Electronics Guide

Mode Definition and Excitation

The differential mode represents the intended signal in most high-speed differential systems. In this mode, port 1 and port 2 of a differential pair are driven with equal amplitude signals that are 180 degrees out of phase. The differential voltage is the difference between these two signals, and ideally, a perfect differential structure responds only to this difference.

The common mode represents noise or interference that appears equally on both conductors of the pair. Common-mode excitation drives both ports with identical amplitude and phase. While undesired in most applications, common-mode behavior must be characterized because asymmetries, imbalances, and electromagnetic coupling inevitably convert some differential energy into common mode and vice versa.

Mathematical mode decomposition uses the concept that any arbitrary excitation of a port pair can be expressed as the sum of a differential-mode component and a common-mode component. The transformation equations relate the single-ended voltages at ports to their mixed-mode equivalents through scaling factors of 1/√2 to preserve power normalization.

Mixed-Mode S-Parameter Matrix Structure

For a two-port differential system (four single-ended ports total), the complete mixed-mode characterization yields four distinct 2×2 matrices. The Sdd matrix describes differential-mode transmission and reflection—how differential signals propagate through the structure. Sdd11 represents differential-mode return loss at port 1, while Sdd21 represents differential-mode insertion loss from port 1 to port 2.

The Scc matrix characterizes purely common-mode behavior. Scc11 indicates how much common-mode energy reflects at port 1, and Scc21 shows common-mode transmission. Good differential designs typically exhibit poor common-mode transmission (high Scc21 loss), as this indicates effective common-mode rejection.

The Sdc matrix quantifies differential-to-common mode conversion. Sdc21, for instance, measures the common-mode signal appearing at port 2 when port 1 is excited with a differential-mode signal. This parameter directly relates to electromagnetic interference generation, as mode conversion produces unbalanced currents that radiate efficiently.

The Scd matrix captures common-to-differential mode conversion. While often reciprocal to Sdc in passive linear networks, Scd parameters are measured independently and reveal how common-mode noise (such as electromagnetic interference picked up by a cable) converts into differential-mode errors that corrupt the intended signal.

Sources of Mode Conversion

Geometric asymmetry represents one of the most common mode conversion mechanisms. When the two traces in a differential pair have different lengths, widths, or spacing to reference planes, the structure cannot maintain perfect differential balance. A purely differential excitation encounters slightly different impedances and propagation velocities on the two conductors, generating a common-mode component.

Via transitions frequently introduce mode conversion in printed circuit board designs. If the two vias in a differential pair have different geometries, different return path configurations, or are placed asymmetrically relative to nearby structures, they convert differential energy to common mode. Stub lengths, antipad sizes, and ground via placement all affect this conversion.

Dielectric inhomogeneity can cause mode conversion even in geometrically symmetric structures. If the two conductors in a differential pair experience different effective dielectric constants—due to different depths in a multilayer stackup or proximity to different materials—their propagation velocities differ, creating time skew that manifests as mode conversion.

Coupling to nearby structures, whether other differential pairs, single-ended signals, or mechanical components, typically affects the two conductors of a pair differently. This asymmetric coupling injects common-mode content and converts differential-mode energy. The quantification of such coupling through Sdc and Scd parameters guides layout optimization and shielding decisions.

Interpreting Mode Conversion Magnitude

The magnitude of Sdc and Scd parameters indicates mode conversion efficiency. Values below -40 dB typically represent acceptable mode conversion for most high-speed digital applications, meaning that less than 1% of the power converts between modes. More stringent applications, particularly those with tight electromagnetic interference budgets, may require mode conversion below -60 dB.

Frequency-dependent mode conversion patterns reveal physical mechanisms. Resonant peaks in Sdc21, for example, often indicate specific structural features causing enhanced conversion at particular frequencies. These might correspond to stub resonances, cavity modes, or lengths that create constructive interference of imbalance effects.

The phase of mode conversion parameters provides additional insight. In-phase mode conversion at both ends of a differential structure can partially cancel, while out-of-phase conversion accumulates. Time-domain analysis of mixed-mode S-parameters reveals whether mode conversion produces discrete impulses (localized discontinuities) or distributed effects (gradual imbalance accumulation).

