogous equation holds for \(I(z)\).

1.2 Wave Solutions and Characteristic Impedance

The general solution to the wave equation for voltage is a superposition of a forward-traveling wave and a backward-traveling wave:

\[ V(z) = V_0^+ e^{-\gamma z} + V_0^- e^{+\gamma z} \]

The corresponding current is obtained by substituting back into the first telegrapher’s equation:

\[ I(z) = \frac{V_0^+}{Z_0} e^{-\gamma z} - \frac{V_0^-}{Z_0} e^{+\gamma z} \]

The characteristic impedance \(Z_0\) is the ratio of voltage to current for a single traveling wave:

\[ Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} \]

The physical interpretation is crucial: \(Z_0\) is not the input impedance of the line (which depends on the load and the line length), but rather an intrinsic property analogous to the wave impedance of free space, \(\eta_0 = \sqrt{\mu_0/\varepsilon_0} \approx 377\,\Omega\).

1.2.1 The Lossless Line

For a lossless line, we set \(R = G = 0\). The propagation constant simplifies to a purely imaginary quantity:

\[ \gamma = j\omega\sqrt{LC} \equiv j\beta \]

The characteristic impedance becomes real:

\[ Z_0 = \sqrt{\frac{L}{C}} \in \mathbb{R} \]

The wave solutions reduce to

\[ V(z) = V_0^+ e^{-j\beta z} + V_0^- e^{+j\beta z} \]

The phase velocity — the speed at which a constant-phase surface travels — is

\[ v_p = \frac{\omega}{\beta} = \frac{1}{\sqrt{LC}} \]

For a line in a medium with relative permittivity \(\varepsilon_r\) and relative permeability \(\mu_r = 1\):

\[ v_p = \frac{c}{\sqrt{\varepsilon_r}}, \qquad \beta = \frac{\omega\sqrt{\varepsilon_r}}{c} \]

where \(c = 3 \times 10^8\) m/s is the speed of light in vacuum.

1.2.2 The Low-Loss Approximation

Practical transmission lines are not perfectly lossless. When losses are small — specifically when \(R \ll \omega L\) and \(G \ll \omega C\) — we may expand \(\gamma\) in a Taylor series to obtain the low-loss approximation:

\[ \alpha \approx \frac{R}{2Z_0} + \frac{G Z_0}{2} \]\[ \beta \approx \omega\sqrt{LC}\left(1 + \frac{1}{8}\left(\frac{R}{\omega L} - \frac{G}{\omega C}\right)^2 + \cdots\right) \approx \omega\sqrt{LC} \]\[ Z_0 \approx \sqrt{\frac{L}{C}}\left(1 - \frac{j}{2}\left(\frac{R}{\omega L} - \frac{G}{\omega C}\right)\right) \]

In practice, for microstrip and coaxial lines at GHz frequencies, \(Z_0\) is very nearly real and equal to its lossless value, justifying the use of real \(Z_0\) in most design calculations.

1.3 Reflection Coefficient and Standing Waves

Consider a transmission line of characteristic impedance \(Z_0\) terminated in a load impedance \(Z_L\) at \(z = 0\). The boundary condition at the load is

\[ \frac{V(0)}{I(0)} = Z_L \]

Applying this to the wave solutions:

\[ \frac{V_0^+ + V_0^-}{V_0^+ - V_0^-} \cdot Z_0 = Z_L \]

Solving for the ratio \(V_0^- / V_0^+\) gives the voltage reflection coefficient at the load:

\[ \Gamma_L = \frac{V_0^-}{V_0^+} = \frac{Z_L - Z_0}{Z_L + Z_0} \]

At a point \(z = -\ell\) (distance \(\ell\) from the load, toward the generator), the reflection coefficient is

\[ \Gamma(-\ell) = \frac{V_0^- e^{-\gamma\ell}}{V_0^+ e^{+\gamma\ell}} = \Gamma_L e^{-2\gamma\ell} \]

For a lossless line:

\[ \Gamma(-\ell) = \Gamma_L e^{-2j\beta\ell} \]

The magnitude of \(\Gamma\) is constant along a lossless line; only its phase rotates at twice the rate \(2\beta\) per unit length (i.e., \(720°\) per wavelength, or equivalently \(180°\) per half-wavelength of line).

1.3.1 Voltage Standing Wave Ratio

The superposition of incident and reflected waves creates a standing wave pattern. The Voltage Standing Wave Ratio (VSWR) characterises the severity of the mismatch:

\[ \text{VSWR} = S = \frac{V_{\max}}{V_{\min}} = \frac{1 + |\Gamma|}{1 - |\Gamma|} \]

The positions of voltage maxima occur where the incident and reflected waves add in phase, and minima occur where they cancel. Successive maxima are separated by \(\lambda/2\).

1.4 Input Impedance of a Terminated Transmission Line

The voltage and current at a distance \(\ell\) from the load (\(z = -\ell\)) on a lossless line are:

\[ V(-\ell) = V_0^+ \left(e^{j\beta\ell} + \Gamma_L e^{-j\beta\ell}\right) \]\[ I(-\ell) = \frac{V_0^+}{Z_0}\left(e^{j\beta\ell} - \Gamma_L e^{-j\beta\ell}\right) \]

The input impedance looking into the line at \(z = -\ell\) is:

\[ Z_{\text{in}}(\ell) = Z_0 \frac{Z_L + jZ_0\tan(\beta\ell)}{Z_0 + jZ_L\tan(\beta\ell)} \]

This is one of the most important formulas in microwave engineering, encapsulating how a transmission line transforms impedances. Several special cases are worth internalizing:

Short-circuit termination (\(Z_L = 0\)):

\[ Z_{\text{in}} = jZ_0\tan(\beta\ell) \]

This is purely reactive. For \(0 < \ell < \lambda/4\), \(Z_{\text{in}}\) is inductive; for \(\lambda/4 < \ell < \lambda/2\), it is capacitive.

Open-circuit termination (\(Z_L \to \infty\)):

\[ Z_{\text{in}} = -jZ_0\cot(\beta\ell) \]

Quarter-wave transformer (\(\ell = \lambda/4\), so \(\beta\ell = \pi/2\)):

\[ Z_{\text{in}} = \frac{Z_0^2}{Z_L} \]

This is the impedance inversion property: a quarter-wave line maps a load \(Z_L\) to its inverse (scaled by \(Z_0^2\)).

Half-wave line (\(\ell = \lambda/2\), so \(\beta\ell = \pi\)):

\[ Z_{\text{in}} = Z_L \]

A half-wave line reproduces the load impedance at its input, regardless of \(Z_0\).

1.5 Power Flow and the Return Loss

The time-averaged power delivered to the load on a lossless line is

\[ P_{\text{del}} = \frac{|V_0^+|^2}{2Z_0}(1 - |\Gamma_L|^2) \]

The first factor is the incident power \(P_{\text{inc}} = |V_0^+|^2 / (2Z_0)\), and the second factor accounts for the fraction reflected. The return loss (RL) is a positive-dB measure of how much power is reflected:

\[ \text{RL} = -20\log_{10}|\Gamma_L| \quad [\text{dB}] \]

A matched load (\(\Gamma_L = 0\)) gives \(\text{RL} = \infty\) dB — no return. A short or open (\(|\Gamma_L| = 1\)) gives \(\text{RL} = 0\) dB — total reflection.

