In the previous article, we explained the integrated role of reference planes. In this article, which is FC37, we will cover S-parameters. We will explain the concept of S-parameters, which describe circuit characteristics in the frequency domain—as opposed to the time domain (voltage and current waveforms) we have discussed so far—and how to practically utilize them in FC design.
Why is a frequency domain description necessary?
In previous articles, we have focused on time-domain concepts such as signal rise time, dV/dt, and loop area. However, the characteristics of high-frequency circuits are often easier to understand fundamentally when described in the frequency domain.
Noise measurement using a spectrum analyzer, as explained in the EMI series, and the PDN impedance measurement in FC35 were already evaluations in the frequency domain. S-parameters are the standard language of high-frequency engineering, which systematizes this frequency-domain circuit description.
Conventional circuit theory (Ohm's Law, Kirchhoff's Laws) is accurate at low frequencies, but in the high-frequency region where signal wavelengths approach the physical dimensions of the circuit (roughly circuits with dimensions 1/10 of the wavelength or greater), the very concepts of voltage and current take on different values depending on the location, causing conventional lumped-element circuit models to fail. S-parameters were developed to accurately describe circuit characteristics in this high-frequency region.
Basic concepts of S-parameters: Incident waves and reflected waves
S-parameters (Scattering Parameters) describe the relationship between incident waves to a circuit and reflected/transmitted waves from the circuit.
Consider a 2-port circuit (input port 1, output port 2). Each port has incident waves (a1, a2) and reflected waves (b1, b2).
S-parameters express the relationship between these waves using a matrix:
b1 = S11×a1 + S12×a2
b2 = S21×a1 + S22×a2
Let's organize the physical meaning of each S-parameter.
S11 (input reflection coefficient) is the ratio of the wave incident on port 1 that reflects back to port 1. If ideal matching (impedance matching) is achieved, S11 = 0 (no reflection).
S21 (forward transmission coefficient) is the signal transmission characteristic from port 1 to port 2. This corresponds to the pass characteristics of a filter or the gain of an amplifier.
S22 (output reflection coefficient) is the reflection characteristic viewed from the port 2 side, and has the same meaning as S11.
S12 (reverse transmission coefficient) is the transmission characteristic from port 2 to port 1. In many passive circuits, S12 = S21 (reciprocity) holds, but this may not hold for active components (such as amplifiers).
Relationship between S-parameters and impedance matching
S11 is directly linked to the concept of impedance matching that we have covered in previous articles.
The reflection coefficient Γ (= S11) is derived from the relationship between the characteristic impedance Z0 and the load impedance ZL as:
Γ = (ZL - Z0) / (ZL + Z0)
When ZL = Z0 (perfect matching), Γ = 0, meaning S11 = 0, and no reflection occurs at all. This is the ultimate goal of the differential impedance management in FC33 and the characteristic impedance design covered in FC39.
S11 in dB is expressed as 20×log10(|S11|). S11 = -10dB means approximately 32% reflection in amplitude (about 10% reflection in power), and S11 = -20dB means approximately 10% reflection in amplitude (about 1% in power). For common high-speed interfaces (USB, PCIe, etc.), it is often required that S11 be -10dB or less (a more negative value).
VSWR (Voltage Standing Wave Ratio) is another way of expressing information equivalent to S11, and it is frequently used in the evaluation of high-frequency circuits:
VSWR = (1+|Γ|) / (1-|Γ|)
For perfect matching (Γ=0), VSWR=1; the greater the mismatch, the higher the VSWR.
Measuring S-Parameters: VNA (Vector Network Analyzer)
S-parameters are measured using a dedicated instrument called a VNA (Vector Network Analyzer).
A VNA obtains S-parameters across the entire frequency band by injecting a known frequency-swept signal into the circuit under test (DUT: Device Under Test) and measuring the amplitude and phase of the reflected and transmitted waves.
The PDN impedance measurement mentioned in FC35 is also performed in principle by converting S11 measurements (using only one port) from a VNA into impedance Z:
Z = Z0 × (1+S11) / (1-S11)
VNA calibration is a critical process for ensuring measurement accuracy. Errors inherent to the measurement system itself (such as cable and connector characteristics) are removed through Open/Short/Load (OSL) calibration or the more precise TRL (Thru-Reflect-Line) calibration.
In practical FC design, full-scale VNA measurement is not performed daily for every FC development because it requires expensive equipment investment. However, VNA measurement is effective in situations such as verifying antenna impedance matching (related to the GNSS antenna in the previous series), evaluating connector and cable characteristics, and verifying filter circuit designs.
Graphical Representation of S-Parameters: The Smith Chart
S-parameters, particularly the reflection coefficient S11, are commonly represented using a specialized polar plot called a Smith Chart.
A Smith Chart is a mapping of the complex impedance plane onto the unit circle of the reflection coefficient Γ. The center of the chart represents perfect matching (Γ=0, Z=Z0), and the circumference represents total reflection (|Γ|=1, pure reactance).
The practical value of the Smith Chart lies in the ability to visually track the impedance trajectory as the frequency is swept. The transition discussed in the FC23 decoupling capacitor section—'capacitive at low frequencies, minimum impedance at SRF, and inductive at high frequencies'—appears as a characteristic trajectory on the Smith Chart.
