Signal Integrity Formulas
Core equations for transmission line analysis, impedance matching, reflections, crosstalk, and high-speed link characterization.
Characteristic Impedance
Microstrip Characteristic Impedance
Z0 = characteristic impedance (Ω)
εr = relative dielectric constant of the substrate
h = height of the dielectric (distance from trace to reference plane)
w = trace width
t = trace thickness (copper weight)
Valid for microstrip geometries where w/h < 2. For wider traces, use the more general Hammerstad-Jensen model. Typical range: 30–120 Ω.
Stripline Characteristic Impedance
Z0 = characteristic impedance (Ω)
εr = relative dielectric constant of the substrate
b = distance between the two reference planes
w = trace width
t = trace thickness
Assumes the trace is centered between two ground planes. For offset stripline, use adjusted formulas or a field solver.
Differential Impedance
Zdiff = differential impedance (Ω)
Z0 = single-ended characteristic impedance of each trace (Ω)
k = coupling coefficient (0 < k < 1), higher values indicate tighter coupling
For typical PCB differential pairs, k ranges from 0.05 to 0.30. USB 3.x targets 90Ω differential; PCIe targets 85Ω.
Propagation Delay
Microstrip Propagation Delay
tpd = propagation delay per unit length (ps/inch)
εr = relative dielectric constant of the substrate material
For FR-4 (εr ≈ 4.2), microstrip propagation delay is approximately 140–150 ps/inch. Microstrip is faster than stripline because the effective dielectric constant is lower (partial air exposure).
Stripline Propagation Delay
tpd = propagation delay per unit length (ps/inch)
εr = relative dielectric constant of the dielectric
For FR-4, stripline propagation delay is approximately 170–180 ps/inch. The signal is fully immersed in the dielectric, so the full εr is used.
Critical Length (Transmission Line Threshold)
Lcrit = critical length — traces longer than this require controlled impedance
tr = signal rise time (10%–90%)
tpd = propagation delay per unit length (ps/inch or ns/inch)
When the trace length exceeds Lcrit, the trace must be treated as a transmission line, and impedance matching becomes essential. Some engineers use a 1/3 or 1/6 factor instead of 1/2 for additional margin.
Bandwidth from Rise Time
BW = bandwidth (Hz) — the knee frequency of the signal
tr = rise time (seconds, 10%–90%)
This approximation assumes a Gaussian response. For 20%–80% rise times, use BW = 0.22/tr. The knee frequency defines the highest significant spectral content of the signal.
Reflections & Return Loss
Reflection Coefficient
Γ = voltage reflection coefficient (dimensionless, range −1 to +1)
ZL = load impedance (Ω)
Z0 = characteristic impedance of the transmission line (Ω)
Γ = 0 means perfect match (no reflection). Γ = +1 is an open circuit; Γ = −1 is a short circuit. In TDR analysis, the reflected waveform reveals the impedance profile along the line.
Return Loss
RL = return loss (dB) — higher values mean better matching
Γ = reflection coefficient (magnitude)
A return loss of 20 dB means only 1% of the power is reflected. Most high-speed interfaces require RL > 10–15 dB across the frequency band of interest.
Voltage Standing Wave Ratio (VSWR)
VSWR = voltage standing wave ratio (dimensionless, range 1 to ∞)
Γ = reflection coefficient (magnitude)
VSWR = 1 indicates a perfect impedance match with no standing wave. VSWR = 2 corresponds to |Γ| = 0.33. Used extensively in RF/microwave design and connector specifications.
Crosstalk
Near-End Crosstalk Coefficient (NEXT / Backward Crosstalk)
Kb = backward (near-end) crosstalk coefficient
Cm = mutual capacitance between the aggressor and victim traces (F/m)
C = self-capacitance per unit length (F/m)
Lm = mutual inductance between the aggressor and victim traces (H/m)
L = self-inductance per unit length (H/m)
NEXT is driven by both capacitive and inductive coupling. In stripline, these contributions partially cancel. In microstrip, they add, making crosstalk worse. Increasing trace spacing reduces both Cm and Lm.
