Sources and References

Primary textbook — A. S. Sedra and K. C. Smith, Microelectronic Circuits, 8th ed., Oxford University Press, 2020. Supplementary texts — B. Razavi, Design of Analog CMOS Integrated Circuits, 2nd ed., McGraw-Hill, 2017; T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits, 2nd ed., Cambridge University Press, 2004; D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012. Online resources — MIT OpenCourseWare 6.776 High Speed Communication Circuits; Stanford EE214B lecture notes (B. Murmann).

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Chapter 1: Device Physics and Modeling

1.1 The PN Junction Diode

The PN junction is the foundational device in all integrated circuit technology. Understanding its physics at RF frequencies requires moving beyond the simple exponential I–V relationship and accounting for charge storage effects.

1.1.1 DC Characteristics and the Ideal Diode Equation

Under the depletion approximation, the built-in potential across a PN junction is

\[ V_{bi} = \frac{kT}{q} \ln\!\left(\frac{N_A N_D}{n_i^2}\right) \]

where \(N_A\) and \(N_D\) are the acceptor and donor concentrations, \(n_i\) is the intrinsic carrier concentration, \(k\) is Boltzmann’s constant, \(T\) is temperature, and \(q\) is the electron charge. The terminal current under forward bias \(V_D\) obeys the Shockley equation

\[ I_D = I_S \left(e^{V_D / V_T} - 1\right), \quad V_T = \frac{kT}{q} \approx 26\,\text{mV at } 300\,\text{K} \]

where \(I_S\) is the reverse saturation current, which scales as \(n_i^2\) and is therefore strongly temperature dependent.

1.1.2 Small-Signal Model and Junction Capacitance

For a diode biased at quiescent current \(I_Q\), the incremental (small-signal) conductance is

\[ g_d = \frac{dI_D}{dV_D}\bigg|_{I_Q} = \frac{I_Q}{V_T} \]

giving a small-signal resistance \(r_d = 1/g_d = V_T / I_Q\).

The junction also exhibits two capacitances critical at RF:

Depletion (junction) capacitance — from the immobile charge in the depletion region:

\[ C_j = \frac{C_{j0}}{\left(1 - V_D / V_{bi}\right)^m} \]

where \(C_{j0}\) is the zero-bias capacitance, \(m\) is the grading coefficient (\(m = 0.5\) for abrupt, \(m = 0.33\) for linearly graded), and \(V_D < V_{bi}\) (reverse or small forward bias).

Diffusion (storage) capacitance — from minority carrier charge stored in neutral regions under forward bias:

\[ C_d = \tau_T g_d = \frac{\tau_T I_Q}{V_T} \]

where \(\tau_T\) is the minority carrier transit time. Under forward bias \(C_d \gg C_j\); under reverse bias \(C_j\) dominates.

The complete small-signal model is therefore \(r_d\) in parallel with \(C_j + C_d\), plus a series resistance \(r_s\) representing contact and bulk resistances. This series resistance is critical at RF because it limits the diode’s quality factor.

1.2 The MOSFET at RF

1.2.1 Long-Channel DC Model

For an NMOS transistor operating in saturation (\(V_{DS} > V_{GS} - V_{th}\)):

\[ I_D = \frac{\mu_n C_{ox}}{2} \cdot \frac{W}{L} \left(V_{GS} - V_{th}\right)^2 \left(1 + \lambda V_{DS}\right) \]

where \(\mu_n\) is electron mobility, \(C_{ox} = \varepsilon_{ox}/t_{ox}\) is the gate oxide capacitance per unit area, \(W/L\) is the aspect ratio, \(V_{th}\) is the threshold voltage, and \(\lambda\) is the channel-length modulation parameter.

1.2.2 Small-Signal Model

The standard hybrid-\(\pi\) small-signal model for MOSFET in saturation includes:

• Transconductance: \(g_m = \sqrt{2 \mu_n C_{ox} (W/L) I_D}\)
• Output resistance: \(r_o = \frac{1}{\lambda I_D}\)
• Gate–source capacitance: \(C_{gs} \approx \frac{2}{3} W L C_{ox} + C_{ov}\) (saturation)
• Gate–drain capacitance: \(C_{gd} \approx C_{ov} = W L_{ov} C_{ox}\) (overlap)
• Drain–bulk capacitance: \(C_{db}\) (reverse-biased PN junction)
• Source–bulk capacitance: \(C_{sb}\) (reverse-biased PN junction)

The gate–drain capacitance \(C_{gd}\) is particularly troublesome at RF because it creates a feedback path from output to input, leading to the Miller effect that degrades amplifier bandwidth.

1.2.3 Transit Frequency \(f_T\) and Maximum Oscillation Frequency \(f_{max}\)

Setting the short-circuit current gain to unity in the small-signal model (with \(r_o \to \infty\)):

\[ f_T = \frac{g_m}{2\pi (C_{gs} + C_{gd})} \approx \frac{g_m}{2\pi C_{gs}} \approx \frac{3 \mu_n (V_{GS} - V_{th})}{4\pi L^2} \]

The approximation reveals that \(f_T\) scales as \(L^{-2}\), which is why aggressive CMOS scaling enables higher RF operating frequencies.

\[ f_{max} = \frac{f_T}{2} \cdot \frac{1}{\sqrt{R_g (g_{ds} + \omega^2 C_{gd}^2 / g_m) + R_g g_m C_{gd}/C_{gs}}} \]

A useful approximation when gate resistance \(R_g\) dominates:

\[ f_{max} \approx \sqrt{\frac{f_T}{8\pi R_g C_{gd}}} \]

Minimizing gate resistance (through multi-finger layouts) and \(C_{gd}\) (through careful device sizing) are key design levers in RF CMOS layout.

1.2.4 Noise in MOSFETs

Two primary noise sources are relevant at RF:

Drain current thermal noise:

\[ \overline{i_{nd}^2} = 4kT\gamma g_{d0} \Delta f \]

where \(g_{d0} = g_m / \alpha\) is the zero-bias drain conductance (\(\alpha \approx 1\) for long channel) and \(\gamma\) is the drain noise excess factor (\(\gamma = 2/3\) for long channel, \(\gamma > 1\) for short channel devices due to hot-carrier effects).