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Chapter 2: The Smith Chart

2.1 The Reflection Coefficient Plane

The Smith chart is a graphical tool for performing transmission line calculations, invented by Phillip H. Smith at Bell Labs in 1939. It is constructed by mapping the complex impedance plane to the complex \(\Gamma\)-plane using the bilinear (Möbius) transformation:

\[ \Gamma = \frac{Z_L/Z_0 - 1}{Z_L/Z_0 + 1} = \frac{z - 1}{z + 1} \]

where \(z = Z_L/Z_0 = r + jx\) is the normalized impedance. The inverse mapping is:

\[ z = \frac{1 + \Gamma}{1 - \Gamma} \]

Since passive loads have \(\text{Re}(Z_L) \geq 0\) (i.e., \(r \geq 0\)), all passive impedances map inside or on the unit circle \(|\Gamma| \leq 1\).

2.2 Constant-\(r\) and Constant-\(x\) Circles

Writing \(\Gamma = u + jv\), the condition \(\text{Re}(z) = r\) (constant normalized resistance) maps to:

\[ \left(u - \frac{r}{1+r}\right)^2 + v^2 = \left(\frac{1}{1+r}\right)^2 \]

These are circles centered at \(\left(\frac{r}{1+r}, 0\right)\) with radius \(\frac{1}{1+r}\). All pass through the point \(\Gamma = 1\) (open circuit). Special cases:

• \(r = 0\): unit circle centered at origin (the purely reactive locus)
• \(r = 1\): circle centered at \((1/2, 0)\) with radius \(1/2\)
• \(r \to \infty\): collapses to the point \(\Gamma = 1\)

The condition \(\text{Im}(z) = x\) (constant normalized reactance) maps to:

\[ (u - 1)^2 + \left(v - \frac{1}{x}\right)^2 = \left(\frac{1}{x}\right)^2 \]

These are circles centered at \(\left(1, \frac{1}{x}\right)\) with radius \(\frac{1}{|x|}\). The upper half of the \(\Gamma\)-plane (\(v > 0\)) contains inductive reactances (\(x > 0\)); the lower half contains capacitive reactances (\(x < 0\)).

2.3 Moving Along the Line

A key feature of the Smith chart is the ease with which transmission line effects are incorporated. Moving a distance \(\ell\) from the load toward the generator on a lossless line rotates the \(\Gamma\) vector clockwise by angle \(2\beta\ell\):

\[ \Gamma_{\text{in}} = \Gamma_L e^{-2j\beta\ell} \]

One full revolution (\(360°\) rotation) corresponds to \(\ell = \lambda/2\). The outermost scale of the Smith chart is graduated in “wavelengths toward generator” (WTG) from 0 to 0.5 per half-wavelength.

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Chapter 3: Impedance Matching Networks

Achieving maximum power transfer and minimizing reflections between source and load are central goals in RF circuit design. Impedance matching is the technique of inserting a lossless network between source and load so that the source sees a conjugate-matched impedance.

3.1 Quarter-Wave Transformer

The quarter-wave transformer is the simplest single-frequency matching network. To match a real load \(R_L\) to a real source impedance \(R_s\), one inserts a \(\lambda/4\) section of transmission line with characteristic impedance:

\[ Z_1 = \sqrt{R_s R_L} \]

At the design frequency \(f_0\), the \(\lambda/4\) section transforms \(R_L\) to \(R_s\) exactly. The reflection coefficient as a function of frequency is

\[ \Gamma(f) \approx \frac{R_L - R_s}{2\sqrt{R_L R_s}}\cos\left(\frac{\pi f}{2f_0}\right) \quad (\text{near } f_0) \]

3.2 Single-Stub Tuning

Single-stub tuning uses a section of open- or short-circuited transmission line (a stub) connected in parallel (shunt) or in series with the main line at a specific distance from the load. The design procedure proceeds in two steps:

1. Choose distance \(d\) from the load such that the input admittance \(Y_{\text{in}}(d)\) has real part equal to \(Y_0 = 1/Z_0\).
2. Choose stub length \(\ell_s\) to produce a susceptance equal and opposite to the imaginary part of \(Y_{\text{in}}(d)\).

For a shunt-connected stub on a lossless \(Z_0\) line, the normalized admittance at distance \(d\) is:

\[ y(d) = \frac{1 - |\Gamma_L|^2}{|1 + \Gamma_L e^{-2j\beta d}|^2} + j\,\text{Im}\left(\frac{1 + \Gamma_L e^{-2j\beta d}}{1 - \Gamma_L e^{-2j\beta d}}\right) \]

Setting \(\text{Re}[y(d)] = 1\) determines the required \(d\). The stub susceptance for a short-circuited stub is \(b_s = -\cot(\beta\ell_s)\), and for an open-circuited stub, \(b_s = \tan(\beta\ell_s)\). There are generally two solutions for \(d\) per half-wavelength; both yield valid matching points.

3.3 Double-Stub Tuning

Single-stub matching fails for loads in a certain forbidden region of the Smith chart (for a fixed stub spacing). Double-stub tuning circumvents this by using two stubs at fixed positions (typically \(d = \lambda/8\) apart) whose lengths are independently variable. The design is most naturally performed on the Smith chart:

1. Rotate the load admittance by \(\lambda/8\) toward the generator to the plane of stub 1. Add the susceptance of stub 1.
2. The resulting admittance must lie on one of two circles on the Smith chart that, when rotated a further \(\lambda/8\), brings it to a circle of \(g = 1\). Add stub 2 susceptance to bring the total admittance to \(1 + j0\).

The forbidden region for double-stub tuning (with \(\lambda/8\) spacing) consists of loads whose normalised conductance \(g > 2\). Changing the stub spacing or using a triple-stub configuration eliminates this restriction.

3.4 Lumped-Element Matching: The L-Network

At lower microwave frequencies (say, below 2–3 GHz) or in chip-scale circuits where distributed elements are impractically large, lumped-element matching networks are preferred. The simplest topology is the L-network, consisting of two reactive elements (inductors or capacitors) in an L configuration.

For matching a real source \(R_s\) to a real load \(R_L > R_s\) (the load is the larger resistance), the shunt element is placed across the load and the series element connects source to shunt node. The loaded Q factor of the network is:

\[ Q = \sqrt{\frac{R_L}{R_s} - 1} \]

The required element values satisfy:

\[ X_s = Q \cdot R_s, \qquad B_p = \frac{Q}{R_L} \]

where \(X_s\) is the series reactance and \(B_p\) is the shunt susceptance. Two solutions exist (one with a shunt capacitor and series inductor, one with shunt inductor and series capacitor), yielding highpass and lowpass characteristics respectively.

3.4.1 Pi and T Networks

The π-network (shunt-series-shunt) and T-network (series-shunt-series) allow the designer to specify the Q factor independently of the impedance ratio. For a π-network matching \(R_L\) to \(R_s\) with chosen loaded \(Q\), introduce a virtual resistance \(R_v\) at the junction such that:

\[ R_v = \frac{R_s}{\left(1 + \left(\frac{Q}{2}\right)^2\right)} \quad \text{(for certain configurations)} \]

The left half-section transforms \(R_s \to R_v\) and the right half-section transforms \(R_v \to R_L\), each half being an L-network. The overall Q of the π-network is higher than the minimum, providing better harmonic rejection but narrower bandwidth.