Far-End Crosstalk Coefficient (FEXT)
Kf = forward (far-end) crosstalk coefficient
Cm = mutual capacitance (F/m)
Lm = mutual inductance (H/m)
C, L = self-capacitance and self-inductance per unit length
length = coupled length of the traces
tr = signal rise time
tpd = propagation delay per unit length
FEXT increases with coupled length and is zero in homogeneous media (stripline) where Cm/2C = Lm/2L. This is a key advantage of stripline routing for sensitive signals.
Eye Diagram & Jitter Analysis
Eye Height
Eye Height = vertical opening of the eye at the sampling point (V)
Vswing = peak-to-peak differential voltage swing (V)
VISI = inter-symbol interference voltage penalty (V)
Vxtalk = crosstalk-induced noise voltage (V)
Vnoise = random noise voltage (V, typically 3σ or 6σ)
A larger eye height provides more voltage margin for the receiver. ISI is usually the dominant impairment in long channels. Equalization (FFE, DFE, CTLE) can recover eye height.
Eye Width
Eye Width = horizontal opening of the eye at the threshold crossing (seconds)
UI = unit interval — the bit period (1 / data rate)
TJ = total jitter at the target BER
The eye width determines timing margin for the receiver. It must exceed the receiver setup and hold time requirements. CDR bandwidth affects how much jitter is tracked out.
Total Jitter
TJ = total jitter at a specified BER (seconds, peak-to-peak)
DJ = deterministic jitter (peak-to-peak, bounded)
N(BER) = multiplier from the inverse normal distribution for the target BER (e.g., N = 14.07 for BER = 10−12)
RJ = random jitter (RMS, Gaussian)
DJ includes data-dependent jitter (DDJ), duty-cycle distortion (DCD), periodic jitter (PJ), and bounded uncorrelated jitter (BUJ). RJ is unbounded but Gaussian, so the N factor sets the probability tail.
Q-Factor (Signal Quality)
Q = quality factor (dimensionless)
Veye = eye opening voltage (V)
σnoise = RMS noise voltage at the receiver (V)
Higher Q means better signal quality. Q = 7 corresponds approximately to BER = 10−12. The Q-factor is used in both optical and electrical serial link analysis.
Bit Error Rate from Q-Factor
BER = bit error rate (probability of a bit error)
Q = quality factor
erfc = complementary error function
Example values: Q = 6 → BER ≈ 10−9, Q = 7 → BER ≈ 10−12, Q = 7.9 → BER ≈ 10−15. Modern SerDes use FEC to relax the raw BER requirement.
Power Integrity Formulas
Essential equations for power distribution network (PDN) design, decoupling, target impedance, and voltage regulation.
Target Impedance & PDN
Target Impedance
Ztarget = maximum allowable PDN impedance (Ω)
Vdd = supply voltage (V)
%ripple = allowed voltage ripple as a fraction (e.g., 0.05 for 5%)
ΔI = transient current demand (A)
For a 1.0V supply with 5% ripple tolerance and 10A transient current: Ztarget = 1.0 × 0.05 / 10 = 5 mΩ. This is extremely low and must be met from DC to the highest frequency of the transient.
Ground Bounce (Simultaneous Switching Noise)
Vbounce = ground bounce voltage (V)
L = effective loop inductance of the power/ground path (H)
dI/dt = rate of change of current (A/s)
Ground bounce is proportional to both inductance and switching speed. Reducing L (shorter vias, wider traces, more vias in parallel) and reducing dI/dt (slew rate control) both help. This is a primary concern for BGA packages and dense FPGA designs.
IR Drop (DC Voltage Drop)
Vdrop = voltage drop across the conductor (V)
I = DC current flowing through the conductor (A)
R = resistance of the conductor (Ω)
IR drop must be accounted for in both traces and planes. For power planes, use the plane resistance formula below. Ensure Vdrop stays within the voltage tolerance budget of the IC.
Plane Resistance
R = resistance of the plane segment (Ω)
ρ = resistivity of copper (1.724 × 10−8 Ω·m at 20°C)
L = length of the current path (m)
W = width of the current path (m)
t = copper thickness (m)
For 1 oz copper (35 μm thick), the sheet resistance is approximately 0.49 mΩ/square. Double the copper weight halves the resistance. Temperature increases copper resistivity by about 0.39%/°C.