Gate current noise (due to capacitive coupling of drain noise to the gate):

\[ \overline{i_{ng}^2} = 4kT\delta g_g \Delta f, \quad g_g = \frac{\omega^2 C_{gs}^2}{5 g_{d0}} \]

where \(\delta \approx 4/3\) for long channel. These two sources are partially correlated with correlation coefficient \(c \approx j0.395\).

1.3 The Bipolar Junction Transistor (BJT)

1.3.1 DC Model and Ebers-Moll Equations

In the forward-active region of an NPN BJT:

\[ I_C = I_S e^{V_{BE}/V_T}, \quad I_B = \frac{I_C}{\beta_F}, \quad I_E = I_C + I_B = \frac{I_C (1 + \beta_F)}{\beta_F} \]

where \(I_S\) is the saturation current and \(\beta_F\) is the forward current gain (typically 50–200 for silicon BJTs).

1.3.2 Small-Signal Hybrid-\(\pi\) Model

For a BJT biased at collector current \(I_C\):

• Transconductance: \(g_m = I_C / V_T\) — note this is determined by bias alone, independent of device geometry
• Base–emitter resistance: \(r_\pi = \beta_0 / g_m = \beta_0 V_T / I_C\)
• Output resistance: \(r_o = V_A / I_C\) (Early effect)
• Base–emitter capacitance: \(C_\pi = C_{je} + C_{diff} \approx g_m \tau_F + C_{je}\)
• Base–collector capacitance: \(C_\mu\) (reverse-biased junction)
• Base resistance: \(r_b\) (extrinsic, layout dependent)

The BJT transit frequency:

\[ f_T = \frac{g_m}{2\pi (C_\pi + C_\mu)} \approx \frac{1}{2\pi\left(\tau_F + \frac{kT}{qI_C}(C_{je} + C_\mu) + C_\mu r_c\right)} \]

At optimum \(I_C\), \(f_T\) is limited by the forward transit time \(\tau_F = \tau_b + \tau_c\) (base transit time plus collector depletion region transit time). Modern SiGe HBTs achieve \(f_T > 300\,\text{GHz}\) at optimum bias.

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Chapter 2: Amplifiers — Basic Topologies and Biasing

2.1 Biasing Fundamentals

A stable quiescent operating point is essential before considering any small-signal RF performance. The bias network must:

1. Establish a predictable \(I_C\) (or \(I_D\)) relatively independent of \(\beta\) variation and temperature.
2. Present minimum impedance perturbation to the RF signal path.
3. Not degrade noise performance.

2.1.1 MOSFET Current Mirrors

The basic current mirror forces equal \(V_{GS}\) on two matched transistors, replicating the reference current \(I_{REF}\):

\[ I_{out} = I_{REF} \cdot \frac{(W/L)_2}{(W/L)_1} \cdot \frac{1 + \lambda V_{DS2}}{1 + \lambda V_{DS1}} \]

For RF circuits, cascode current mirrors improve output impedance: \(r_{out,cascode} \approx g_{m2} r_{o2} r_{o4}\).

2.2 Common-Source Amplifier

The common-source (CS) configuration is the workhorse of RF CMOS amplification.

Voltage gain (neglecting \(C_{gd}\)):

\[ A_v = \frac{v_{out}}{v_{in}} = -g_m (r_o \| R_L) \]

Input impedance: \(Z_{in} = 1/j\omega C_{gs}\) (purely capacitive, not matched to 50 \(\Omega\))

Output impedance: \(Z_{out} = r_o \| R_L\)

Miller effect on \(C_{gd}\): The gate–drain capacitance appears at the input as

\[ C_{Miller} = C_{gd}(1 + g_m R_L) \]

This dramatically increases effective input capacitance and limits bandwidth.

2.3 Common-Gate and Common-Drain Configurations

The common-gate (CG) amplifier is naturally suited for RF applications because it offers:

• Non-inverting gain
• Input impedance \(Z_{in} \approx 1/g_m\), facilitating 50-\(\Omega\) matching without inductors
• No Miller multiplication of \(C_{gd}\)

Voltage gain of CG stage:

\[ A_v = g_m (R_D \| r_o) \approx g_m R_D \]

The common-drain (source follower) has near-unity voltage gain, low output impedance \(Z_{out} \approx 1/g_m\), and high input impedance — used as output buffers.

2.4 Wideband Amplifier Techniques

Shunt–series feedback: Resistive feedback from output to input simultaneously sets input and output impedances and gain. For an inverting amplifier with gain \(-A\) and feedback resistor \(R_F\):

\[ Z_{in} \approx \frac{R_F}{1 + A}, \quad Z_{out} \approx \frac{R_F}{1 + A}, \quad A_v \approx -A \]

Cherry–Hooper topology: Alternating transresistance and transconductance stages provide wideband gain with predictable bandwidth.

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Chapter 3: Passive Components — RLC, Resonators, Transformers, and Transmission Lines

3.1 On-Chip Resistors

Integrated resistors (polysilicon, n-well, or metal) have significant parasitic capacitance to the substrate. At RF, the capacitance creates a high-frequency pole and limits the useful frequency range. For a resistor with resistance \(R\) and shunt capacitance \(C_p\):

\[ Z_R(\omega) = \frac{R}{1 + j\omega R C_p} \]

The self-resonant frequency is \(f_{res} = 1/(2\pi\sqrt{L_{lead} C_p})\) where \(L_{lead}\) is the parasitic lead inductance.

3.2 On-Chip Inductors

Spiral inductors are realized as patterned metal coils in the upper metal layers. The quality factor \(Q\) is the key figure of merit:

Physical loss mechanisms in on-chip spirals:

1. Metal ohmic loss — skin effect causes resistance to increase as \(\sqrt{f}\) above the skin-effect frequency \(f_{skin} = \rho/(\pi \mu t^2)\)
2. Substrate eddy current loss — image currents in the conductive silicon substrate reduce effective inductance and add loss
3. Capacitive coupling — oxide and substrate capacitance creates a self-resonance above which the spiral becomes capacitive

A lumped model for the spiral inductor consists of inductance \(L_s\), series resistance \(R_s\), oxide capacitance \(C_{ox}\) from metal to substrate (split equally at each port), substrate resistance \(R_{sub}\), and substrate capacitance \(C_{sub}\).