The loaded Q in terms of component values for a π-network is:

\[ Q_L \approx \frac{f_0}{\Delta f_{-3\,\text{dB}}} \]

Higher Q implies smaller \(-3\,\text{dB}\) bandwidth and steeper out-of-band roll-off — useful for frequency selectivity near an antenna interface.

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Chapter 4: Microwave Network Analysis

4.1 Limitations of Z, Y, and ABCD Parameters at Microwave Frequencies

At low frequencies, the behavior of multiport networks is completely characterized by their \(Z\), \(Y\), or ABCD matrices, obtained by setting terminal voltages or currents to specific values. At microwave frequencies, two practical difficulties arise:

1. Short and open circuits are impractical. Achieving a true short at 10 GHz requires a quarter-wave stub with zero contact resistance — any small inductance or resistance will detune it. Similarly, an open circuit radiates and couples to the environment.

2. Voltages and currents are not unique. For non-TEM waveguide modes, voltage and current cannot be defined unambiguously.

The solution is scattering parameters (S-parameters), which relate incident and reflected power wave amplitudes at each port, and are defined with all other ports terminated in their reference impedances — conditions that are straightforward to realize experimentally.

4.2 S-Parameter Definition

For an \(N\)-port network, define the power wave variables at port \(i\):

\[ a_i = \frac{V_i + Z_0 I_i}{2\sqrt{R_0}}, \qquad b_i = \frac{V_i - Z_0 I_i}{2\sqrt{R_0}} \]

where \(V_i\) and \(I_i\) are the voltage and current at port \(i\), and \(R_0 = \text{Re}(Z_0)\) is the reference resistance. The quantity \(|a_i|^2/2\) is the power delivered toward the network at port \(i\), and \(|b_i|^2/2\) is the power traveling away from the network at port \(i\).

The S-matrix relates these:

\[ \mathbf{b} = \mathbf{S}\,\mathbf{a} \]

The individual S-parameters are:

\[ S_{ij} = \left.\frac{b_i}{a_j}\right|_{a_k = 0 \text{ for } k \neq j} \]

4.3 Properties of the S-Matrix

For a two-port network, the unitary condition yields three independent equations:

\[ |S_{11}|^2 + |S_{21}|^2 = 1 \]\[ |S_{12}|^2 + |S_{22}|^2 = 1 \]\[ S_{11}^* S_{12} + S_{21}^* S_{22} = 0 \]

4.4 Signal Flow Graphs

Signal flow graphs (SFGs) are a graphical technique for evaluating complex S-parameter expressions in cascaded and interconnected networks, using Mason’s gain rule.

Each port quantity (\(a_i\), \(b_i\)) is represented as a node. Each S-parameter \(S_{ij}\) is a directed branch from node \(a_j\) to node \(b_i\) with gain \(S_{ij}\). A reflection from a load \(\Gamma_L\) contributes a branch from \(b_i\) back to \(a_i\) with gain \(\Gamma_L\).

Mason’s rule states that the transfer function from source node \(X_s\) to output node \(X_o\) is:

\[ H = \frac{\sum_k P_k \Delta_k}{\Delta} \]

where:

• \(P_k\) = gain of the \(k\)-th forward path from \(X_s\) to \(X_o\)
• \(\Delta = 1 - \sum L_1 + \sum L_2 - \cdots\) = graph determinant (\(L_n\) are sums of products of \(n\) non-touching loops)
• \(\Delta_k\) = cofactor of \(P_k\) (determinant of the graph with all nodes touching path \(k\) removed)

SFGs are particularly powerful when computing the input/output reflection and transmission of amplifiers with feedback.

4.5 Converting Between S and Z/Y Parameters

For a two-port with reference impedance \(Z_0\):

\[ \mathbf{Z} = Z_0(\mathbf{I} + \mathbf{S})(\mathbf{I} - \mathbf{S})^{-1} \]\[ \mathbf{S} = (\mathbf{Z} - Z_0\mathbf{I})(\mathbf{Z} + Z_0\mathbf{I})^{-1} \]

These can be verified by substituting the power wave definitions into the S-matrix relation and solving for \(\mathbf{Z}\) from \(\mathbf{V} = \mathbf{Z}\mathbf{I}\).

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Chapter 5: Planar Microwave Technology

5.1 Microstrip

Microstrip is the most widely used planar transmission line structure in microwave engineering. It consists of a metallic strip of width \(W\) on top of a dielectric substrate of thickness \(h\) and relative permittivity \(\varepsilon_r\), with a ground plane on the bottom surface.

Because the fields exist partly in the dielectric (below the strip) and partly in air (above the strip), microstrip does not support a pure TEM mode. The dominant mode is a quasi-TEM mode, and the line is characterized by an effective relative permittivity \(\varepsilon_{\text{eff}}\) that lies between 1 (air) and \(\varepsilon_r\) (full dielectric):

\[ \varepsilon_{\text{eff}} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2}\,F(W/h) \]

where, for \(W/h \leq 1\):

\[ F(W/h) = \left(1 + 12\frac{h}{W}\right)^{-1/2} + 0.04\left(1 - \frac{W}{h}\right)^2 \]

and for \(W/h \geq 1\):

\[ F(W/h) = \left(1 + 12\frac{h}{W}\right)^{-1/2} \]

The characteristic impedance formulas (Hammerstad approximation) are:

For \(W/h \leq 1\):

\[ Z_0 = \frac{60}{\sqrt{\varepsilon_{\text{eff}}}}\ln\left(\frac{8h}{W} + \frac{W}{4h}\right) \]

For \(W/h \geq 1\):

\[ Z_0 = \frac{120\pi}{\sqrt{\varepsilon_{\text{eff}}}\left[\frac{W}{h} + 1.393 + 0.667\ln\left(\frac{W}{h} + 1.444\right)\right]} \]

The synthesis problem (finding \(W/h\) given \(Z_0\) and \(\varepsilon_r\)) also has closed-form approximations and is implemented in all microwave CAD tools.

5.2 Microstrip Losses

Attenuation in microstrip arises from three mechanisms:

Conductor loss (dominant at lower frequencies):

\[ \alpha_c \approx \frac{R_s}{Z_0 W} \quad [\text{Np/m}] \]

where \(R_s = \sqrt{\pi f \mu_0 / \sigma}\) is the surface resistance of the conductor (skin effect).

Dielectric loss:

\[ \alpha_d = \frac{\pi f}{c}\frac{\varepsilon_r(\varepsilon_{\text{eff}} - 1)\tan\delta}{\sqrt{\varepsilon_{\text{eff}}}(\varepsilon_r - 1)} \quad [\text{Np/m}] \]

where \(\tan\delta\) is the loss tangent of the substrate.

Radiation loss becomes significant at higher frequencies and for discontinuities. Well-designed layouts minimize radiation by avoiding sharp bends and using rounded corners or mitered bends.