Four-Port Measurement Approach

The most direct method for obtaining mixed-mode S-parameters involves measuring all 16 single-ended S-parameters of a four-port network (representing both ports of a differential pair at each end) and then mathematically transforming the results into mixed-mode format. Modern vector network analyzers with four test ports can perform this measurement in a single connection, reducing setup time and improving accuracy by maintaining consistent fixturing.

The measurement proceeds by exciting each single-ended port while terminating the others in the characteristic impedance and measuring the response at all four ports. This yields the complete 4×4 single-ended S-parameter matrix. Software then applies the mixed-mode transformation to compute Sdd, Scc, Sdc, and Scd matrices.

Calibration for mixed-mode measurements typically employs standard single-ended calibration techniques such as SOLT (Short-Open-Load-Thru) or TRL (Thru-Reflect-Line) applied to each of the four ports individually. The mixed-mode transformation propagates through the calibration mathematically, ensuring that mode conversion artifacts from the test system itself are removed.

Two-Port Measurement with Mode Excitation

Alternatively, some vector network analyzers support true differential-mode and common-mode excitation using internal or external baluns. This approach directly stimulates the device under test with the desired mode, measuring Sdd and Scc parameters without requiring mathematical transformation. However, measuring the cross-mode parameters Sdc and Scd still requires either mathematical transformation or switching between mode excitation types.

This method offers intuitive interpretation, as the measured results directly represent differential-mode or common-mode behavior without post-processing. However, the baluns themselves must exhibit very low mode conversion and must be carefully characterized and de-embedded to avoid introducing measurement errors.

Measurement Accuracy Considerations

Mode conversion measurements are particularly sensitive to fixture asymmetry and ground loop effects. Even small imbalances in the test cables, probe ground connections, or fixture launch transitions can introduce artificial mode conversion that appears as device characteristics. Careful attention to symmetric fixturing is essential.

Dynamic range limitations affect mode conversion measurements more severely than other S-parameter measurements. Because mode conversion is typically small (often -40 dB or lower), noise floor and cross-talk in the measurement system can obscure the actual device performance. Using longer averaging, optimizing VNA power levels, and ensuring high-quality connections all improve measurement reliability.

Port impedance definitions in mixed-mode measurements must be carefully considered. While single-ended ports typically reference 50 Ω, differential-mode impedance might be 100 Ω and common-mode impedance might be 25 Ω. The mixed-mode transformation accounts for these different impedance environments, but proper test system matching ensures accurate measurements.

Common-Mode Rejection Ratio

The common-mode rejection ratio (CMRR) compares the differential-mode transmission to common-mode transmission through a structure. Mathematically, CMRR equals 20·log₁₀(|Sdd21|/|Scc21|), expressing how many decibels more attenuation common-mode signals experience compared to differential-mode signals. High CMRR values indicate superior common-mode suppression.

Ideal differential structures with perfect symmetry exhibit infinite CMRR—common-mode signals completely cancel at differential receivers. Real structures typically achieve CMRR between 20 and 60 dB, depending on frequency, layout quality, and component tolerances. CMRR generally degrades at higher frequencies as wavelength-scale asymmetries become more significant.

Common-mode rejection protects against both external interference and power supply noise. External electromagnetic fields couple common-mode currents onto differential pairs, but high CMRR ensures these currents produce minimal differential-mode interference. Similarly, simultaneous switching noise on power rails injects common-mode content that well-designed differential circuits reject.

Design Techniques for Enhanced Rejection

Tight coupling between differential pair conductors improves common-mode rejection. When traces are closely spaced, common-mode currents flow in the same direction on both conductors, producing fields that tend to cancel. Differential-mode currents flow in opposite directions, producing fields that reinforce. This asymmetric field interaction naturally favors differential-mode propagation.

Symmetric routing maintains common-mode rejection across the entire signal path. Matched trace lengths, symmetric via structures, and identical coupling to nearby features ensure that any common-mode content experiences identical propagation conditions on both conductors, preventing conversion to differential mode. Layout tools often provide differential pair routing features that automate this symmetry.

Common-mode filtering can be intentionally introduced to improve rejection beyond what passive routing symmetry provides. Common-mode chokes, which present high impedance to common-mode currents but low impedance to differential-mode currents, effectively block common-mode propagation. Mixed-mode S-parameters of such filters show very low Scc21 (strong common-mode attenuation) while maintaining high Sdd21 (minimal differential-mode loss).