The effective quality factor accounting for substrate:

\[ Q_{eff} = \frac{\omega L_s}{R_s} \cdot \frac{R_p}{R_p + \left(\omega L_s / R_s\right)^2 \cdot R_s + R_s} \cdot \left(1 - \frac{R_s^2 C_p}{L_s} - \omega^2 L_s C_p\right) \]

Typical values: \(Q = 5\)–\(15\) at 2–5 GHz in standard CMOS. Patterned ground shields (PGS) interrupt eddy currents and improve \(Q\).

3.3 On-Chip Capacitors

MIM (metal–insulator–metal) capacitors offer the best RF performance: high capacitance density, low bottom-plate capacitance, and high self-resonant frequency. The quality factor is limited by the series resistance of the metal plates.

MOS varactors exploit the voltage-dependent capacitance of MOS gate oxide and are used in voltage-controlled oscillators (VCOs). An inversion-mode MOS varactor has:

\[ C_{var}(V) \approx \frac{C_{ox}}{1 + \left|V_{GS} - V_{th}\right| \cdot k} \]

The capacitance ratio \(C_{max}/C_{min}\) sets VCO tuning range.

3.4 LC Resonators

A parallel LC tank with inductor \(L\), capacitor \(C\), and inductor quality factor \(Q_L\) (series resistance \(r = \omega L / Q_L\)) has a resonant frequency and bandwidth:

\[ \omega_0 = \frac{1}{\sqrt{LC}}, \quad BW = \frac{\omega_0}{Q_{tank}} \]

The parallel equivalent resistance at resonance:

\[ R_p = Q_L^2 r = Q_L \omega_0 L = \frac{Q_L}{\omega_0 C} \]

This parallel resistance \(R_p\) directly loads tuned amplifiers and oscillators, determining gain and phase noise performance.

3.5 Transformers

On-chip transformers (coupled spiral inductors) are used for impedance transformation, signal balancing (single-ended to differential), and AC coupling without DC paths. The coupling coefficient \(k\) is:

\[ k = \frac{M}{\sqrt{L_1 L_2}}, \quad 0 \leq k \leq 1 \]

For an ideal 1:n turns-ratio transformer, the impedance transformation ratio is \(n^2\). In practice, \(k = 0.7\)–\(0.9\) is achievable on silicon.

The transformer’s Y-parameter model (considering \(k\)):

\[ \begin{pmatrix} I_1 \\ I_2 \end{pmatrix} = \frac{1}{j\omega(L_1 L_2 - M^2)} \begin{pmatrix} L_2 & M \\ M & L_1 \end{pmatrix} \begin{pmatrix} V_1 \\ V_2 \end{pmatrix} \]

3.6 Transmission Lines

3.6.1 Distributed Circuit Model and Telegrapher’s Equations

A transmission line is modeled as a distributed network with per-unit-length resistance \(R'\), inductance \(L'\), capacitance \(C'\), and conductance \(G'\). The telegrapher’s equations are:

\[ \frac{\partial V}{\partial z} = -(R' + j\omega L') I, \qquad \frac{\partial I}{\partial z} = -(G' + j\omega C') V \]

The complex propagation constant is:

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

For a lossless line (\(R' = G' = 0\)):

\[ \gamma = j\omega\sqrt{L'C'} = j\beta, \quad v_p = \frac{\omega}{\beta} = \frac{1}{\sqrt{L'C'}}, \quad Z_0 = \sqrt{\frac{L'}{C'}} \]

3.6.2 Reflection Coefficient and VSWR

The voltage reflection coefficient at a load \(Z_L\):

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

The voltage standing wave ratio (VSWR):

\[ \text{VSWR} = \frac{1 + |\Gamma_L|}{1 - |\Gamma_L|} \]

3.6.3 Input Impedance of a Terminated Line

For a line of length \(\ell\) terminated in \(Z_L\):

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

Special cases:

• Quarter-wave transformer (\(\beta\ell = \pi/2\)): \(Z_{in} = Z_0^2 / Z_L\) — transforms impedance by \(Z_0^2\)
• Half-wave line (\(\beta\ell = \pi\)): \(Z_{in} = Z_L\) — transparent
• Short-circuit stub (\(Z_L = 0\)): \(Z_{in} = jZ_0 \tan(\beta\ell)\) — inductive for \(0 < \beta\ell < \pi/2\)
• Open-circuit stub (\(Z_L \to \infty\)): \(Z_{in} = -jZ_0 \cot(\beta\ell)\) — capacitive for \(0 < \beta\ell < \pi/2\)

3.6.4 S-Parameters

Scattering parameters describe network behavior in terms of traveling waves, making them ideal for RF characterization. For a two-port:

\[ \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix} \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} \]

where \(a_i = V_i^+ / \sqrt{Z_0}\) are incident waves and \(b_i = V_i^- / \sqrt{Z_0}\) are reflected waves. Key parameters:

• \(S_{11}\): input reflection coefficient (port 2 matched)
• \(S_{21}\): forward transmission (voltage gain for matched system)
• \(S_{22}\): output reflection coefficient (port 1 matched)
• \(S_{12}\): reverse isolation

Power gain: \(G = |S_{21}|^2\) when both ports are matched to \(Z_0\).

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Chapter 4: Impedance Matching and Filters

4.1 Why Matching Matters

Maximum power transfer from source with impedance \(Z_S\) to load \(Z_L\) occurs when \(Z_L = Z_S^*\) (complex conjugate match). At RF, the standard reference impedance is \(Z_0 = 50\,\Omega\). Unmatched interfaces cause:

• Power reflection loss: \(P_{reflected}/P_{available} = |\Gamma|^2\)
• Frequency-dependent gain variation
• Increased noise figure (when input is mismatched)

4.2 L-Network Matching

The L-network uses two reactive elements to transform a real load resistance \(R_L\) to a real source resistance \(R_S < R_L\). Define the Q of the network:

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

For a shunt-C, series-L topology (low-pass L-network):

\[ X_C = -\frac{R_L}{Q} = -\frac{R_L}{\sqrt{R_L/R_S - 1}}, \quad X_L = Q \cdot R_S \]

The matching bandwidth is inversely proportional to \(Q\): narrowband matching has high Q and small bandwidth; broadband matching requires reducing Q via multi-section networks.