5.3 Stripline

Stripline consists of a conductor strip centered between two ground planes in a homogeneous dielectric. Unlike microstrip, stripline supports a true TEM mode, making it non-dispersive and easier to analyze. The effective permittivity equals the substrate \(\varepsilon_r\), so the phase velocity is \(c/\sqrt{\varepsilon_r}\).

The characteristic impedance of stripline (conductor width \(W\), substrate thickness \(b\), centered strip thickness \(t \to 0\)) is approximately:

\[ Z_0 = \frac{30\pi}{\sqrt{\varepsilon_r}} \frac{b}{W_e + 0.441b} \]

where \(W_e = W - (t/\pi)\ln(4e/\sqrt{(t/b)^2 + (t/(\pi W))^2})\) is an effective width accounting for finite strip thickness. Stripline is used when isolation from radiation and a homogeneous medium are required, but it is harder to fabricate and integrate with active components than microstrip.

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Chapter 6: Microwave Passive Components

6.1 Microwave Resonators

A microwave resonator is an energy-storage element that supports oscillating fields at discrete frequencies (resonant modes). At microwave frequencies, lumped \(LC\) resonators become impractical due to parasitic effects, and distributed resonators are used instead.

6.1.1 Transmission Line Resonators

A short-circuited half-wave (\(\lambda/2\)) section of transmission line resonates when its input impedance is zero (series resonance) or infinite (parallel resonance). Near the resonant frequency \(\omega_0\), the input impedance of a \(\lambda/2\) resonator behaves as:

\[ Z_{\text{in}} \approx R_s + j\frac{\pi}{2\omega_0}(\omega - \omega_0) \cdot \frac{2Z_0}{\pi} \]

mimicking a series \(RLC\) circuit. The unloaded Q factor measures the resonator’s internal energy loss:

\[ Q_0 = \frac{\omega_0 W_{\text{stored}}}{P_{\text{loss}}} = \frac{\beta}{2\alpha} \]

6.1.2 Cavity Resonators

Metallic cavity resonators achieve very high Q values (\(10^3\)–\(10^5\)), far exceeding what is achievable with planar resonators. A rectangular cavity of dimensions \(a \times b \times d\) (with \(a \geq d \geq b\)) supports TE and TM modes with resonant frequencies:

\[ f_{mnp} = \frac{c}{2\sqrt{\mu_r\varepsilon_r}}\sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2 + \left(\frac{p}{d}\right)^2} \]

The dominant mode is TE\(_{101}\) (for \(a \geq d\)), with resonant frequency \(f_{101} = c/(2\sqrt{\mu_r\varepsilon_r})\sqrt{1/a^2 + 1/d^2}\).

6.2 Directional Couplers

A directional coupler is a four-port network that samples a fraction of the power from a transmission line while maintaining directivity — passing the sampled signal to one output port and a small residual to another. The four ports are: input, direct (through), coupled, and isolated.

For a lossless, reciprocal, matched four-port (all ports matched: \(S_{ii} = 0\)), the S-matrix takes the form:

\[ \mathbf{S} = \begin{pmatrix} 0 & \alpha & j\beta & 0 \\ \alpha & 0 & 0 & j\beta \\ j\beta & 0 & 0 & \alpha \\ 0 & j\beta & \alpha & 0 \end{pmatrix} \]

with \(\alpha^2 + \beta^2 = 1\) (losslessness). Here \(\alpha = \cos\theta\) and \(\beta = \sin\theta\) for a coupled-line coupler with coupling angle \(\theta\).

6.2.1 Coupled-Line Coupler (Backward Wave)

A backward-wave coupler uses two closely spaced parallel transmission lines. The coupled and isolated ports are defined by the direction of power flow in the coupled arm. For a lossless line coupler:

\[ C = 20\log_{10}\left(\frac{Z_{0e} - Z_{0o}}{Z_{0e} + Z_{0o}}\right) \quad [\text{dB}] \]

where \(Z_{0e}\) and \(Z_{0o}\) are the even- and odd-mode characteristic impedances of the coupled pair. These are related to the desired coupling coefficient \(k\) by:

\[ Z_{0e} = Z_0\sqrt{\frac{1+k}{1-k}}, \qquad Z_{0o} = Z_0\sqrt{\frac{1-k}{1+k}} \]

Tighter coupling (smaller \(C\) in dB, e.g., 3 dB) requires larger gap dimensions, which may be impractical for planar realizations.

6.2.2 Bethe-Hole and Multi-Hole Waveguide Couplers

For waveguide structures, directional coupling is achieved by small apertures (holes) in the common wall between two waveguides. A single circular aperture (Bethe hole) couples electric and magnetic dipoles; these two coupling mechanisms cancel in one direction (directivity), but the inherent directivity of a single hole is limited. Multi-hole couplers (Moreno or Schwinger designs) use multiple apertures spaced \(\lambda_g/4\) apart to achieve high directivity over a broad band.

6.3 Power Dividers and Combiners

A power divider splits an input signal into two or more output signals. A power combiner performs the reverse operation.

6.3.1 T-Junction Power Divider

The simplest power divider is a T-junction (a three-port formed by splitting one transmission line into two). For a lossless, matched, reciprocal three-port to exist, it can be shown from the lossless condition that it is impossible: at least one port must be mismatched or the device must be non-reciprocal or lossy. This is a fundamental constraint:

The T-junction divides power but creates a mismatch at the input when the output ports are not loaded identically.

6.3.2 Wilkinson Power Divider

The Wilkinson divider resolves the impossibility theorem by introducing a resistor between the two output ports. It achieves:

• All ports matched at the design frequency
• Reciprocal
• Isolation between output ports (due to the resistor)
• Low loss when the outputs are equal (resistor carries no current for matched symmetric excitation)

For a two-way equal-split Wilkinson divider matching to \(Z_0\):

\[ Z_1 = Z_2 = \sqrt{2}\,Z_0 \quad (\text{quarter-wave arms}) \]\[ R = 2Z_0 \quad (\text{isolation resistor between outputs}) \]

The S-matrix at the design frequency (\(\lambda/4\) arm length) is:

\[ \mathbf{S} = \frac{-j}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} \]

6.4 Hybrid Junctions

A hybrid junction is a four-port that divides power equally (\(-3\,\text{dB}\) split) with a specific phase relationship.

6.4.1 Branch-Line Hybrid (90° Hybrid)

The branch-line coupler consists of two transmission line sections in a square loop with two shunt branches. At the design frequency, input at port 1 produces equal-amplitude outputs at ports 2 and 3 with a 90° phase difference, and no output at port 4 (isolated). The S-matrix is:

\[ \mathbf{S} = \frac{-1}{\sqrt{2}}\begin{pmatrix} 0 & j & 1 & 0 \\ j & 0 & 0 & 1 \\ 1 & 0 & 0 & j \\ 0 & 1 & j & 0 \end{pmatrix} \]

The series arms have \(Z_0/\sqrt{2}\) impedance and the shunt arms have \(Z_0\).

6.4.2 Rat-Race Ring Hybrid (180° Hybrid)

The rat-race (ring) coupler consists of a circular loop of total circumference \(3\lambda/2\), with four ports attached at intervals of \(\lambda/4\), \(\lambda/4\), \(\lambda/4\), and \(3\lambda/4\). Input at port 1 produces in-phase equal-split outputs at ports 2 and 4, with isolation at port 3. The ring characteristic impedance is \(Z_0\sqrt{2}\). The 180° hybrid is used as a balanced-to-unbalanced transformer (balun), in balanced mixers, and in phase-coherent combining.