Capacitor & Resonance
Capacitor Impedance (Real Model)
Z = impedance magnitude of the capacitor (Ω)
ω = angular frequency = 2πf (rad/s)
C = capacitance (F)
L = equivalent series inductance, ESL (H)
ESR = equivalent series resistance (Ω)
Below the SRF, the capacitor behaves capacitively. Above SRF, it behaves inductively. At SRF, the impedance equals ESR. This is why multiple capacitor values are used in PDN design to cover different frequency ranges.
Self-Resonant Frequency (SRF)
SRF = self-resonant frequency (Hz)
L = equivalent series inductance (ESL) (H)
C = capacitance (F)
At the SRF, the capacitor provides minimum impedance (equal to ESR). A 100 nF MLCC with 0.5 nH ESL has SRF ≈ 22 MHz. Smaller packages (0201, 0402) have lower ESL and thus higher SRF.
Decoupling Capacitance
C = required decoupling capacitance (F)
ΔI = transient current step (A)
Δt = duration of the transient (s)
ΔV = maximum allowable voltage droop (V)
This gives the minimum bulk capacitance needed. In practice, you also need high-frequency capacitors close to the IC for fast transient response. The effective capacitance is limited by ESL and mounting inductance.
Anti-Resonance Frequency
far = anti-resonance frequency (Hz)
L1 = ESL of the larger capacitor (H)
C2 = capacitance of the smaller capacitor (F)
Anti-resonance creates an impedance peak between two capacitors of different values. The peak occurs where the inductive impedance of the larger cap equals the capacitive impedance of the smaller cap. This must be managed to keep PDN impedance below Ztarget.
Power Planes & Loop Inductance
Plane Pair Capacitance
Cplane = capacitance of the plane pair (F)
ε0 = permittivity of free space (8.854 × 10−12 F/m)
εr = relative dielectric constant
A = area of the overlapping planes (m2)
d = dielectric thickness between the planes (m)
Thin dielectrics between power/ground plane pairs provide high-frequency decoupling capacitance. A 2-mil core with εr = 4 over a 4″ × 4″ area provides about 2 nF. Effective up to the plane resonance frequency.
Plane Resonance Frequencies
fmn = resonant frequency for mode (m, n) (Hz)
c = speed of light (3 × 108 m/s)
εr = relative dielectric constant of the substrate
m, n = mode indices (integers, m=0,1,2... and n=0,1,2...)
a, b = dimensions of the rectangular plane (m)
Plane resonances create impedance peaks that can couple noise. For a 10 cm × 10 cm FR-4 plane, the (1,0) mode resonates at approximately 730 MHz. Plane stitching capacitors or resistive termination can dampen resonance peaks.
PDN Loop Inductance (Parallel Plate)
Lloop = loop inductance of the power/ground plane pair (H)
μ0 = permeability of free space (4π × 10−7 H/m)
d = dielectric spacing between power and ground planes (m)
l = length of the current path (m)
w = width of the current path (m)
Minimizing plane spacing (d) is the most effective way to reduce loop inductance. Using a thin core (2–3 mil) between adjacent power/ground planes dramatically reduces noise coupling and improves transient response.
Voltage Regulator Output Impedance
ZVRM = output impedance of the voltage regulator module (Ω)
Rout = DC output resistance (Ω)
Lout = effective output inductance (H) — dominates above the regulator bandwidth
fcrossover = control loop crossover frequency (Hz)
Below the crossover frequency, the VRM actively regulates and output impedance is low. Above crossover, the VRM behaves as an inductor, and bulk capacitors must take over. Typical crossover: 10–100 kHz for switching regulators.
EMI / EMC Formulas
Formulas for electromagnetic interference and compatibility analysis: radiation, shielding, skin effect, and spectral content of digital signals.
Radiation & Emissions
Electric Field from a Small Loop Source
E = electric field strength (V/m)
f = frequency (MHz)
A = loop area (cm2)
I = current flowing in the loop (mA)
r = distance from the source (m)
Radiated emissions from PCB current loops increase with f2. Reducing loop area is the most effective mitigation. A 1 cm2 loop carrying 10 mA at 100 MHz produces about 13 μV/m at 3 m distance.