4.3 Pi and T Networks

The \(\pi\)-network consists of a series arm between two shunt elements. It provides an additional degree of freedom: independent control of \(Q\) and impedance transformation ratio. This allows the designer to set network Q (and hence bandwidth) independently of the impedance transformation ratio — a major advantage over the L-network.

For a \(\pi\)-network matching \(R_S\) to \(R_L\) with desired \(Q\):

\[ Q_{net} \geq \sqrt{\frac{R_{high}}{R_{low}} - 1} \]

The network is synthesized as two back-to-back L-networks sharing an intermediate virtual resistance \(R_V = R_{high} / (Q^2 + 1)\).

4.4 Inductor-Based Matching with Quality Factor Degradation

When a lossy inductor (quality factor \(Q_L\)) is used in a matching network, the effective resistance transformation is degraded. The parallel equivalent resistance of inductor \(L\) with series loss \(r_s = \omega L / Q_L\):

\[ R_{p,L} = r_s (Q_L^2 + 1) \approx Q_L^2 r_s = Q_L \omega L \]

Insertion loss of a matching network due to finite component Q:

\[ IL = 1 - \frac{Q_{net}}{Q_L} \]

Valid when \(Q_L \gg Q_{net}\). This emphasizes the importance of maximizing \(Q_L\) in on-chip spiral inductors.

4.5 Filter Fundamentals

RF filters (bandpass, low-pass, notch) use LC ladders realized with transmission line stubs, spiral inductors, and MIM capacitors. Key filter prototypes:

Butterworth — maximally flat passband, monotonically decreasing stopband. The normalized transfer function has poles equally spaced on the left half of the unit circle.

Chebyshev — equiripple passband, steeper rolloff than Butterworth for same order. The transfer function uses Chebyshev polynomials \(T_n(\omega)\):

\[ |H(j\omega)|^2 = \frac{1}{1 + \varepsilon^2 T_n^2(\omega/\omega_c)} \]

where \(\varepsilon\) sets the ripple: passband ripple \(= 10\log_{10}(1 + \varepsilon^2)\,\text{dB}\).

Bandpass filter from lowpass prototype: Apply the lowpass-to-bandpass transformation:

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

where \(Q_{BPF} = \omega_0 / BW\).

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Chapter 5: Tuned Amplifiers and Narrowband LNAs

5.1 Tuned Amplifier Fundamentals

A tuned (narrowband) amplifier replaces the resistive drain load with a parallel LC resonator. At resonance \(\omega_0 = 1/\sqrt{LC}\), the tank presents a high impedance \(R_p\), giving large voltage gain. Away from resonance, the tank impedance falls rapidly, providing frequency selectivity.

Voltage gain at resonance:

\[ A_v(\omega_0) = -g_m R_p = -g_m Q_L \omega_0 L \]

3-dB bandwidth:

\[ BW_{-3dB} = \frac{\omega_0}{Q_{loaded}} \]

where the loaded Q accounts for source loading, device output conductance, and the resonator’s own losses:

\[ \frac{1}{Q_{loaded}} = \frac{1}{Q_L} + \frac{g_{ds}}{\omega_0 C} + \frac{1}{Q_{source}} \]

5.2 Noise Figure of RF Amplifiers

Friis formula for cascaded stages:

\[ F_{total} = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \cdots \]

This shows that the first stage (LNA) dominates the overall noise figure, motivating the need for an ultra-low-noise first stage.

5.3 LNA Design Topologies

5.3.1 Common-Gate LNA

The CG topology offers inherent 50-\(\Omega\) input matching (\(Z_{in} \approx 1/g_m\)), but its noise figure is limited by the minimum achievable NF:

\[ F_{CG} = 1 + \frac{\gamma}{\alpha} \approx 1 + \frac{2}{3} \cdot \frac{1}{\alpha} \]

where \(\alpha = g_m/g_{d0}\). For long-channel devices, \(F_{CG,min} \approx 2.2\,\text{dB}\), which is too high for modern receivers.

5.3.2 Inductively Degenerated CS LNA

The inductively degenerated common-source LNA is the most widely used narrowband LNA topology. A source inductor \(L_s\) and gate inductor \(L_g\) are added to simultaneously achieve:

• Input matching to 50 \(\Omega\)
• Minimum noise figure

The source inductor creates a real part in the input impedance without adding noise (unlike a resistor):

\[ Z_{in} = j\omega(L_g + L_s) + \frac{1}{j\omega C_{gs}} + \frac{g_m L_s}{C_{gs}} = \frac{g_m L_s}{C_{gs}} + j\left[\omega(L_g + L_s) - \frac{1}{\omega C_{gs}}\right] \]

At the resonance frequency where the imaginary part vanishes:

\[ \omega_0 = \frac{1}{\sqrt{(L_g + L_s)C_{gs}}} \]

The real part equals the transistor’s transit-frequency product:

\[ \text{Re}\left\{Z_{in}\right\} = \frac{g_m L_s}{C_{gs}} = \omega_T L_s \]

Setting \(\omega_T L_s = 50\,\Omega\) achieves input matching.

Voltage gain of inductively degenerated CS LNA at resonance:

\[ A_v = \frac{v_{out}}{v_{in}} = \frac{-g_m R_L}{1 + g_m Z_s} \approx \frac{-g_m R_L \cdot Q_{in}}{1} \]

where \(Q_{in} = 1/(2\omega_0 C_{gs} R_S)\) is the input Q-factor (voltage amplification at the gate terminal due to resonance).

Noise Figure of the inductive-degeneration LNA:

The minimum noise figure of the inductively degenerated CS LNA (Shaeffer and Lee, 1997):

\[ F_{min} \approx 1 + 2\omega_0 / \omega_T \cdot \sqrt{\gamma \delta (1 - |c|^2)} \]

where \(\delta\) is the gate noise coefficient and \(c\) is the correlation coefficient. The ratio \(\omega_0/\omega_T\) emphasizes the importance of using a transistor well below its \(f_T\) for low NF.

5.4 Stability Considerations

An amplifier is unconditionally stable if it does not oscillate for any passive source and load termination. The Rollett stability factor:

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

and the auxiliary condition \(|\Delta| = |S_{11}S_{22} - S_{12}S_{21}| < 1\) are jointly sufficient for unconditional stability.