6.5 Circulators and Isolators

A circulator is a three-port non-reciprocal device (using ferrite material biased by a static magnetic field) in which power circulates in only one direction: from port 1 to 2, 2 to 3, and 3 to 1. The ideal circulator S-matrix is:

\[ \mathbf{S} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \]

An isolator is a two-port non-reciprocal device that passes signals in one direction with low loss and blocks signals in the reverse direction with high attenuation. It is realized by terminating one port of a circulator in a matched load.

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Chapter 7: Microwave Filter Design

7.1 The Insertion Loss Method

Microwave filters are designed systematically using the insertion loss method, which starts from a prototype low-pass filter response and applies frequency and impedance transformations to reach the target specifications.

The insertion loss (IL) of a two-port network is:

\[ IL(\omega) = -10\log_{10}\left(1 - |\Gamma(\omega)|^2\right) = -10\log_{10}|S_{21}(\omega)|^2 \quad [\text{dB}] \]

Four standard response types are used:

Maximally flat (Butterworth): The magnitude squared of the transmission coefficient is

\[ |S_{21}(j\omega)|^2 = \frac{1}{1 + \left(\omega/\omega_c\right)^{2N}} \]

where \(N\) is the filter order and \(\omega_c\) is the \(-3\,\text{dB}\) cutoff frequency. The passband is maximally flat; out-of-band roll-off is \(-6N\) dB/octave.

Equal-ripple (Chebyshev): The transmission squared is

\[ |S_{21}(j\omega)|^2 = \frac{1}{1 + \varepsilon^2 T_N^2(\omega/\omega_c)} \]

where \(T_N(x)\) is the \(N\)-th order Chebyshev polynomial and \(\varepsilon\) controls the passband ripple magnitude. The Chebyshev filter achieves a steeper roll-off than Butterworth for the same order, at the cost of passband ripple. The passband ripple (in dB) is \(L_{\text{Ar}} = 10\log_{10}(1 + \varepsilon^2)\).

Elliptic (Cauer): Achieves the steepest roll-off for a given order by placing finite transmission zeros in the stopband. The prototype is more complex to synthesize.

Linear phase (Bessel): Optimized for constant group delay (linear phase), at the cost of gradual roll-off. Used when pulse fidelity is important.

7.2 Low-Pass Prototype Filters

The low-pass prototype is a ladder network of normalized \(g\)-values (element values scaled to \(\omega_c = 1\,\text{rad/s}\), \(Z_0 = 1\,\Omega\)). For a Butterworth prototype of order \(N\):

\[ g_k = 2\sin\left(\frac{(2k-1)\pi}{2N}\right), \quad k = 1, 2, \ldots, N; \quad g_0 = g_{N+1} = 1 \]

For a Chebyshev prototype with passband ripple \(L_{\text{Ar}}\,\text{dB}\):

\[ \beta = \ln\left(\coth\frac{L_{\text{Ar}}}{17.37}\right), \quad \gamma = \sinh\frac{\beta}{2N} \]\[ a_k = \sin\left(\frac{(2k-1)\pi}{2N}\right), \quad b_k = \gamma^2 + \sin^2\left(\frac{k\pi}{N}\right) \]\[ g_1 = \frac{2a_1}{\gamma}, \quad g_k = \frac{4a_{k-1}a_k}{b_{k-1}g_{k-1}} \quad (k = 2,\ldots,N) \]

The source and load resistances are \(g_0 = 1\) and \(g_{N+1} = 1\) (for odd \(N\)) or a calculated value (for even \(N\), since Chebyshev with even order requires \(R_L \neq R_s\) for maximum selectivity).

7.3 Frequency Transformations

The low-pass prototype is transformed to the desired filter type by frequency substitutions:

Low-pass to high-pass:

\[ \omega \to -\frac{\omega_c}{\omega} \]

Series inductors become series capacitors, and shunt capacitors become shunt inductors (dual transformation).

Low-pass to bandpass (center frequency \(\omega_0\), bandwidth \(\Delta\omega\)):

\[ \omega \to \frac{\omega_0}{\Delta\omega}\left(\frac{\omega}{\omega_0} - \frac{\omega_0}{\omega}\right) \]

Each prototype element maps to a series or parallel \(LC\) resonator, doubling the number of elements.

Low-pass to bandstop:

\[ \omega \to \frac{\Delta\omega}{\omega_0}\left(\frac{\omega_0}{\omega} - \frac{\omega}{\omega_0}\right)^{-1} \]

7.4 Microwave Filter Realizations

7.4.1 Stepped-Impedance Filters

The simplest realization of a microstrip low-pass filter uses alternating sections of high- and low-impedance transmission lines as approximations to series inductors and shunt capacitors respectively. A short section of high-impedance line (\(Z_h\)) approximates a series inductor:

\[ L \approx \frac{Z_h}{\omega_0}\,\beta\ell, \quad \text{so } \ell = \frac{L\omega_0}{Z_h\beta} \]

and a short section of low-impedance line (\(Z_\ell\)) approximates a shunt capacitor:

\[ C \approx \frac{1}{\omega_0 Z_\ell}\,\beta\ell, \quad \text{so } \ell = \frac{C\omega_0 Z_\ell}{\beta} \]

The design is straightforward but the response deviates from ideal for longer sections (higher frequencies). Practical designs use \(Z_h/Z_0 \approx 2\)–\(4\) and \(Z_0/Z_\ell \approx 2\)–\(4\).

7.4.2 Coupled-Line Bandpass Filters

Coupled-line filters are the most common microstrip bandpass topology. Each section consists of two parallel coupled microstrip lines of length \(\lambda/4\) at the center frequency. An \(N\)-pole filter requires \(N+1\) coupled-line sections.

The even- and odd-mode impedances of each section are:

\[ Z_{0e}^{(j)} = \frac{1}{Y_0}\left(1 + J_j/Y_0 + (J_j/Y_0)^2\right), \quad Z_{0o}^{(j)} = \frac{1}{Y_0}\left(1 - J_j/Y_0 + (J_j/Y_0)^2\right) \]

where the admittance inverter values \(J_j\) are derived from the prototype \(g\)-values:

\[ \frac{J_1}{Y_0} = \sqrt{\frac{\pi \Delta\omega}{2g_1\omega_0}}, \qquad \frac{J_j}{Y_0} = \frac{\pi\Delta\omega}{2\omega_0\sqrt{g_{j-1}g_j}} \quad (j = 2,\ldots,N), \qquad \frac{J_{N+1}}{Y_0} = \sqrt{\frac{\pi\Delta\omega}{2g_N g_{N+1}\omega_0}} \]

7.4.3 Stub Filters

Open- or short-circuited stubs can serve as resonators in bandpass or bandstop filters. End-coupled, parallel-coupled, and interdigital configurations are common. The interdigital filter alternates open-circuited stubs pointing in opposite directions, providing a compact layout. The combline filter uses capacitively loaded stubs shorter than \(\lambda/4\), allowing even more compact implementations, particularly at X-band and above.