Differential Impedance and Return Loss

Differential impedance, typically 100 Ω for many high-speed serial standards, determines how differential-mode energy reflects at impedance discontinuities. The Sdd11 parameter represents differential return loss—energy reflected back toward the source when a differential signal encounters an impedance mismatch.

Unlike single-ended impedance, which references the signal conductor to ground, differential impedance describes the relationship between the two conductors of the pair. A structure can simultaneously present 50 Ω single-ended impedance and 100 Ω differential impedance if properly designed. The differential impedance determines differential-mode reflections even when single-ended impedances are well-controlled.

Common differential impedance discontinuities include connector transitions, via field transitions, and changes in trace geometry. Each discontinuity creates reflections quantified by Sdd11, and multiple discontinuities can create resonances that degrade signal integrity. Time-domain reflectometry analysis of Sdd11 localizes these impedance variations along the transmission path.

Differential Insertion Loss and Dispersion

The Sdd21 parameter characterizes differential insertion loss—how much differential signal energy successfully transmits from input to output. This loss includes both resistive dissipation and reflection losses. Frequency-dependent Sdd21 reveals dispersion, showing how different frequency components experience different attenuation.

Differential structures typically exhibit different loss characteristics than single-ended structures of similar geometry because the field distribution differs. Tightly coupled differential pairs concentrate more field energy in the dielectric between the traces and less in the substrate, potentially reducing dielectric losses while maintaining or increasing conductor losses due to proximity effects.

Group delay derived from the phase of Sdd21 indicates signal propagation time as a function of frequency. Differential-mode group delay variation across a signal's bandwidth causes pulse spreading and inter-symbol interference. Applications requiring precise timing, such as high-speed serial links exceeding several gigabits per second, are particularly sensitive to group delay flatness.

Amplitude and Phase Imbalance

Amplitude imbalance quantifies the difference in magnitude between the two single-ended signals comprising a differential pair. Perfect differential signaling requires equal amplitudes; any amplitude difference represents a common-mode component. Amplitude imbalance is often expressed as a percentage or in decibels, calculated from the ratio of the larger to smaller amplitude.

Phase imbalance measures deviation from the ideal 180-degree phase relationship between differential pair conductors. Ideally, when one conductor is at +V, the other is at -V. Any phase error reduces the effective differential amplitude while creating common-mode content. Phase imbalance is typically specified in degrees, with values below 5 degrees considered acceptable for most applications.

These imbalance metrics can be derived from mixed-mode S-parameters by examining the single-ended parameters before transformation. The difference in magnitude and phase between S21 and S43 (the two single-ended transmission paths in a four-port differential measurement) directly indicates amplitude and phase imbalance.

Longitudinal Conversion Loss

Longitudinal conversion loss (LCL), commonly used in telecommunications, measures the effectiveness with which a balanced structure converts between metallic (differential) mode and longitudinal (common) mode. LCL is essentially the inverse of mode conversion efficiency, with higher LCL values indicating better balance and less mode conversion.

LCL in decibels equals 20·log₁₀(|Sdd21|/|Scd21|), comparing differential-mode transmission to common-to-differential mode conversion. This metric is particularly relevant for understanding how external electromagnetic interference (which couples as common mode) affects the differential signal. LCL values above 40 dB indicate good balance; values above 60 dB indicate excellent balance.

Regulatory standards for telecommunications equipment often specify minimum LCL values to ensure that balanced transmission systems do not pick up or radiate excessive interference. Mixed-mode S-parameter measurements provide the data needed to verify LCL compliance across the operating frequency range.

Radiated Emissions from Mode Conversion

Differential-mode currents, flowing in opposite directions on closely spaced conductors, produce fields that largely cancel in the far field. Common-mode currents, flowing in the same direction, produce fields that reinforce, creating efficient antenna structures. Mode conversion therefore directly determines radiated emission levels.

The Sdc21 parameter quantifies how much differential-mode excitation converts to common mode, providing the source term for electromagnetic interference calculations. A cable or trace pair with Sdc21 of -40 dB converts 1% of differential power to common mode, and this common-mode current drives radiation. Combining Sdc21 with cable length, frequency, and common-mode impedance enables prediction of radiated field strength.

Frequency-domain electromagnetic interference predictions use mixed-mode S-parameters to identify problematic frequencies where mode conversion peaks coincide with high differential signal content. Time-domain analysis can predict radiated emissions from specific data patterns by computing the differential signal spectrum, applying the Sdc transfer function, and modeling the resulting common-mode current distribution as a radiating antenna.