Available gain circles and noise figure circles on the Smith chart guide the designer in trading gain, noise, and stability margin.

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Chapter 6: Feedback, Oscillators, and PLLs

6.1 Feedback Theory Review

Negative feedback in amplifiers reduces gain by the factor \((1 + A\beta)\) but simultaneously reduces sensitivity to component variations, extends bandwidth, and modifies impedance levels. For an inverting amplifier with open-loop gain \(A\) and feedback factor \(\beta\):

\[ A_f = \frac{A}{1 + A\beta} \approx \frac{1}{\beta} \quad (A\beta \gg 1) \]

Input impedance with series-series feedback: \(Z_{in,f} = Z_{in}(1 + A\beta)\) Output impedance with shunt feedback: \(Z_{out,f} = Z_{out}/(1 + A\beta)\)

6.2 Oscillator Theory

6.2.1 Barkhausen Criteria

A linear oscillator requires that at the oscillation frequency \(\omega_{osc}\), the loop gain satisfies:

\[ A(\omega_{osc}) \beta(\omega_{osc}) = 1\angle 0° \]

i.e., unity magnitude and zero (or integer multiple of \(360°\)) phase shift around the loop. In practice, the loop gain is designed slightly greater than unity, and amplitude is limited by nonlinearity.

6.2.2 LC Oscillators — the Cross-Coupled VCO

The most common RF LC oscillator is the cross-coupled NMOS VCO. The cross-coupled pair provides a negative resistance of \(-2/g_m\) to compensate tank losses. The start-up condition requires:

\[ \frac{2}{g_m} < R_p \]

i.e., the transistor negative resistance must exceed the tank parallel resistance. A safety margin of 2–3× is designed in to guarantee start-up over process, voltage, and temperature (PVT) variations.

Oscillation frequency:

\[ \omega_{osc} = \frac{1}{\sqrt{L C_{total}}}, \quad C_{total} = C_{fixed} + C_{var}(V_{tune}) \]

Tuning range:

\[ \Delta\omega / \omega_0 = \frac{1}{2} \Delta C / C_{total} \]

6.2.3 Phase Noise

Phase noise is the fundamental performance metric of oscillators. The Leeson–Cutler model gives the single-sideband phase noise spectral density \(\mathcal{L}(f_m)\) at an offset \(f_m\) from the carrier:

\[ \mathcal{L}(f_m) = 10\log_{10}\left[\frac{2FkT}{P_s}\left(1 + \frac{f_0^2}{4Q^2 f_m^2}\right)\left(1 + \frac{f_{1/f}}{f_m}\right)\right] \]

where \(F\) is an empirical device noise factor, \(P_s\) is the signal power in the resonator, \(Q\) is the loaded quality factor, and \(f_{1/f}\) is the transistor’s flicker noise corner frequency.

The Hajimiri–Lee linear time-variant (LTV) model provides a more rigorous analysis. The phase noise is expressed through the impulse sensitivity function (ISF) \(\Gamma(\omega_0 \tau)\):

\[ \mathcal{L}(\Delta\omega) = 10\log_{10}\left(\frac{\overline{\Gamma^2}}{2q_{max}^2} \cdot \frac{\overline{i_n^2}/\Delta f}{\Delta\omega^2}\right) \]

where \(q_{max}\) is the maximum charge displacement in the tank and \(\overline{\Gamma^2}\) is the RMS value of the ISF. The LTV model explains why noise injected near the zero-crossings of the oscillation waveform contributes most to phase noise.

6.2.4 Ring Oscillators

A ring oscillator consists of an odd number \(N\) of inverter stages. Each stage contributes phase shift of \(180°/N\), and the oscillation frequency is:

\[ f_{osc} = \frac{1}{2N \cdot t_{pd}} \]

where \(t_{pd}\) is the propagation delay of each stage. Ring oscillators occupy no area (no inductors) but exhibit substantially worse phase noise than LC oscillators for a given power dissipation. They are used in digital PLLs and clock synthesis applications where area is more constrained than phase noise.

6.3 Phase-Locked Loops

6.3.1 PLL Architecture

A basic charge-pump PLL consists of:

1. Phase-Frequency Detector (PFD) — compares phase/frequency of reference \(f_{ref}\) and feedback signals, produces UP/DN pulses
2. Charge Pump (CP) — converts pulse widths to charge injection into the loop filter
3. Loop Filter (LF) — a passive RC network converting charge to a control voltage
4. Voltage-Controlled Oscillator (VCO) — outputs frequency proportional to control voltage: \(f_{out} = f_0 + K_{VCO} V_{ctrl}\)
5. Frequency Divider — divides VCO output by \(N\), feeding back to PFD

At lock, \(f_{out} = N f_{ref}\), synthesizing frequencies in multiples of the reference.

6.3.2 Linear PLL Model and Transfer Function

In the s-domain, the PLL components are modeled as:

• PFD + CP: \(K_{PD} = I_{CP} / (2\pi)\,\text{[A/rad]}\)
• Loop filter (passive second-order): \(Z(s) = (1 + sR_1 C_1)/(s(C_1 + C_2)(1 + s\tau_2))\) where \(\tau_2 = R_1 C_1 C_2/(C_1+C_2)\)
• VCO: \(K_{VCO}/s\,\text{[rad/s/V]}\)
• Divider: \(1/N\)

Open-loop transfer function:

\[ T(s) = \frac{K_{PD} K_{VCO}}{N} \cdot \frac{Z(s)}{s} \]

Closed-loop transfer function (reference to output):

\[ H(s) = \frac{N \cdot T(s)}{1 + T(s)} = \frac{N \cdot K_{PD} K_{VCO} Z(s)}{s + K_{PD} K_{VCO} Z(s)/N} \]

The closed-loop bandwidth (loop bandwidth) \(\omega_n\) and damping ratio \(\zeta\) for a second-order PLL with a simple RC filter:

\[ \omega_n = \sqrt{\frac{K_{PD} K_{VCO}}{N C_1}}, \quad \zeta = \frac{R_1}{2}\sqrt{\frac{K_{PD} K_{VCO} C_1}{N}} \]

Noise filtering properties:

• Reference phase noise is low-pass filtered: appears at output below \(\omega_n\) with gain \(N\)
• VCO phase noise is high-pass filtered: suppressed below \(\omega_n\), passes above it

The loop bandwidth \(\omega_n\) is typically set at 1/10 to 1/5 of \(f_{ref}\) to maintain stability and adequate reference spur rejection.