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Chapter 8: Microwave Amplifier Design

8.1 Two-Port Amplifier with Source and Load

A microwave amplifier consists of a transistor (typically a FET or BJT) biased for RF operation, flanked by input and output matching networks. The source presents impedance \(Z_s\) and the load presents \(Z_L\). Designing for maximum gain, specified noise figure, or a combination of both requires working with the transistor S-parameters and carefully tracking power flow.

8.2 Amplifier Power Gain Definitions

Three gain definitions are standard:

Transducer power gain \(G_T\): ratio of power delivered to load to power available from source.

\[ G_T = \frac{P_L}{P_{\text{avs}}} = \frac{(1 - |\Gamma_s|^2)|S_{21}|^2(1 - |\Gamma_L|^2)}{|1 - S_{11}\Gamma_s|^2\,|1 - S_{22}'\Gamma_L|^2} \]

where \(S_{22}' = S_{22} + S_{12}S_{21}\Gamma_s/(1 - S_{11}\Gamma_s)\) is the modified output reflection.

Operating power gain \(G_P\): ratio of power delivered to load to power input to two-port (independent of source matching).

Available power gain \(G_A\): ratio of power available from two-port to power available from source (independent of load matching).

For a unilateral transistor (\(S_{12} = 0\)), the transducer gain separates into input, transistor, and output matching factors:

\[ G_T = G_s \cdot |S_{21}|^2 \cdot G_L \]\[ G_s = \frac{1 - |\Gamma_s|^2}{|1 - S_{11}\Gamma_s|^2}, \quad G_L = \frac{1 - |\Gamma_L|^2}{|1 - S_{22}\Gamma_L|^2} \]

Maximum \(G_s\) is achieved by conjugate matching at the input: \(\Gamma_s = S_{11}^*\), giving \(G_{s,\max} = 1/(1 - |S_{11}|^2)\). Similarly \(\Gamma_L = S_{22}^*\).

8.3 Stability Analysis

A transistor amplifier is conditionally stable if it can oscillate for some combination of source and load impedances. Stable amplifier design requires either ensuring unconditional stability or restricting the impedances to the stable region.

8.3.1 Stability Circles

The boundary in the \(\Gamma_s\) (or \(\Gamma_L\)) plane where \(|\Gamma_{\text{in}}| = 1\) (or \(|\Gamma_{\text{out}}| = 1\)) defines the stability circle. Points outside the unit circle in the \(\Gamma_s\) plane are passive (physical), so we only care about stability circles within the unit circle.

The output stability circle in the \(\Gamma_L\) plane has center and radius:

\[ C_L = \frac{(S_{22} - \Delta S_{11}^*)^*}{|S_{22}|^2 - |\Delta|^2}, \qquad r_L = \left|\frac{S_{12}S_{21}}{|S_{22}|^2 - |\Delta|^2}\right| \]

where \(\Delta = S_{11}S_{22} - S_{12}S_{21}\) is the determinant of the S-matrix.

The input stability circle in the \(\Gamma_s\) plane has:

\[ C_s = \frac{(S_{11} - \Delta S_{22}^*)^*}{|S_{11}|^2 - |\Delta|^2}, \qquad r_s = \left|\frac{S_{12}S_{21}}{|S_{11}|^2 - |\Delta|^2}\right| \]

8.3.2 Unconditional Stability Conditions

A device is unconditionally stable if and only if both of the following conditions are satisfied:

\[ K = \frac{1 - |S_{11}|^2 - |S_{22}|^2 + |\Delta|^2}{2|S_{12}S_{21}|} > 1 \]\[ |\Delta| = |S_{11}S_{22} - S_{12}S_{21}| < 1 \]

A single-parameter test due to Edwards and Sinsky uses the \(\mu\) factor:

\[ \mu = \frac{1 - |S_{11}|^2}{|S_{22} - \Delta S_{11}^*| + |S_{12}S_{21}|} > 1 \iff \text{unconditionally stable} \]

8.4 Maximum Gain and the Unilateral Figure of Merit

For an unconditionally stable device, the maximum transducer gain under simultaneous conjugate matching is:

\[ G_{T,\max} = \frac{|S_{21}|}{|S_{12}|}(K - \sqrt{K^2 - 1}) \]

The unilateral transducer gain (\(S_{12} = 0\) assumption) gives:

\[ G_{TU} = \frac{1 - |\Gamma_s|^2}{|1 - S_{11}\Gamma_s|^2}|S_{21}|^2\frac{1 - |\Gamma_L|^2}{|1 - S_{22}\Gamma_L|^2} \]

The unilateral figure of merit bounds the error introduced by the unilateral assumption:

\[ U = \frac{|S_{12}||S_{21}||S_{11}||S_{22}|}{(1 - |S_{11}|^2)(1 - |S_{22}|^2)} \]

The ratio of true transducer gain to unilateral gain is bounded by:

\[ \frac{1}{(1 + U)^2} \leq \frac{G_T}{G_{TU}} \leq \frac{1}{(1 - U)^2} \]

If \(U \ll 1\), the unilateral design approximation introduces negligible error.

8.5 Noise Figure and Noise Circles

The noise figure (NF) of a two-port characterizes how much it degrades the signal-to-noise ratio:

\[ F = \frac{S/N_{\text{in}}}{S/N_{\text{out}}} \geq 1 \]\[ \text{NF} = 10\log_{10} F \geq 0\,\text{dB} \]

The noise figure of a transistor depends on the source impedance presented to its input. The minimum noise figure \(F_{\min}\) is achieved at the optimum noise source impedance \(Z_{\text{opt}} = R_{\text{opt}} + jX_{\text{opt}}\). The general noise figure for source admittance \(Y_s = G_s + jB_s\) is:

\[ F = F_{\min} + \frac{R_n}{G_s}|Y_s - Y_{\text{opt}}|^2 \]

where \(R_n\) is the noise resistance of the device, a parameter describing the sensitivity of noise figure to source admittance mismatch.

In terms of the source reflection coefficient \(\Gamma_s\):

\[ F = F_{\min} + \frac{4R_n}{Z_0}\frac{|\Gamma_s - \Gamma_{\text{opt}}|^2}{(1 - |\Gamma_s|^2)|1 + \Gamma_{\text{opt}}|^2} \]

Noise circles (constant-F contours in the \(\Gamma_s\) plane) are obtained by setting \(F =\) constant. They are circles centered at:

\[ C_F = \frac{\Gamma_{\text{opt}}}{1 + N}, \quad r_F = \frac{\sqrt{N^2 + N(1 - |\Gamma_{\text{opt}}|^2)}}{1 + N} \]

where \(N = (F - F_{\min})|1 + \Gamma_{\text{opt}}|^2 / (4R_n/Z_0)\). The noise circles are centered on the line connecting the center of the Smith chart to \(\Gamma_{\text{opt}}\), with \(\Gamma_{\text{opt}}\) lying on the innermost circle (\(F = F_{\min}\)).