Cable Radiation — Differential Mode
EDM = differential-mode radiated electric field (V/m)
f = frequency (Hz)
I = differential-mode current (A)
A = loop area formed by the cable pair (m2)
L = cable length (m)
r = measurement distance (m)
Differential-mode radiation is typically lower than common-mode. Twisted pair cables reduce A, and shielding reduces I on the external surface. Route cables close together to minimize loop area.
Cable Radiation — Common Mode
ECM = common-mode radiated electric field (V/m)
f = frequency (Hz)
ICM = common-mode current on the cable (A)
L = cable length acting as an antenna (m)
r = measurement distance (m)
Common-mode radiation is usually the dominant emission mechanism in cables. Just a few μA of CM current can cause compliance failures. Common-mode chokes and proper cable termination are key mitigations.
Trapezoidal Waveform Harmonic Amplitude (Fourier)
cn = amplitude of the n-th harmonic (V)
V = peak voltage of the waveform (V)
d = duty cycle (ratio of pulse width to period)
n = harmonic number (integer)
f = fundamental frequency (Hz)
tr = rise time (seconds)
The spectral envelope of a trapezoidal waveform has two breakpoints: at 1/(π×pulse width) where amplitude rolls off at −20 dB/decade, and at 1/(π×tr) where it rolls off at −40 dB/decade. Slower rise times reduce high-frequency harmonic content.
dBμV to Volts Conversion
V = voltage in volts
dBμV = voltage level in decibels referenced to 1 microvolt
Common EMI measurement unit. 0 dBμV = 1 μV, 60 dBμV = 1 mV, 120 dBμV = 1 V. FCC Class B conducted emission limits are typically 48–56 dBμV in the 150 kHz–30 MHz range.
Field Strength vs. Distance
E2 = field strength at distance r2 (V/m or dBμV/m)
E1 = known field strength at distance r1 (V/m or dBμV/m)
r1, r2 = distances from the source (m)
In the far field, E decreases as 1/r (−6 dB per doubling of distance). Converting between 3 m and 10 m measurements: E10m = E3m − 10.46 dB. CISPR standards commonly use 3 m and 10 m distances.
Shielding & Skin Effect
Shielding Effectiveness
SE = total shielding effectiveness (dB)
R = reflection loss (dB) — impedance mismatch at the shield boundary
A = absorption loss (dB) — attenuation through the shield material
B = re-reflection correction factor (dB) — usually negligible if A > 10 dB
Reflection loss dominates at low frequencies for electric fields. Absorption loss increases with frequency. A 1 mm aluminum sheet provides > 100 dB SE at 1 MHz. Seams, apertures, and penetrations degrade SE significantly.
Skin Depth
δ = skin depth (m) — depth at which current density falls to 1/e (37%)
ρ = resistivity of the conductor (Ω·m)
ω = angular frequency = 2πf (rad/s)
μ = magnetic permeability (H/m), μ = μ0μr
For copper at 1 GHz: δ ≈ 2.1 μm. At 1 MHz: δ ≈ 66 μm. Skin effect increases AC resistance of traces and planes at high frequencies, which impacts both signal loss and power dissipation. Shield thickness should be ≥ 3δ for effective shielding.
Absorption Loss
A = absorption loss (dB)
t = shield thickness (m)
δ = skin depth (m)
Each skin depth of material provides 8.686 dB of absorption loss. Three skin depths provide 26 dB. At high frequencies, absorption loss is the dominant shielding mechanism.
Antenna Resonance Frequency
f = resonant frequency (Hz)
c = speed of light (3 × 108 m/s)
length = physical length of the conductor or aperture (m)
Cables, traces, and enclosure slots become efficient radiators when their length approaches λ/2. A 15 cm cable resonates at 1 GHz. Shield apertures should be kept well below λ/20 at the highest frequency of concern.
Maximum Aperture Leakage
SEaperture = shielding effectiveness limit from an aperture (dB)
λ = wavelength at the frequency of interest (m)
lmax = maximum dimension of the aperture (m)
A single 1 cm slot limits SE to 20 dB at 1.5 GHz. Multiple apertures further degrade SE. Keep all seams and openings electrically small. For N identical apertures: SE degrades by 20×log10(√N).
PCB Design Formulas
Practical equations for PCB trace design, via modeling, thermal management, and wavelength calculations.