6.3.3 Integer-N vs Fractional-N PLLs

In an integer-N PLL, channel spacing equals \(f_{ref}\). For narrowly spaced channels (e.g., 200 kHz GSM channels), a low \(f_{ref}\) forces a narrow loop bandwidth, slowing settling time and worsening VCO phase noise suppression.

A fractional-N PLL uses a dual-modulus divider that alternates between divide values \(N\) and \(N+1\), achieving a non-integer average division ratio:

\[ \bar{N} = N + \frac{k}{M}, \quad f_{out} = \left(N + \frac{k}{M}\right) f_{ref} \]

This allows a larger \(f_{ref}\) (wider loop bandwidth, better phase noise) for the same channel resolution. The primary penalty is fractional spurs from periodic modulator patterns, mitigated using \(\Sigma\Delta\) modulation of the divide ratio.

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Chapter 7: Communication and Computing Systems

7.1 Receiver Architectures

7.1.1 Heterodyne Receiver

The superheterodyne receiver translates the RF signal to an intermediate frequency (IF) using a mixer and local oscillator (LO). The IF is chosen to place image signals away from the desired band, where an image-reject filter can attenuate them.

Image rejection ratio (IRR) required:

\[ IRR = 20\log_{10}\left(\frac{1 + \sqrt{A^2 + \theta^2}}{|1 - A e^{j\theta}|}\right) \approx 20\log_{10}\left(\frac{2}{\sqrt{A^2 + \theta^2}}\right) \]

where \(A\) is the amplitude mismatch and \(\theta\) is the phase mismatch between I and Q paths. For 40-dB IRR, amplitude mismatch must be below 1% and phase mismatch below 0.57°.

7.1.2 Direct-Conversion (Zero-IF) Receiver

The direct-conversion receiver eliminates the IF stage by mixing directly to baseband. Advantages: no external IF filter, compatible with highly integrated CMOS. Key challenges:

• DC offset: LO self-mixing produces a DC component that saturates subsequent baseband amplifiers. Mitigated by AC coupling or DC calibration.
• Flicker (1/f) noise: In MOSFET circuits, 1/f noise is significant at the DC–MHz range occupied by the baseband signal. Minimized by large device area, current-mode topologies, or chopper stabilization.
• I/Q mismatch: Amplitude and phase mismatch between I and Q paths cause constellation distortion. Corrected with digital calibration.
• Even-order distortion: AM interferers create a baseband signal through second-order nonlinearity (IIP2 critical).

7.1.3 System Link Budget

The overall receiver noise figure:

\[ NF_{rx} = NF_{LNA} + \frac{NF_{mixer} - 1}{G_{LNA}} + \frac{NF_{IF} - 1}{G_{LNA} G_{mixer}} + \cdots \]

The required minimum detectable signal (MDS):

\[ MDS = -174\,\text{dBm/Hz} + NF_{rx} + 10\log_{10}(BW) + SNR_{min} \]

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Chapter 8: Signal Processing and Modulation Techniques

8.1 Modulation Review

8.1.1 Amplitude and Phase Modulation

AM modulation — the carrier amplitude is varied by the information signal:

\[ x_{AM}(t) = A_c\left[1 + m \cdot m(t)\right]\cos(\omega_c t) \]

where \(m\) is the modulation index and \(m(t)\) is the baseband message. The AM signal bandwidth is \(2W\) where \(W\) is the message bandwidth.

FM modulation — the instantaneous frequency is varied:

\[ x_{FM}(t) = A_c \cos\!\left(\omega_c t + 2\pi k_f \int_0^t m(\tau)\,d\tau\right) \]

FM bandwidth (Carson’s rule): \(BW_{FM} \approx 2(\Delta f + W) = 2W(1 + \beta)\) where \(\beta = \Delta f / W\) is the modulation index.

8.1.2 Digital Modulation and IQ Representation

Modern RF systems use quadrature modulation, where independent information streams are carried on orthogonal (I and Q) carriers:

\[ x(t) = I(t)\cos(\omega_c t) - Q(t)\sin(\omega_c t) \]

The complex envelope (phasor) representation: \(\tilde{x}(t) = I(t) + jQ(t)\).

QAM constellation: For M-QAM, each symbol carries \(\log_2 M\) bits. Higher-order QAM (64-QAM, 256-QAM) achieves higher spectral efficiency but requires higher SNR and tighter phase noise specification.

Error Vector Magnitude (EVM): measures the deviation of received symbols from ideal constellation points; sets upper bounds on phase noise, I/Q mismatch, and nonlinearity requirements.

8.1.3 OFDM and Multicarrier Systems

Orthogonal Frequency Division Multiplexing (OFDM) divides the channel bandwidth into \(N\) narrowband subcarriers, each modulated by a QAM symbol. The OFDM baseband signal:

\[ x(t) = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} X_k e^{j2\pi k \Delta f t}, \quad 0 \leq t \leq T_s \]

where \(\Delta f = 1/T_s\) ensures orthogonality. OFDM is robust to multipath channels (each subcarrier experiences flat fading) but is sensitive to phase noise and has a high peak-to-average power ratio (PAPR).

PAPR of OFDM: For \(N\) uncorrelated subcarriers, the theoretical PAPR approaches \(10\log_{10} N\) dB. High PAPR forces the power amplifier to operate at large power backoff, degrading efficiency — this couples directly to PA design constraints.

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Chapter 9: Mixers and Multipliers

9.1 Mixer Fundamentals

A mixer performs frequency translation by multiplying the RF signal with an LO signal. For an ideal multiplier:

\[ v_{out}(t) = v_{RF}(t) \cdot v_{LO}(t) = A_{RF}\cos(\omega_{RF} t) \cdot A_{LO}\cos(\omega_{LO} t) \]\[ = \frac{A_{RF} A_{LO}}{2}\left[\cos\!\left((\omega_{RF} - \omega_{LO})t\right) + \cos\!\left((\omega_{RF} + \omega_{LO})t\right)\right] \]

The desired term at the difference frequency (downconversion to IF = RF − LO) is selected by a bandpass or lowpass filter.