Susceptibility to External Interference

External electromagnetic fields couple into differential structures as common-mode currents. The Scd21 parameter determines how much of this common-mode coupling converts to differential-mode interference that corrupts the intended signal. Low Scd21 magnitude indicates good immunity to external interference.

Electromagnetic compatibility testing often involves subjecting equipment to radiated or conducted electromagnetic interference while monitoring system functionality. Mixed-mode S-parameters of the victim circuits, combined with knowledge of the interfering field strength, enable prediction of induced differential-mode voltage and assessment of whether the system will maintain correct operation.

Common-mode rejection ratio and longitudinal conversion loss both relate to susceptibility. High CMRR indicates that even if external fields couple common-mode currents onto a differential pair, the receiving circuitry will reject most of this interference. High LCL indicates that common-mode currents convert inefficiently to differential mode, preventing coupled interference from reaching the receiver.

Ideal Balun Properties

An ideal balun perfectly transforms a single-ended input signal into a differential output with zero common-mode content, or vice versa. In mixed-mode S-parameter terms, a balun should exhibit excellent Sdd21 (low differential-mode insertion loss), very poor Scc21 (high common-mode rejection), and minimal Sdc21 and Scd21 (low mode conversion in both directions).

The impedance transformation ratio of a balun relates single-ended impedance (often 50 Ω) to differential impedance (often 100 Ω) and common-mode impedance. Proper impedance transformation ensures that the balun presents matched impedances to both the single-ended and differential domains, minimizing reflections. Mixed-mode S-parameters reveal impedance matching through Sdd11, Scc11, and their single-ended counterparts.

Phase balance is critical in balun design. The two differential outputs must maintain precise 180-degree phase relationship across the operating bandwidth. Phase imbalance in a balun directly creates common-mode content and mode conversion. The single-ended S-parameters before mixed-mode transformation reveal phase tracking between the two differential ports.

Real-World Balun Performance

Practical baluns exhibit frequency-dependent performance. Transformer-based baluns typically operate from a lower cutoff frequency (where inductive reactance becomes significant) to an upper cutoff (where parasitic capacitance and leakage inductance limit bandwidth). Mixed-mode S-parameters across the full frequency range reveal these limitations and identify the usable bandwidth.

Active baluns, constructed from differential amplifiers or specialized integrated circuits, can achieve wider bandwidths than passive designs, but introduce noise, linearity limits, and power consumption. Mixed-mode S-parameters characterize their small-signal linear behavior, while additional measurements (noise figure, harmonic distortion, power consumption) complete the performance picture.

Hybrid baluns and marchand baluns use transmission line techniques to achieve broadband transformation. Their mixed-mode S-parameters reveal the degree to which physical layout symmetry translates to electrical balance. Even small geometric asymmetries in planar balun structures can create mode conversion that degrades performance.

Balun Selection and Application

Selecting an appropriate balun requires matching its mixed-mode S-parameter characteristics to application requirements. High-speed serial communication might prioritize flat Sdd21 across multiple gigahertz of bandwidth and low group delay variation, while radio frequency applications might emphasize mode conversion suppression to maximize electromagnetic interference rejection.

De-embedding balun effects from measurements requires careful characterization. When using baluns to enable differential measurements with single-ended equipment, or vice versa, the balun's own S-parameters must be measured and mathematically removed to obtain the true device-under-test characteristics. This de-embedding process relies on cascading S-parameter matrices and requires stable, repeatable balun performance.

Transformation Matrix Formulation

For a four-port network representing a two-port differential system, the 4×4 single-ended S-parameter matrix relates incident waves a₁, a₂, a₃, a₄ to reflected waves b₁, b₂, b₃, b₄. The transformation to mixed-mode representation involves defining new incident and reflected wave variables representing differential and common modes.

The mixed-mode wave variables are defined as: differential-mode incident wave ad₁ = (a₁ - a₂)/√2, common-mode incident wave ac₁ = (a₁ + a₂)/√2, and similarly for the reflected waves and the second port pair. The factor 1/√2 ensures power normalization—the total power in the mixed-mode representation equals the total power in the single-ended representation.