8.5.1 Cascaded Noise Figure — Friis Formula

For a cascade of two-ports with noise figures \(F_1, F_2, \ldots, F_N\) and available power gains \(G_{A1}, G_{A2}, \ldots\), the overall noise figure is:

\[ F_{\text{total}} = F_1 + \frac{F_2 - 1}{G_{A1}} + \frac{F_3 - 1}{G_{A1}G_{A2}} + \cdots \]

This is the Friis noise formula. The first stage dominates the system noise figure when \(G_{A1} \gg 1\), which is why the LNA (first stage) must have both low noise figure and sufficient gain.

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Chapter 9: RF Oscillators

9.1 Feedback Oscillator Conditions

An oscillator is a self-sustaining signal source. In feedback terms, an amplifier with gain \(A\) and feedback network with transfer function \(\beta_f(\omega)\) will oscillate when the Barkhausen criterion is satisfied:

\[ A(\omega_0)\,\beta_f(\omega_0) = 1 \]

This requires simultaneously: (i) the loop gain magnitude equals unity, and (ii) the total loop phase shift is a multiple of \(360°\). In practice, the loop gain starts above unity (to ensure oscillation build-up from noise) and is then limited by device nonlinearity as the amplitude grows.

Common feedback oscillator topologies at RF include:

• Colpitts oscillator: capacitive voltage divider in the feedback path; tank circuit uses an inductor and two capacitors.
• Hartley oscillator: inductive voltage divider; tank uses a tapped inductor and a capacitor.
• Clapp oscillator: Colpitts modification with an additional series capacitor for improved frequency stability.
• Crystal oscillator: uses a piezoelectric resonator (crystal) for extremely high \(Q\) (\(10^4\)–\(10^6\)) and frequency stability.

9.2 Negative-Resistance Oscillators

At microwave frequencies, the feedback oscillator model becomes difficult to apply because transistors are characterized by S-parameters rather than voltage/current gains. Instead, the negative-resistance oscillator model is preferred.

A one-port negative-resistance device (or a two-port loaded to appear as one-port) has an input impedance:

\[ Z_{\text{in}}(\omega, A) = R_{\text{in}}(\omega, A) + jX_{\text{in}}(\omega, A) \]

where \(A\) is the signal amplitude. Oscillation requires \(R_{\text{in}} < 0\) at small signal. The oscillator circuit also has a resonator (load) with impedance \(Z_L(\omega) = R_L(\omega) + jX_L(\omega)\).

The steady-state oscillation conditions are found from the Kirchhoff loop equation:

\[ Z_{\text{in}}(\omega_0, A_0) + Z_L(\omega_0) = 0 \]

which gives two conditions:

\[ R_{\text{in}}(\omega_0, A_0) + R_L(\omega_0) = 0 \]\[ X_{\text{in}}(\omega_0, A_0) + X_L(\omega_0) = 0 \]

9.2.1 Kurokawa Stability Conditions

Oscillation conditions alone do not guarantee a stable, single-frequency oscillation. The Kurokawa stability condition ensures that the oscillation is stable against small perturbations in both amplitude and frequency:

\[ \frac{\partial R_{\text{in}}}{\partial A}\frac{\partial X_L}{\partial \omega} - \frac{\partial X_{\text{in}}}{\partial A}\frac{\partial R_L}{\partial \omega} > 0 \]

at the operating point \((A_0, \omega_0)\). For a resonator with \(\partial X_L/\partial\omega > 0\) (inductively dominant near resonance) and a device where \(\partial R_{\text{in}}/\partial A > 0\) (negative resistance decreases in magnitude as amplitude grows), this condition is typically satisfied.

9.3 Phase Noise

Practical oscillators exhibit random fluctuations in their output frequency and phase, termed phase noise. It manifests as energy spread around the carrier in the frequency spectrum. The Leeson model gives an empirical formula for single-sideband phase noise spectral density (in dBc/Hz) at offset frequency \(f_m\) from the carrier:

\[ \mathcal{L}(f_m) = 10\log_{10}\left[\frac{2FkT}{P_s}\left(1 + \frac{f_0^2}{(2Q_L f_m)^2}\right)\left(1 + \frac{f_c}{f_m}\right)\right] \]

where:

• \(F\) = device noise figure (linear)
• \(k\) = Boltzmann’s constant, \(T\) = temperature
• \(P_s\) = carrier power (signal power into device)
• \(f_0\) = carrier frequency
• \(Q_L\) = loaded Q of the resonator
• \(f_c\) = flicker noise corner frequency of the device

9.4 Microwave Oscillator Realizations

Several device technologies support microwave oscillation:

Gunn diode: Exhibits bulk negative differential resistance via the Gunn effect (intervalley electron transfer in GaAs or InP) for CW oscillation at mm-wave frequencies.

IMPATT diode: Impact ionization avalanche transit-time device. Provides negative resistance via a phase delay between voltage and current. Higher power than Gunn but noisier.

FET oscillators: A microwave FET (GaAs MESFET, InP HEMT, GaN HEMT) with appropriate termination can exhibit negative resistance at one port. These are the most common sources in planar microwave circuits.

Dielectric Resonator Oscillators (DRO): High-Q ceramic resonators (resonant mode in high-permittivity, low-loss cylinders, \(\varepsilon_r \sim 20\)–80) coupled to a FET oscillator provide excellent phase noise performance at X-band and above.

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Chapter 10: Mixers

10.1 Mixer Fundamentals

A mixer is a frequency-translation device: it combines two signals at frequencies \(f_{\text{RF}}\) (Radio Frequency input) and \(f_{\text{LO}}\) (Local Oscillator) to produce an output at the intermediate frequency \(f_{\text{IF}} = |f_{\text{RF}} \pm f_{\text{LO}}|\). The mixer performs multiplication in the time domain:

\[ v_{\text{out}}(t) \propto v_{\text{RF}}(t) \cdot v_{\text{LO}}(t) \]\[ = A_{\text{RF}}\cos(2\pi f_{\text{RF}} t)\cdot A_{\text{LO}}\cos(2\pi f_{\text{LO}} t) = \frac{A_{\text{RF}}A_{\text{LO}}}{2}[\cos(2\pi(f_{\text{RF}} - f_{\text{LO}})t) + \cos(2\pi(f_{\text{RF}} + f_{\text{LO}})t)] \]

A filter following the mixer selects either the sum (upconversion) or difference (downconversion) frequency component.

10.2 Mixer Performance Metrics

Conversion loss (CL): The ratio of available IF output power to available RF input power:

\[ CL = -10\log_{10}\frac{P_{\text{IF}}}{P_{\text{RF}}} \quad [\text{dB}] \]

For an ideal diode mixer, minimum conversion loss is \(-3\,\text{dB}\) (actually \(2/\pi^2 \approx -3.9\,\text{dB}\) for a switching mixer with ideal diodes).

Noise figure: Mixers add noise, primarily from diode shot noise and conversion to image frequency. For a DSB (double-sideband) mixer, NF ≥ CL; for SSB (single-sideband) mixers, NF is approximately 3 dB higher than the DSB case.

Image frequency: The RF and image frequencies (\(f_{\text{im}} = 2f_{\text{LO}} - f_{\text{RF}}\)) both mix with the LO to produce the same IF. Unless the image is filtered before the mixer, image noise and signals degrade performance.