9.1.1 Conversion Gain

9.2 Single-Balanced and Double-Balanced Mixers

9.2.1 Gilbert Cell Mixer

The Gilbert cell is the most common active mixer topology. It consists of a transconductor stage (driven by \(v_{RF}\)) followed by a differential switching quad (driven by \(v_{LO}\)). The transconductance converts the RF voltage to a current, which is then commutated at the LO frequency.

Conversion gain of Gilbert cell:

\[ CG = \frac{2}{\pi} g_{m1} (R_D \| r_{o3,4}) \]

where \(g_{m1}\) is the transconductance of the input pair and \(R_D\) is the drain load resistance.

Noise figure of Gilbert cell mixer:

The mixer noise figure is dominated by:

1. Thermal noise from the input transconductor pair: contribution \(\approx 4kT\gamma g_{m1}\)
2. Noise from the switching quad during LO transitions (commutation noise)
3. Noise from load resistors \(R_D\)

The double-sideband (DSB) noise figure:

\[ NF_{DSB} = 10\log_{10}\left(1 + \frac{\gamma}{\alpha} + \frac{R_S g_{m1}^2}{\left(\frac{2}{\pi}\right)^2 g_{m1}} \cdot \frac{1}{R_D}\right) \]

The single-sideband (SSB) NF is 3 dB higher than DSB NF because the image band contributes noise but not signal.

9.3 Linearity Metrics

9.3.1 Second and Third-Order Intercept Points

For a weakly nonlinear system with input–output relationship:

\[ y(t) = a_1 x(t) + a_2 x^2(t) + a_3 x^3(t) + \cdots \]

with input \(x(t) = A\cos(\omega t)\):

• Fundamental: amplitude \(a_1 A\)
• Second harmonic: amplitude \(\frac{1}{2} a_2 A^2\) at \(2\omega\)
• Third harmonic: amplitude \(\frac{1}{4} a_3 A^3\) at \(3\omega\)

For a two-tone input \(x(t) = A_1\cos(\omega_1 t) + A_2\cos(\omega_2 t)\), third-order intermodulation products appear at \(2\omega_1 - \omega_2\) and \(2\omega_2 - \omega_1\), which fall in-band for narrowband systems.

The output-referred OIP3:

\[ OIP3 = IIP3 + G \quad \text{(in dB)} \]

Cascaded IIP3:

\[ \frac{1}{IIP3_{total}^2} \approx \frac{1}{IIP3_1^2} + \frac{G_1^2}{IIP3_2^2} + \frac{G_1^2 G_2^2}{IIP3_3^2} + \cdots \]

This shows that later stages with high gain drive the overall system IIP3, making linearity of the IF chain critical.

9.3.2 1-dB Compression Point

The 1-dB compression point is the input power at which gain drops by 1 dB due to gain compression:

\[ P_{in,1dB} = IIP3 - 9.6\,\text{dB} \approx IIP3 - 10\,\text{dB} \]

(for a memoryless third-order nonlinearity). At this point:

\[ a_1 A - \frac{3}{4} |a_3| A^3 = 0.891 \cdot a_1 A \]

yielding \(A_{1dB} = 0.145\sqrt{|a_1/a_3|}\).

9.3.3 Second-Order Intercept Point (IIP2)

For direct-conversion receivers, the IIP2 determines susceptibility to AM interference via second-order nonlinearity. The output second-order term from two-tone input:

\[ y_{IM2} = a_2 A^2 \cos\!\left((\omega_1 - \omega_2)t\right) \]

falls at DC or low frequency, directly corrupting the baseband signal.

\[ IIP2 = \frac{a_1^2}{a_2}, \quad \text{or equivalently } P_{IIP2} = 2 P_{fund} - P_{IM2} \]

Improving IIP2 requires circuit symmetry (differential topologies, well-matched components) and calibration.

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Chapter 10: Power Amplifiers, Nonlinearity, and Distortion

10.1 PA Efficiency

The efficiency of a power amplifier is the primary design metric, as PAs dominate battery life in portable transmitters.

Drain efficiency:

\[ \eta_D = \frac{P_{out}}{P_{DC}} = \frac{P_{out}}{V_{DD} I_{DD}} \]

Power-Added Efficiency (PAE):

\[ PAE = \frac{P_{out} - P_{in}}{P_{DC}} = \eta_D \left(1 - \frac{1}{G}\right) \]

PAE is more relevant than drain efficiency for moderate-gain PAs.

10.2 PA Classes

10.2.1 Class A

In Class A, the transistor is biased for conduction over the full \(360°\) of the RF cycle. Maximum theoretical efficiency is 50% (when the output swing just reaches the supply rails), but practical efficiency is much lower due to power backoff.

At output power \(P_{out}\) backed off by \(\Delta\) dB from maximum:

\[ \eta_{class A}(P_{out}) = \eta_{max} \cdot 10^{-\Delta/10} \]

Efficiency degrades linearly with power backoff in Class A — severe penalty for OFDM signals with high PAPR.

10.2.2 Class B

Transistor conducts for half the RF cycle (\(180°\) conduction angle). The maximum theoretical drain efficiency is \(\pi/4 \approx 78.5\%\). Efficiency at power backoff \(P_{out}/P_{out,max} = x\):

\[ \eta_{class B}(x) = \frac{\pi}{4}\sqrt{x} \]

Class B efficiency decreases more slowly with backoff than Class A.

10.2.3 Class AB

The conduction angle is between \(180°\) and \(360°\). Class AB is the most common practical PA mode, offering a tradeoff between linearity (better than Class B) and efficiency (better than Class A).

10.2.4 Class C, D, E, and F

• Class C (\(<180°\) conduction): Higher efficiency (\(>78.5\%\)) but lower output power capability; highly nonlinear, suitable only for constant-envelope modulations.
• Class D (switching PA): Theoretically 100% efficient using complementary switches; practical operation limited by switch transition times and parasitics.
• Class E (resonant switching): Zero-voltage switching (ZVS) eliminates capacitive switching loss; designed so that \(v_{DS} = 0\) and \(dv_{DS}/dt = 0\) at the switching instant. Maximum theoretical efficiency: 100%.
• Class F (harmonic tuning): Uses harmonic resonators to shape the drain voltage waveform toward a square wave, reducing overlap between \(v_{DS}\) and \(i_D\) to improve efficiency.

