Transmission Lines
At high frequencies, interconnects must be treated as transmission lines with distributed inductance, capacitance, resistance, and conductance.
$$Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}} \approx \sqrt{\frac{L}{C}}$$
Characteristic impedance of a transmission line
Microstrip
Signal trace on top of a dielectric with ground plane below. Most common for PCB and package routing.
Stripline
Signal trace embedded between two ground planes. Better shielding but harder to manufacture.
S-Parameters
Scattering parameters describe how RF signals propagate through a network. For a 2-port network:
$$\begin{bmatrix} b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$$
S₁₁ - Input Reflection Coefficient
Ratio of reflected to incident wave at port 1. Related to return loss: RL = -20log|S₁₁|
S₂₁ - Forward Transmission
Ratio of transmitted to incident wave. For passive networks, insertion loss: IL = -20log|S₂₁|
S₁₂ - Reverse Transmission
Isolation in reverse direction. For reciprocal networks, S₁₂ = S₂₁
S₂₂ - Output Reflection
Reflection coefficient at output port with input terminated
Smith Chart
A graphical tool for solving transmission line problems. Maps complex impedance to reflection coefficient.
$$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} = |\Gamma|e^{j\theta}$$
Reflection coefficient relates load impedance to characteristic impedance
- Center: Matched load (Z = Z₀, Γ = 0)
- Right edge: Open circuit (Z = ∞, Γ = +1)
- Left edge: Short circuit (Z = 0, Γ = -1)
- Constant resistance circles: Pass through right edge
- Constant reactance arcs: Pass through right edge
Impedance Matching
Maximum power transfer occurs when source impedance equals complex conjugate of load impedance.
$$P_{delivered} = P_{available} \cdot (1 - |\Gamma|^2)$$
Power transfer efficiency depends on reflection coefficient
L-Match
Two reactive elements. Simple but narrow bandwidth. Q = √(R_high/R_low - 1)
Pi/T Match
Three elements allow Q control. Wider bandwidth than L-match.
Quarter-Wave
Transmission line transformer. Z₀ = √(Z_in × Z_load)
Antenna Theory
Antennas convert guided electromagnetic waves to free-space radiation and vice versa.
$$G = \frac{4\pi A_e}{\lambda^2} = \eta D$$
Gain relates effective aperture to wavelength; η = antenna efficiency
Directivity
$$D = \frac{U_{max}}{U_{avg}} = \frac{4\pi U_{max}}{P_{rad}}$$
Array Factor
$$AF = \sum_{n=1}^{N} a_n e^{j(n-1)(kd\cos\theta + \beta)}$$
Friis Equation
$$\frac{P_r}{P_t} = G_t G_r \left(\frac{\lambda}{4\pi R}\right)^2$$
Signal Integrity
High-speed digital signals require careful attention to transmission line effects.
Rise Time vs Bandwidth
$$BW_{knee} = \frac{0.35}{t_r}$$
Signals with faster rise times contain higher frequency content
Critical Length
$$\ell_{crit} = \frac{t_r}{2 \cdot t_{pd}}$$
Traces longer than critical length require termination
Eye Diagram
Overlay of all bit transitions shows timing margin (eye width) and voltage margin (eye height)
Power Distribution
The power distribution network must maintain low impedance across all frequencies of interest.
$$Z_{target} = \frac{\Delta V_{allowed}}{\Delta I_{max}}$$
Target impedance based on voltage ripple budget
- VRM: Provides DC and handles low-frequency transients (1 Hz - 10 kHz)
- Bulk capacitors: Handle mid-frequency transients (10 kHz - 1 MHz)
- MLCCs: Decouple high-frequency noise (1 MHz - 100 MHz)
- On-die capacitance: Handles highest frequencies (>100 MHz)