1 dB compression point (P\(_{1\text{dB}}\)) and third-order intercept point (IP3): Metrics for linearity. When \(P_{\text{RF}}\) is large, the mixer saturates. Third-order intermodulation products appear at \(2f_{\text{RF1}} - f_{\text{RF2}}\) and can fall in-band.

10.3 Diode Mixer Topologies

Single-ended (single-diode) mixer: Simplest configuration. Poor isolation; LO leaks into IF and RF ports. Used only when simplicity outweighs performance.

Single-balanced mixer: Uses two diodes with a hybrid (180° or 90°). Provides LO-to-IF isolation and LO-to-RF isolation (depending on configuration). Common in practice for moderate performance.

Double-balanced mixer (DBM): Uses four diodes in a ring (diode ring or “star” configuration) with two balanced hybrids. Provides LO, RF, and IF port isolation (all three pairs); rejects even-order intermodulation products; widely used in wideband systems. The ideal DBM with switching diodes achieves \(-3.9\,\text{dB}\) conversion loss and high dynamic range.

10.4 FET Mixers

FET (especially HEMT) mixers offer conversion gain (active mixing), lower noise figure, and easy integration in MMIC processes. The drain current of a FET is a nonlinear function of gate-source voltage; the LO modulates the transconductance at LO frequency, and the RF signal multiplied by this time-varying \(g_m\) produces an IF output:

\[ i_d(t) \approx g_m(t)\cdot v_{\text{gs,RF}}(t) \]\[ g_m(t) = G_{m0} + G_{m1}\cos(2\pi f_{\text{LO}} t) + \cdots \]

The IF component is \(G_{m1}\cdot v_{\text{RF}}\cos(2\pi f_{\text{IF}} t)/2\), giving conversion gain \(G_c = (G_{m1}/2)^2 R_{\text{IF}}/G_{\text{RF}}\).

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Chapter 11: Computer-Aided Design of RF Circuits

11.1 Role of CAD in Microwave Design

Modern RF and microwave circuits are almost exclusively designed and verified using commercial CAD tools before fabrication. The complexity of distributed circuits (transmission line effects, coupling between adjacent lines, parasitic modes) makes hand analysis insufficient except for first-order estimates. The primary CAD tool used in industry and taught in ECE 373 labs is Keysight Advanced Design System (ADS).

11.2 Types of Simulation

Circuit simulation (SPICE-like, harmonic balance): The circuit is described by lumped elements, transmission line elements, and S-parameter blocks. For linear circuits, frequency-domain analysis yields S-parameters, gain, noise figure. For nonlinear circuits (amplifiers near compression, mixers, oscillators), harmonic balance solves for steady-state periodic solutions in the frequency domain.

Electromagnetic (EM) simulation (Method of Moments): For planar circuits, the substrate and metal layers are meshed and Maxwell’s equations are solved numerically. ADS’s Momentum is a 2.5-D EM simulator based on the Method of Moments (MoM), suitable for microstrip and stripline layouts. Full 3-D EM simulators (HFSS, CST) are used for cavity structures, connectors, and complex 3-D geometries.

System-level simulation (envelope, Ptolemy): For communication system analysis — BER, EVM, ACPR — signal-level simulations propagate modulated waveforms through nonlinear blocks.

11.3 EM Simulation Workflow

A typical EM simulation workflow in ADS:

1. Define substrate stackup (\(\varepsilon_r\), \(\tan\delta\), layer thicknesses, metal conductivity).
2. Draw the layout geometry (import from schematic or draw manually).
3. Set simulation frequency range and mesh density.
4. Run Momentum; extract S-parameter results.
5. Export S-parameter data (Touchstone .s2p format) for use in circuit simulations.
6. Compare with ideal transmission line models and iterate layout until specifications are met.

The Touchstone (.snp) format is the standard exchange format for S-parameters between different CAD tools and between simulation and measurement (VNA).

11.4 Vector Network Analyzer Measurements

The Vector Network Analyzer (VNA) is the instrument for measuring S-parameters of microwave devices. It simultaneously measures the magnitude and phase of incident, reflected, and transmitted waves at each port, over a specified frequency range.

Calibration is critical: systematic errors from connectors, cables, and imperfect directional couplers in the VNA test set are removed by measuring known calibration standards (short, open, load, through — SOLT calibration; or thru-reflect-line — TRL calibration for non-coaxial environments). After calibration, the reference planes are moved to the device-under-test (DUT) ports and corrected S-parameters are displayed.

Lab 4 in ECE 373 uses the VNA to measure filter and amplifier S-parameters and compare with simulation results.

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Chapter 12: RF System Integration

12.1 Hybrid and Monolithic Microwave Integrated Circuits

Hybrid MIC (Microwave Integrated Circuit): Passive circuit elements (microstrip lines, coupled lines, stubs) are etched on a dielectric substrate (often alumina, \(\varepsilon_r = 9.8\), or PTFE-based material), and discrete active components (chip transistors, diodes) are bonded or soldered onto the substrate. Hybrid MICs offer design flexibility, allow high-performance transistors to be used, and are suitable for low- to medium-volume production.

Monolithic MIC (MMIC): All circuit elements — active and passive — are fabricated on a single semiconductor chip (typically GaAs, InP, or GaN). MMICs offer reproducibility, compactness, and suitability for high-volume production, but require longer design cycles (no-touch-up after fabrication) and have lower Q passive elements due to lossy semiconductor substrates.

12.2 RF Wireless System Context

Understanding where ECE 373 components fit in a wireless communication system contextualizes the design objectives:

Receive chain (simplified): Antenna → LNA → Image Reject Filter → Downconverter Mixer → IF filter → IF amplifier → ADC

• The LNA must have minimum noise figure (first-stage dominance) and sufficient gain to suppress the noise contribution of subsequent stages (Friis formula).
• The image reject filter must attenuate the image frequency before the mixer.
• The mixer must have low conversion loss, low noise figure, and high IIP3 (input-referred third-order intercept).

Transmit chain (simplified): DAC → IF amplifier → Upconverter Mixer → Bandpass Filter → Power Amplifier → Antenna

• The power amplifier (PA) must deliver high output power with good efficiency (not covered in ECE 373 in depth but part of the broader active circuits module).
• The bandpass filter suppresses harmonics and spurious outputs from the mixer and PA.

12.3 Link Budget Analysis

A link budget accounts for all gains and losses between transmitter and receiver:

\[ P_r = P_t + G_t - L_{\text{path}} + G_r - L_{\text{misc}} \]

(all in dB/dBm). The free-space path loss is:

\[ L_{\text{path}} = 20\log_{10}\left(\frac{4\pi d f}{c}\right) \quad [\text{dB}] \]

The minimum detectable signal at the receiver is set by the receiver noise floor:

\[ P_{\text{noise}} = kT_0 B \cdot F_{\text{sys}} = -174\,\text{dBm/Hz} + 10\log_{10}(B) + \text{NF}_{\text{sys}}\,[\text{dBm}] \]

where \(B\) is the noise bandwidth and \(\text{NF}_{\text{sys}}\) is the system noise figure in dB. The link margin is \(P_r - P_{\text{noise}} - \text{SNR}_{\text{min}}\), which must be positive for reliable communication.

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Appendix: Key Formulas Reference

Transmission Lines

S-Parameters

Amplifier Design