Trace Properties
Trace Resistance (DC)
R = DC resistance of the trace (Ω)
ρ = resistivity of copper (1.724 × 10−8 Ω·m at 20°C)
L = trace length (m)
W = trace width (m)
t = trace thickness (m), typically 1 oz = 35 μm
At high frequencies, the effective cross-section is reduced by the skin effect, increasing AC resistance. The AC resistance can be approximated as RAC ≈ RDC × t/(2δ) when t > 2δ.
Current Capacity (IPC-2221 / IPC-2152)
I = maximum current (A)
k = constant: 0.048 for outer layers, 0.024 for inner layers
ΔT = temperature rise above ambient (°C)
A = cross-sectional area of the trace (mils2)
IPC-2221 is a conservative guideline. For a 10 mil wide, 1 oz copper outer trace with 10°C rise: I ≈ 1.0 A. IPC-2152 provides more accurate charts based on actual test data. Always verify with thermal simulation for critical designs.
Skin Depth in Copper
δCu = skin depth in copper (μm)
f = frequency (MHz)
Simplified formula specific to copper at 20°C. At 100 MHz: δ ≈ 6.6 μm. At 1 GHz: δ ≈ 2.1 μm. At 10 GHz: δ ≈ 0.66 μm. Since 1 oz copper is 35 μm thick, skin effect becomes significant above roughly 4 MHz.
Wavelength in PCB Medium
λ = wavelength in the PCB dielectric (m)
c = speed of light in vacuum (3 × 108 m/s)
f = frequency (Hz)
εr = effective dielectric constant of the medium
At 5 GHz in FR-4 (εr ≈ 4.0): λ ≈ 30 mm. Traces longer than λ/10 require transmission line treatment. Stubs should be kept shorter than λ/20 to avoid resonance issues.
Via Modeling
Via Inductance
L = inductance of the via (nH)
h = via height / board thickness (mm)
d = via drill diameter (mm)
A through-hole via in a 1.6 mm board with a 0.3 mm drill has approximately 0.9 nH of inductance. Via inductance adds to decoupling capacitor ESL, reducing effectiveness. Use multiple vias in parallel for power pins to reduce inductance (Ltotal ≈ L/N for N vias).
Via Capacitance (Pad-to-Plane)
C = capacitance between the via pad and the surrounding plane (pF)
εr = relative dielectric constant
T = thickness of the plane layer (mm)
D = via pad diameter (mm)
d = via anti-pad (clearance hole) diameter (mm)
Via capacitance is typically 0.3–0.7 pF per plane layer traversed. For high-speed signals, via stubs create capacitive loading and resonances. Back-drilling or using blind/buried vias reduces the stub effect.
Via Characteristic Impedance
Zvia = characteristic impedance of the via (Ω)
εr = dielectric constant of the surrounding material
Dclearance = anti-pad (clearance hole) diameter (m)
Ddrill = via drill diameter (m)
Via impedance is typically 30–50 Ω, which is lower than most trace impedances. This creates a capacitive discontinuity. Adjusting the anti-pad diameter and using back-drill can improve via impedance matching for signals above 10 Gbps.
Via Stub Resonance Frequency
fstub = quarter-wave resonance frequency of the via stub (Hz)
c = speed of light (3 × 108 m/s)
lstub = length of the unused via barrel (stub length) (m)
εr = effective dielectric constant
A 1 mm via stub in FR-4 resonates at approximately 37.5 GHz. At this frequency, the stub acts as an open-circuit quarter-wave transformer, creating a short circuit (notch) in the signal path. Back-drilling is standard practice for 25+ Gbps links.
Supplemental Formulas
Additional equations for specialized analysis including insertion loss, thermal calculations, and frequency-domain conversions.
Loss & Attenuation
Conductor Loss (per unit length)
αc = conductor attenuation constant (Np/m)
Rs = surface resistance = √(πfμρ) (Ω/sq)
Z0 = characteristic impedance (Ω)
w = trace width (m)
Conductor loss increases with √f due to skin effect. Rough copper surfaces increase loss further by 20–60%. HVLP (Hyper Very Low Profile) copper foil reduces surface roughness effects for high-speed designs.