The transformation matrix M converts the single-ended S-parameter matrix Sse to mixed-mode S-parameter matrix Smm through the relationship Smm = M · Sse · M⁻¹, where M contains the linear combinations that define differential and common modes. This transformation is invertible, allowing conversion in either direction, and preserves network properties like reciprocity and passivity.

Reciprocity and Symmetry

Reciprocal networks (those constructed from passive, linear, bilateral components) satisfy Sij = Sji in single-ended representation. This reciprocity extends to mixed-mode parameters: Sdd12 = Sdd21, Scc12 = Scc21, and Sdc21 = Scd12 in reciprocal networks. This symmetry reduces the number of independent parameters that must be measured.

Geometric symmetry in differential structures creates additional relationships among mixed-mode S-parameters. If the two ports of a differential pair are perfectly symmetric, then Sdd11 = Sdd22, Scc11 = Scc22, and mode conversion parameters from port 1 to port 2 equal those from port 2 to port 1. Deviations from these relationships quantify asymmetry.

Port ordering and polarity definitions affect the transformation results. Consistent definition of which conductor is the positive and which is the negative terminal of each differential pair is essential. Reversing polarity changes the sign of differential-mode parameters but not common-mode parameters, potentially causing confusion if not carefully documented.

Practical Computation Considerations

Software tools for vector network analyzers and signal integrity simulation typically include built-in mixed-mode transformation capabilities. These tools handle the matrix mathematics automatically and often provide visualization of differential return loss, differential insertion loss, and mode conversion on logarithmic magnitude and phase plots familiar to RF engineers.

Numerical precision matters in mixed-mode transformations, particularly for mode conversion parameters. Since mode conversion is typically small compared to the main diagonal terms, subtraction of nearly equal single-ended S-parameters can lead to loss of precision. Using high-resolution measurement data and maintaining computational precision through the transformation ensures accurate results.

Frequency interpolation and extrapolation of mixed-mode S-parameters requires care. While single-ended S-parameters can often be interpolated between measured frequency points using spline or rational function fitting, mixed-mode parameters derived from interpolated single-ended data may exhibit artifacts. Performing interpolation before or after transformation can yield different results, with transformation of interpolated single-ended data generally providing smoother results.

High-Speed Serial Link Design

Modern serial communication standards such as PCIe Gen4/Gen5, USB 3.x/4.x, and Ethernet 10GBASE-T specify differential signaling with strict requirements on mode conversion. These standards often mandate maximum Sdd11 (differential return loss), minimum Sdd21 (differential insertion loss), and maximum Sdc21 and Scd21 (mode conversion) to ensure interoperability and electromagnetic compliance.

Channel simulation for compliance testing uses measured or simulated mixed-mode S-parameters of the complete channel—connectors, cables, printed circuit board traces, and vias—to predict eye diagram opening, jitter accumulation, and bit error rate. Statistical analysis of multiple channel samples, each with slightly different mixed-mode S-parameters due to manufacturing variation, enables yield prediction and margin analysis.

Equalization techniques (pre-emphasis, de-emphasis, decision feedback equalization) modify the effective channel frequency response. Mixed-mode S-parameters provide the channel characterization that equalization algorithms optimize against, enabling calculation of optimal tap coefficients for finite impulse response filters that compensate for frequency-dependent loss and dispersion.

Radio Frequency and Microwave Applications

Differential low-noise amplifiers, mixers, and other radio frequency components benefit from mixed-mode characterization. The Sdd parameters describe signal path performance, while Scc and mode conversion parameters reveal how well the circuit rejects local oscillator feedthrough, power supply noise, and substrate coupling.

Balanced antennas and transmission lines in radio systems require matched differential impedance and minimal mode conversion to avoid pattern distortion and polarization errors. Mixed-mode S-parameters of antenna feeds, baluns, and transmission line transitions guide the design of feeding networks that maintain pattern symmetry and minimize cross-polarization.

Differential filters for radio frequency applications exhibit different passband and stopband characteristics for differential and common modes. Intentional design of asymmetric common-mode response (high Scc21 in the passband, low Scc21 in stopband) provides additional filtering of power supply noise and electromagnetic interference without compromising differential signal transmission.

Design Optimization Strategies

Iterative design optimization uses mixed-mode S-parameter simulations to evaluate trade-offs. Tightening differential pair spacing improves coupling and common-mode rejection (better CMRR) but may increase crosstalk to adjacent traces. Widening traces reduces conductor loss (improves Sdd21) but may cause impedance mismatches. