Electronics Guide

Resistance (R)

Series resistance per unit length, measured in ohms per meter (Ω/m), represents the conductor's opposition to current flow due to the finite conductivity of the metal. At DC and low frequencies, resistance is determined primarily by the conductor's cross-sectional area and material resistivity. However, at higher frequencies, the skin effect causes current to concentrate near the conductor surface, effectively reducing the cross-sectional area and increasing resistance. Surface roughness further increases resistance by increasing the effective path length for current flow.

Inductance (L)

Series inductance per unit length, measured in henries per meter (H/m), arises from the magnetic field surrounding current-carrying conductors. For a transmission line, both the external inductance (due to magnetic flux between conductors) and internal inductance (due to magnetic flux within the conductors themselves) contribute to the total value. The inductance depends on the geometry of the conductors and their separation, and at high frequencies, skin effect reduces the internal inductance component as current becomes concentrated at the surface.

Conductance (G)

Shunt conductance per unit length, measured in siemens per meter (S/m), represents leakage current flowing through the dielectric material between conductors. This parameter accounts for losses in the insulating material and is related to the dielectric loss tangent. In good dielectrics, conductance is typically very small at DC but increases with frequency due to polarization losses in the dielectric material. For high-quality transmission lines, G is often negligible compared to other loss mechanisms.

Capacitance (C)

Shunt capacitance per unit length, measured in farads per meter (F/m), represents the ability to store electric charge between the conductors. This parameter depends on the geometry of the conductors, their separation, and the permittivity of the dielectric material between them. Unlike resistance and inductance, capacitance is largely independent of frequency for most practical transmission lines, though the effective dielectric constant may show some frequency dependence in certain materials.

Lossless Line (R = 0, G = 0)

For an ideal lossless transmission line:

Z₀ = √(L/C)

This fundamental result shows that the characteristic impedance of a lossless line is real, frequency-independent, and depends only on the ratio of inductance to capacitance per unit length. This is the impedance most commonly specified for transmission lines (50Ω for RF systems, 75Ω for video, etc.).

Low-Loss Line at High Frequency

For practical transmission lines with small losses (R << ωL and G << ωC), which is typical at high frequencies:

Z₀ ≈ √(L/C) × [1 + j(R/2ωL - G/2ωC)]

This approximation shows that the characteristic impedance becomes slightly complex, but the magnitude remains close to √(L/C), with a small phase shift.

Lossy Line at Low Frequency

At very low frequencies where R >> ωL and G >> ωC:

Z₀ ≈ √(R/G)

In this regime, resistance dominates, and the characteristic impedance is again real but now determined by the resistance and conductance rather than the reactive elements.

Phase Velocity

The phase velocity v_p describes how fast a point of constant phase (such as a wave peak) travels along the transmission line. For a lossless line, the phase velocity is:

v_p = ω/β = 1/√(LC)

Since the inductance and capacitance per unit length depend on the geometry and the permittivity and permeability of the dielectric material, the phase velocity is related to the speed of light c by:

v_p = c/√(ε_rμ_r)

where ε_r is the relative permittivity and μ_r is the relative permeability of the dielectric. For most practical dielectrics, μ_r ≈ 1, so:

v_p ≈ c/√ε_r

Common PCB materials like FR-4 have ε_r ≈ 4.0 to 4.5, giving propagation velocities around 50-53% of the speed of light, or approximately 15-16 cm per nanosecond.

Group Velocity

For signals containing multiple frequency components (such as digital pulses), the group velocity v_g describes how fast the signal envelope propagates. It is defined as:

v_g = ∂ω/∂β

For lossless, non-dispersive transmission lines, the group velocity equals the phase velocity. However, in dispersive media where the phase velocity depends on frequency, the group velocity differs from the phase velocity. This dispersion causes pulse spreading and is a critical consideration in high-speed digital systems and fiber optic communications.

Time Delay

The propagation delay per unit length τ_d is the reciprocal of the propagation velocity:

τ_d = 1/v_p = √(LC)

For a line of length ℓ, the total time delay is:

T_d = ℓ × τ_d = ℓ/v_p

This delay is critical in high-speed digital design for timing budget analysis, clock distribution networks, and length matching of differential pairs or parallel buses. A timing mismatch of even a few millimeters can cause signal skew problems at multi-gigahertz data rates.

Electrical Length

The electrical length of a transmission line is often more important than its physical length. It is expressed in terms of wavelength or degrees of phase shift:

θ = βℓ = 2πℓ/λ

where λ is the wavelength at the operating frequency. A quarter-wavelength line (θ = 90°) has special impedance-transforming properties, while a half-wavelength line (θ = 180°) appears electrically transparent, replicating the load impedance at the input.