10.2.5 Load-Pull Analysis

PA efficiency and output power depend critically on the load impedance presented to the transistor. The optimum load resistance for maximum output power (Class A):

\[ R_{opt} = \frac{V_{DD} - V_{knee}}{I_{max}/2} \]

This is typically much lower than 50 \(\Omega\), requiring an output matching network to transform 50 \(\Omega\) to \(R_{opt}\). Load-pull measurements (varying the load impedance and recording contours of constant \(P_{out}\) and PAE) characterize the PA transistor and guide matching network design.

10.3 Linearization Techniques

10.3.1 Predistortion

Digital predistortion (DPD) applies an inverse nonlinearity to the transmit signal before the PA, compensating for the PA’s AM-AM and AM-PM distortion in the digital domain. Modern DPD uses adaptive algorithms and lookup tables, achieving 20–30 dB of linearity improvement.

10.3.2 Envelope Tracking

Envelope tracking (ET) dynamically adjusts the PA supply voltage \(V_{DD}(t)\) to follow the envelope of the transmitted signal. This keeps the PA near compression (high efficiency) even during low-power portions of an OFDM waveform, significantly improving average efficiency.

10.3.3 LINC (Linear Amplification with Nonlinear Components)

In LINC, the amplitude-modulated signal is decomposed into two constant-envelope (phase-modulated) components, each amplified by a nonlinear (efficient) PA, then recombined:

\[ x(t) = s_1(t) + s_2(t) \]

where \(|s_1(t)| = |s_2(t)| = \text{const}\). The recombination network must be a power combiner with good isolation.

10.4 Spectral Regrowth and ACPR

Nonlinear distortion in the PA causes spectral regrowth: energy spreads from the intended channel into adjacent channels, violating spectrum regulations.

Adjacent Channel Power Ratio (ACPR) measures how much power leaks into the adjacent channel relative to the in-channel power:

\[ ACPR = 10\log_{10}\frac{\int_{adjacent\, channel} S_{out}(f)\,df}{\int_{main\, channel} S_{out}(f)\,df} \]

For WCDMA, ACPR must be below \(-45\,\text{dBc}\) at \(\pm 5\,\text{MHz}\) offset. ACPR directly relates to the PA’s IIP3 and IIP5 through the Volterra series expansion of its nonlinear transfer characteristic.

10.5 Volterra Series Analysis

For a weakly nonlinear system with memory (frequency-dependent nonlinearity), the Volterra series generalizes the power series:

\[ y(t) = \sum_{n=1}^{\infty} \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} h_n(\tau_1,\ldots,\tau_n) \prod_{k=1}^n x(t - \tau_k)\,d\tau_k \]

The nth-order Volterra kernel \(h_n\) characterizes nth-order nonlinear behavior with memory. The first-order kernel is the ordinary impulse response. The third-order kernel \(H_3(\omega_1, \omega_2, \omega_3)\) (its 3D Fourier transform) determines IM3 products at any combination of input frequencies — critical for analyzing PA distortion with wideband (OFDM) input signals.

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Chapter 11: Guest Lectures and Advanced Topics

11.1 System-Level RF Design Considerations

Modern RF ICs must meet stringent co-existence requirements as wireless standards proliferate. Key system considerations:

Blocker tolerance: A receiver must maintain adequate sensitivity in the presence of large out-of-band interferers (blockers). A large blocker desensitizes the receiver by compressing the LNA or generating intermodulation with a co-channel interferer. The blocker compression requirement drives LNA IIP3 and dynamic range.

Spurious-free dynamic range (SFDR):

\[ SFDR = \frac{2}{3}\left(IIP3 - P_{noise}\right) \]

where \(P_{noise} = -174 + NF + 10\log_{10}(BW)\,\text{dBm}\). SFDR is the input power range over which the system is limited by noise (at low input) or by third-order distortion (at high input), in dBm.

11.2 Silicon Process Technologies for RF

11.2.1 CMOS Scaling and RF Performance

As CMOS feature size decreases (from 180 nm → 65 nm → 28 nm → 7 nm FinFET):

• \(f_T\) increases (now exceeding 300 GHz in 28 nm)
• Supply voltage decreases (1.8 V → 1.0 V → 0.8 V), limiting headroom for PA design
• Gate oxide thickness decreases, increasing gate leakage and complicating bias circuits
• Flicker noise corner frequency increases, worsening phase noise in VCOs

11.2.2 SiGe BiCMOS

Silicon-Germanium (SiGe) heterojunction bipolar transistors (HBTs) offer higher \(f_T/f_{max}\), lower noise, and better linearity than CMOS at the same feature size. SiGe HBT \(f_T\) up to 300–500 GHz enables operation into the millimeter-wave range (>30 GHz).

BiCMOS processes integrate SiGe HBTs with CMOS transistors, allowing the designer to use BJTs for high-performance analog/RF functions (LNA, VCO) and CMOS for digital baseband — combining the best of both.

11.3 Millimeter-Wave and 5G/6G Considerations

At millimeter-wave frequencies (24–100 GHz for 5G NR, >100 GHz for 6G research):

• On-chip transmission lines replace lumped LC elements (wavelength on-chip becomes comparable to circuit dimensions)
• Antenna-in-package (AiP) and antenna-on-chip become practical
• Phased arrays with per-element phase shifting enable beam steering
• Path loss scales as \(f^2\) (Friis formula), requiring high EIRP and antenna gain

The beam-forming gain of a phased array with \(N\) elements:

\[ G_{array} = 10\log_{10}(N^2) = 20\log_{10}(N) \]

(N² gain: N from coherent addition, N from increased aperture). A 64-element array provides 36 dB of beam gain, compensating for increased path loss at mmWave.

11.4 Summary of Key RF Design Specifications

These specifications are not independent. For example, reducing NF (by increasing LNA transconductance) typically degrades IIP3 (more nonlinearity at higher bias current). Achieving all specifications simultaneously within a power budget is the central challenge of RF IC design — the subject of ongoing research and the skill developed through this course.