In GNSS antenna impedance matching design, the antenna's impedance trajectory is observed on the Smith Chart, and a matching circuit (LC matching) is designed to bring it closer to the center of the chart (the 50Ω matching point) at the target frequency (e.g., L1 band 1575MHz). This is a more advanced, quantitative approach to the GNSS antenna design explained in the previous series.
S-Parameters and Filter Design
The characteristics of the filter circuits (LC filters, common-mode chokes) explained in the series before last can also be quantitatively evaluated using S-parameters.
Insertion Loss is expressed by the magnitude of S21 (in dB) and indicates how much the filter attenuates signals in the target frequency band. In EMI filter design, the goal is for S21 to show significant attenuation (e.g., -20dB or less) in the noise frequency band.
Loss in the passband is also evaluated using S21, where it is desirable for S21 to be as close to 0dB as possible (minimal loss) in the signal band.
Return Loss is expressed by the magnitude of S11 (in dB, often represented as a positive value by inverting the sign) and indicates how well the filter's input impedance matches the ideal 50Ω (or system impedance). If the return loss is small (poor matching), signal reflection to the preceding stage of the filter can become a problem.
By measuring and simulating these characteristics as S-parameters, the filter design qualitatively explained in the previous series can be quantitatively verified.
Utilizing S-parameters in High-Speed Digital Signals
In recent high-speed digital interface standards (USB 3.0 and above, PCIe, etc.), S-parameter-based design verification has become standard.
As an insertion loss mask, each standard specifies that S21 (insertion loss) must be below a certain value at specific frequencies. For example, the USB 3.0 standard sets an upper limit for insertion loss near the Nyquist frequency (half the signal speed).
Regarding S-parameter evaluation of crosstalk, the NEXT and FEXT discussed in FC34 can also be quantified as multi-port S-parameters (evaluation of differential pairs with 4 or more ports).
Regarding the relationship with the Eye Diagram, a standard workflow for high-speed interface design involves performing time-domain simulations (using IBIS-AMI models, etc.) from S-parameters to predict the final signal quality (eye opening).
The SPI communication between the MCU and sensors, which is central to FC design (tens of MHz), is much slower than these cutting-edge standards (several Gbps), so such strict S-parameter-based verification is usually unnecessary. However, when dealing with high-speed interfaces (USB 3.0, MIPI CSI, etc.) mounted on industrial FCs, knowledge of these evaluation methods becomes necessary.
S-Parameter Simulation: Collaboration with Electromagnetic Field Analysis
In addition to actual measurements, S-parameter simulation at the design stage is also a practical tool.
3D electromagnetic field solvers (Ansys HFSS, Keysight ADS, etc.) calculate S-parameters based on electromagnetic field equations from the actual structure of the PCB (wiring shape, vias, planes). This is a more generalized high-frequency version of the PDN analysis tool mentioned in FC35.
Regarding the S-parameters of via models (in relation to the discussion in FC32), the effects of via parasitic inductance and stubs can be quantified as S-parameters to evaluate their impact on signal quality.
For the S-parameters of differential pairs (in relation to FC33), 4-port S-parameters (differential mode notation such as SDD11, SDD21, etc.) are used to evaluate the degree of differential impedance matching and transmission characteristics.
At the scale of FC design (mainly several hundred MHz or less), the introduction of such advanced simulation tools is not necessarily required, but understanding that S-parameters are a framework that uniformly describes the previous articles (impedance, reference planes, filters) is useful for more advanced design and troubleshooting.
Relationship between S-parameters and previous articles: Integrated understanding
We will organize how the concepts covered in the PCB Design series so far are integrated within the framework of S-parameters.
Characteristic impedance (to be detailed in FC39) is a basic quantity of S-parameters as S11 (reflection coefficient). Differential impedance (FC33) is extended as SDD parameters.
Crosstalk (FC34) is quantified as NEXT/FEXT using S-parameters between multiple ports (S31, S41, etc.).
If the continuity of the reference plane (FC36) is impaired, the impedance discontinuity at that location appears as a degradation of S11.
PDN impedance (FC35) is the impedance itself converted from 1-port S-parameters (S11).
In this way, S-parameters serve as an integrated framework that describes the phenomena explained in individual articles using the common language of reflection and transmission in the frequency domain.
Summary
S-parameters are a frequency-domain method that describes the characteristics of high-frequency circuits through the relationship between incident and reflected waves (reflection coefficient S11, transmission coefficient S21, etc.), playing an essential role in the high-frequency region where conventional voltage/current-based circuit theory fails. Practical evaluation methods include actual measurements using a VNA and visualization via Smith charts, which can be directly applied to evaluating filter insertion loss and return loss, as well as GNSS antenna matching design. The characteristic impedance, differential pairs, crosstalk, reference planes, and PDN covered in previous PCB Design series are all related phenomena that can be described within the unified framework of S-parameters. In the next installment, FC38, we will explain transmission lines—the physics of how signals propagate as 'waves' and the theory of reflection and termination.