Lossless Line Model

The lossless transmission line model assumes R = 0 and G = 0, leaving only the reactive elements L and C. While no real transmission line is truly lossless, this approximation is excellent for many practical situations, particularly:

• Short interconnects in digital systems where propagation time matters more than attenuation
• High-quality RF transmission lines over moderate distances at frequencies well below cutoff
• Initial design and analysis where loss effects can be added as perturbations later

For a lossless line, the propagation constant simplifies to:

γ = jβ = jω√(LC)

This means α = 0 (no attenuation) and the phase constant β = ω√(LC) is purely real. The characteristic impedance is real and frequency-independent:

Z₀ = √(L/C)

Waves propagate without amplitude decay, and reflections at impedance discontinuities are purely geometric, depending only on impedance ratios. The lossless model enables clean analytical solutions for reflection coefficients, standing wave patterns, and impedance transformations.

Low-Loss Line Model

For most practical transmission lines at high frequencies, losses are small but not negligible. The low-loss approximation assumes R << ωL and G << ωC, which allows perturbative treatment of losses while maintaining analytical tractability.

Under these conditions, the attenuation constant can be approximated as:

α ≈ (R/2Z₀) + (GZ₀/2)

where Z₀ ≈ √(L/C). This shows that attenuation comes from two sources: conductor losses (first term) and dielectric losses (second term). The phase constant remains approximately:

β ≈ ω√(LC)

This model is widely used for cable design, PCB trace analysis, and RF system link budgets where losses must be accounted for but don't fundamentally change the wave behavior.

General Lossy Line Model

When losses are significant (such as long cables at high frequencies, poor conductors, or lossy dielectrics), the full lossy transmission line equations must be used without approximation. The propagation constant retains both substantial real and imaginary parts:

γ = √[(R + jωL)(G + jωC)]

In this regime, several important effects occur:

• Frequency-dependent attenuation: Higher frequencies experience greater loss, causing signal distortion and pulse spreading
• Complex characteristic impedance: The impedance becomes frequency-dependent and complex, affecting matching networks
• Dispersion: Different frequency components travel at different velocities, causing pulse broadening
• Reduced reflection coefficients: Losses damp reflections, making mismatches less severe but also making time-domain reflectometry less effective

Analyzing lossy lines often requires numerical methods or computer-aided tools, particularly for time-domain analysis of complex signals.

Resistance Variation

The series resistance R exhibits the strongest frequency dependence due to two primary mechanisms:

Skin Effect: At DC and low frequencies, current distributes uniformly across the conductor's cross-section. As frequency increases, current concentrates exponentially near the surface within a depth called the skin depth δ:

δ = √[2/(ωμσ)]

where μ is the permeability and σ is the conductivity of the conductor. For copper at room temperature, δ ≈ 66/√f mm, where f is in Hz. At 1 GHz, the skin depth is only about 2 micrometers.

As the effective conducting area decreases with frequency, resistance increases proportionally to √f at high frequencies:

R_AC ≈ R_DC × √(f/f_c)

where f_c is the frequency at which skin depth equals the conductor radius.

Proximity Effect: In transmission lines with closely spaced conductors, the current distribution is further modified by magnetic fields from neighboring conductors. This proximity effect can increase resistance beyond the skin effect alone, particularly in closely coupled differential pairs or multi-conductor cables.

Inductance Variation

The total inductance consists of external inductance (from magnetic flux between conductors) and internal inductance (from flux within conductors). While external inductance remains essentially constant with frequency, internal inductance decreases as skin effect concentrates current near the surface.

At low frequencies where current is uniform, the internal inductance contributes significantly. As frequency increases and skin effect dominates, internal inductance approaches zero, and the total inductance approaches the external inductance value:

L(f) → L_external as f → ∞

For most high-frequency applications, this frequency dependence of inductance is much weaker than that of resistance and can often be neglected.

Capacitance Variation

Capacitance is generally the most frequency-independent of the four parameters for good quality dielectrics. However, some frequency dependence exists due to:

• Dielectric dispersion: The relative permittivity of some materials varies slightly with frequency due to molecular polarization mechanisms
• Surface roughness effects: At very high frequencies, field penetration into surface roughness features can effectively increase capacitance

For most practical purposes and common PCB dielectrics, capacitance can be treated as frequency-independent up to many gigahertz.