Dielectric Loss (per unit length)
αd = dielectric attenuation constant (Np/m)
f = frequency (Hz)
εr = relative dielectric constant
tanδ = loss tangent (dissipation factor, Df)
c = speed of light (m/s)
Dielectric loss increases linearly with frequency. At high frequencies (above approximately 5–10 GHz), dielectric loss typically dominates over conductor loss. Low Df materials (Megtron 6, Rogers) are essential for 25+ Gbps channels.
Total Channel Insertion Loss
IL = total insertion loss at frequency f (dB, negative value)
αc, αd = conductor and dielectric attenuation (dB/m or dB/inch)
L = trace length
ILvia = via transition insertion loss (dB)
ILconnector = connector insertion loss (dB)
The insertion loss budget determines whether a channel will work at the target data rate. For PCIe Gen 5 (32 GT/s), the total channel loss budget at Nyquist (16 GHz) is typically 28–36 dB depending on the link configuration.
Thermal & Reliability
Thermal Resistance (Junction to Ambient)
Tj = junction temperature (°C)
Ta = ambient temperature (°C)
Pd = power dissipation (W)
θJA = thermal resistance, junction to ambient (°C/W)
Ensure Tj stays below the IC maximum rating (typically 85–125 °C for commercial parts). For more accurate analysis, use the full thermal resistance network: θJC (junction-to-case) + θCA (case-to-ambient) with heatsink considerations.
Trace Temperature Rise
ΔT = temperature rise above ambient (°C)
I = current (A)
k = layer constant (0.048 outer, 0.024 inner)
A = cross-sectional area (mils2)
Inverse of the IPC-2221 current capacity formula. Use this to verify that power traces do not exceed the acceptable temperature rise (typically 10–20 °C for most designs, 40–45 °C maximum).
Frequency Domain
Nyquist Frequency
fNyquist = Nyquist frequency (Hz) — the fundamental frequency of an NRZ signal
DataRate = bit rate (bits/s)
For NRZ signaling, the Nyquist frequency is where the channel insertion loss is most critical. For PAM4, the Nyquist frequency is DataRate/4 (since each symbol carries 2 bits). PCIe Gen 5 at 32 GT/s has fNyquist = 16 GHz.
S-parameter to dB Conversion
S21 (dB) = insertion loss in decibels (negative for lossy channels)
|S21| = magnitude of the forward transmission coefficient (dimensionless, 0 to 1)
S11 (return loss) uses the same conversion. A lossless, perfectly matched channel has S21 = 0 dB and S11 = −∞ dB. Practical channels show S21 decreasing (more negative) with frequency.
Group Delay
τg = group delay (seconds)
φ = phase of the transfer function (radians)
ω = angular frequency (rad/s)
Constant group delay means all frequency components arrive at the same time (no dispersion). Variation in group delay across the signal bandwidth causes pulse distortion and ISI. Measured from the phase of S21.
Common Constants & Conversions
| Constant | Symbol | Value | Unit |
|---|---|---|---|
| Speed of light | c | 2.998 × 108 | m/s |
| Permittivity of free space | ε0 | 8.854 × 10−12 | F/m |
| Permeability of free space | μ0 | 4π × 10−7 | H/m |
| Copper resistivity (20°C) | ρCu | 1.724 × 10−8 | Ω·m |
| FR-4 dielectric constant | εr | 4.0–4.5 (varies with frequency) | — |
| 1 oz copper thickness | t | 35 / 1.37 | μm / mils |
| 1 inch | — | 25.4 | mm |
| 1 mil | — | 25.4 | μm |
Common dB Conversions
| dB Value | Voltage Ratio | Power Ratio | Significance |
|---|---|---|---|
| 0 dB | 1.000 | 1.000 | No change |
| 3 dB | 1.414 | 2.000 | Double power |
| 6 dB | 2.000 | 4.000 | Double voltage |
| 10 dB | 3.162 | 10.00 | Order of magnitude (power) |
| 20 dB | 10.00 | 100.0 | Order of magnitude (voltage) |
| −3 dB | 0.707 | 0.500 | Half power (bandwidth point) |
| −6 dB | 0.500 | 0.250 | Half voltage |
| −20 dB | 0.100 | 0.010 | 10% voltage (1% power) |