Conductance Variation

Dielectric conductance increases approximately linearly with frequency due to dielectric loss mechanisms. This is commonly expressed through the loss tangent tan(δ):

G = ωC × tan(δ)

The loss tangent represents the ratio of energy dissipated to energy stored in the dielectric per cycle. For low-loss materials like PTFE, tan(δ) ≈ 0.0002, while standard FR-4 has tan(δ) ≈ 0.02 at 1 GHz. Because G is proportional to frequency, dielectric losses become increasingly important at higher frequencies.

Classical Skin Effect Theory

Skin effect arises from the interaction between current flow and the magnetic field it creates. The time-varying magnetic field induces eddy currents that oppose the main current in the conductor's interior, forcing current toward the surface. This self-induction effect strengthens with frequency, progressively excluding current from the conductor core.

For a cylindrical conductor, the current density decreases exponentially from the surface:

J(x) = J_surface × e^-x/δ

where x is the depth below the surface and δ is the skin depth. The skin depth represents the depth at which current density has fallen to 1/e (about 37%) of its surface value.

The AC resistance of a conductor with skin effect can be expressed in terms of a resistance ratio:

R_AC/R_DC = (a/2δ) × [ber'(a/δ)bei(a/δ) - bei'(a/δ)ber(a/δ)] / [ber'²(a/δ) + bei'²(a/δ)]

where a is the conductor radius and ber, bei are Kelvin functions. For the common case where a >> δ, this simplifies to the square-root frequency dependence mentioned earlier.

For rectangular conductors like PCB traces, the analysis is more complex, but the fundamental behavior remains: resistance increases as √f when skin depth is much smaller than conductor dimensions.

Surface Roughness Effects

Real conductors are not perfectly smooth. PCB copper foils, in particular, have intentional surface roughness to promote adhesion to the dielectric substrate. This roughness significantly affects high-frequency resistance beyond what classical skin effect predicts.

When skin depth becomes comparable to or smaller than the roughness feature size, current must travel a longer path following the surface contours, increasing the effective length and hence resistance. Several models describe this effect:

Hammerstad-Bekkadal Model: One of the earliest practical models relates the roughness-enhanced resistance to RMS surface roughness R_q:

K_rough = 1 + (2/π) × arctan[1.4(R_q/δ)²]

where K_rough is the resistance multiplication factor. This model shows that roughness effects become significant when δ ≈ R_q and saturate when δ << R_q.

Snowball Model: A more sophisticated approach recognizes that roughness exists at multiple spatial scales (from large peaks to fine texture). The model treats roughness as a fractal and includes both the increased path length and the local current redistribution around roughness features.

Cannonball-Huray Model: This modern model represents rough surfaces as spherical protrusions and accounts for both the geometric increase in path length and the field concentration effects. It has shown good agreement with measurements across wide frequency ranges.

Practical implications include:

• Standard copper foil (R_q ≈ 2-5 μm) can increase loss by 1.5-2× at 10 GHz compared to smooth copper
• Low-profile or reverse-treat foils (R_q ≈ 0.5-1.5 μm) significantly reduce this penalty
• The roughness penalty is most severe in the frequency range where skin depth is comparable to roughness size (typically 1-20 GHz for typical PCB copper)
• Very high data rate signals (56 Gbps and beyond) require careful selection of low-roughness copper to maintain acceptable channel loss

Design Considerations

Managing skin effect and roughness losses requires several strategies:

• Conductor selection: Use low-roughness copper for high-frequency applications; accept higher roughness for better adhesion in lower-frequency designs
• Wider traces: At high frequencies where current is confined to the surface, increasing trace width proportionally decreases resistance
• Plating quality: Smooth plating reduces roughness penalty, but plating itself adds resistance if poorly controlled
• Material choice: Lower-loss dielectrics become more important at higher frequencies where conductor losses are already significant
• Accurate modeling: Modern signal integrity tools include roughness models; ensure your simulator accounts for these effects with parameters matched to your fabrication process

Polarization and Loss Tangent

When an electric field is applied to a dielectric, the material becomes polarized as charges and dipoles respond to the field. In AC conditions, the polarization must continuously reverse direction, and the energy required for this reorientation is partially dissipated as heat.

The complex permittivity captures both energy storage and dissipation:

ε = ε' - jε''

where ε' is the real part (energy storage) and ε'' is the imaginary part (energy loss). The loss tangent is the ratio:

tan(δ) = ε''/ε'

A low loss tangent (< 0.001) indicates a high-quality dielectric; higher values (> 0.01) indicate significant losses. The loss tangent directly relates to the conductance parameter:

G = ωCtan(δ)

This shows that dielectric loss increases linearly with frequency, unlike conductor loss which increases as √f.

Physical Loss Mechanisms

Dipolar Relaxation: Polar molecules or groups within the dielectric have permanent dipole moments that attempt to align with an applied electric field. At lower frequencies, dipoles can follow field oscillations, but as frequency increases, the inertia of molecular reorientation causes a phase lag between the field and polarization. This lag means energy is absorbed from the field and converted to molecular kinetic energy (heat). Each type of dipole has a characteristic relaxation frequency where loss peaks.

Ionic Conduction: Mobile ions in the dielectric (even in very small concentrations) contribute to loss by drifting in response to the applied field. This mechanism is essentially a very small DC conductivity that adds to the loss. It's typically more important at lower frequencies and in materials with higher moisture content.

Electronic Polarization: At very high frequencies (optical and above), electronic cloud distortion in atoms becomes the dominant polarization mechanism. The losses associated with electronic polarization are usually negligible at RF and microwave frequencies.

Interfacial Polarization: In composite or heterogeneous dielectrics (like fiber-reinforced materials), charge accumulation at interfaces between different phases causes additional loss. This Maxwell-Wagner polarization is significant in glass-reinforced PCB materials and can vary with fiber weave pattern.

Common Dielectric Materials

Understanding typical material properties helps in design choices:

PTFE (Teflon): Excellent high-frequency properties with ε_r ≈ 2.1 and tan(δ) ≈ 0.0002-0.0004. Non-polar molecular structure minimizes dipolar losses. Used in demanding RF applications, but expensive and mechanically soft.

FR-4: The ubiquitous PCB material has ε_r ≈ 4.2-4.5 and tan(δ) ≈ 0.015-0.025 at 1 GHz. Relatively high loss limits use to moderate frequencies (below ~5 GHz for long traces). Loss varies significantly between manufacturers and resin formulations.

Rogers RO4000 series: Mid-range hydrocarbon ceramic materials with ε_r ≈ 3.3-3.5 and tan(δ) ≈ 0.002-0.004. Good balance of cost and performance for microwave applications up to 40 GHz.

Polyimide: Used in flexible circuits and high-temperature applications. ε_r ≈ 3.4-3.6 and tan(δ) ≈ 0.002-0.008 depending on formulation. Polar groups in molecular structure lead to higher loss than PTFE.

Low-loss FR-4 variants: Modern formulations (Megtron, N4000, etc.) achieve tan(δ) ≈ 0.002-0.008 while maintaining FR-4 processability. Enable digital data rates beyond 25 Gbps without resorting to exotic materials.

Frequency Dependence

Dielectric properties vary with frequency due to dispersion. The Debye model describes relaxation for a single dipole species:

ε(ω) = ε_∞ + (ε_s - ε_∞)/(1 + jωτ)

where ε_s is the static permittivity, ε_∞ is the high-frequency limit, and τ is the relaxation time. Real materials often show multiple relaxation times, requiring more complex models (Cole-Cole, Havriliak-Negami, etc.).

For PCB materials in the microwave range, the permittivity typically decreases slightly with frequency (1-5% variation from DC to 10 GHz), while loss tangent may increase gradually. Accurate modeling for broadband digital signals requires frequency-dependent material parameters, usually obtained from vendor data or measurements.

Environmental Effects

Dielectric properties are sensitive to environmental conditions:

• Moisture absorption: Water has high permittivity (ε_r ≈ 80) and loss tangent. Even small amounts of absorbed moisture significantly increase both ε_r and tan(δ). Conformal coating or hermetic packaging protects against this.
• Temperature: Molecular mobility increases with temperature, generally increasing loss tangent while decreasing permittivity. Temperature coefficient of permittivity (TCε) is an important specification for temperature-stable designs.
• Aging: Some materials degrade over time through oxidation or chemical changes, increasing loss. Glass transition temperature and thermal cycling accelerate aging.

Causality Principle

Causality states that the impulse response h(t) of any physical system must be zero for t < 0. In other words, there is no output before an input is applied. This requirement is so fundamental that violations would contradict basic physical law.

When we transform to the frequency domain, causality places strict constraints on the mathematical form of the frequency response. Specifically, if a frequency response is to represent a physically realizable system, its real and imaginary parts (or equivalently, magnitude and phase) cannot vary independently—they must be related through specific integral transforms.

Kramers-Kronig Relations

The Kramers-Kronig relations are mathematical expressions of causality in the frequency domain. They state that the real and imaginary parts of any causal, linear frequency response are Hilbert transforms of each other.

For a complex response function H(ω) = H'(ω) + jH''(ω):

H'(ω) = (1/π) × P∫[H''(ω')/(ω' - ω)]dω'
H''(ω) = -(1/π) × P∫[H'(ω')/(ω' - ω)]dω'

where P denotes the Cauchy principal value of the integral, and the integration is over all frequencies from -∞ to +∞.

These relations mean that if you know the real part of the response at all frequencies, you can calculate the imaginary part (and vice versa). This is not merely a mathematical curiosity—it has practical implications for measurement and modeling.

Application to Transmission Lines

Complex Permittivity: The complex permittivity ε(ω) = ε'(ω) - jε''(ω) must satisfy Kramers-Kronig relations:

ε'(ω) - ε_∞ = (2/π) × P∫[ω'ε''(ω')/(ω'² - ω²)]dω'
ε''(ω) = -(2ω/π) × P∫[(ε'(ω') - ε_∞)/(ω'² - ω²)]dω'

This means you cannot arbitrarily choose both the dielectric constant and loss tangent as functions of frequency—if one is specified, the other is determined by causality. This constraint is important when developing broadband dielectric models from limited measurement data.

Propagation Constant: The complex propagation constant γ(ω) = α(ω) + jβ(ω) must also obey Kramers-Kronig relations. This means attenuation and phase response are not independent. For example, a transmission line with frequency-dependent attenuation will necessarily exhibit dispersion (frequency-dependent phase velocity), even if the dielectric is non-dispersive.

Characteristic Impedance: The complex characteristic impedance Z₀(ω) similarly obeys these relations. A frequency-dependent resistance necessarily implies a reactive component, which is why lossy lines always have complex characteristic impedance.

Practical Implications

Model Validation: Any electrical model that violates Kramers-Kronig relations is non-causal and therefore unphysical. When developing or using transmission line models, verifying KK compliance ensures physical consistency. Models that fit measured data well at discrete frequencies but violate KK relations will produce non-physical results in time-domain simulation.

Measurement Extrapolation: If you measure one component of a complex parameter (say, dielectric loss) over a limited frequency range, Kramers-Kronig relations can help extrapolate the real part (permittivity) beyond the measured range, or validate the consistency of measurements.

Dispersion and Loss Connection: The relations prove that any frequency-dependent loss must produce dispersion. This is why lossy transmission lines always exhibit pulse spreading—the absorption that causes loss necessarily makes different frequencies travel at different velocities. You cannot have loss without dispersion (though you can have dispersion without loss in lossless but dispersive media).

Minimum Phase Systems: Causal systems with no zeros in the right half of the complex frequency plane are called minimum phase systems. For such systems, the phase response is uniquely determined by the magnitude response. Many transmission line models approximate minimum phase behavior, allowing phase to be calculated from measured magnitude data alone.

Physical Interpretation

Why does causality create these mathematical constraints? Consider a pulse propagating through a transmission line. The leading edge of the pulse must be affected by the line's properties at all frequencies, because even a sharp edge has a broad spectrum. The way the line attenuates different frequency components (loss) necessarily affects how those components add up in phase (dispersion) to maintain the constraint that the pulse cannot arrive before it was sent.

In information theory terms, causality means you cannot have perfect knowledge of a signal at any time without having observed it for infinite time. This fundamental limitation manifests in the frequency domain as the Kramers-Kronig relations linking magnitude and phase, or equivalently, real and imaginary parts of any transfer function.

Verification in Practice

When working with measured or modeled transmission line data:

• Verify that tabulated frequency-dependent parameters satisfy KK relations (many simulation tools include KK checking)
• Be suspicious of material data showing loss without corresponding dispersion, or vice versa
• When curve-fitting models to measurements, impose KK constraints to ensure physical realizability
• For broadband digital signal integrity, ensure that your transmission line model is causal across the entire frequency range of interest (DC to several times the Nyquist frequency)
• In time-domain simulation, causality violations may appear as pre-shoot (signal response before the stimulus arrives), indicating a non-physical model

High-Speed Digital Design

• Impedance control: Maintaining characteristic impedance along signal paths minimizes reflections and ensures signal integrity
• Length matching: Using propagation delay calculations to match trace lengths in differential pairs and parallel buses
• Loss budgeting: Accounting for both conductor and dielectric losses when predicting channel performance at high data rates
• Dispersion management: Understanding how frequency-dependent parameters affect rise time and eye diagram closure

RF and Microwave Systems

• Impedance matching networks: Using transmission line transformations and quarter-wave sections
• Filter design: Exploiting distributed element behavior to create filters, couplers, and power dividers
• Resonant structures: Using half-wave and quarter-wave resonances
• Calibration and de-embedding: Removing transmission line effects from measurements

Power Integrity

• Power distribution network design: Managing impedance of power delivery traces at high frequencies
• Decoupling strategy: Accounting for inductance and propagation delay in capacitor placement
• Return path management: Understanding how current return paths affect effective inductance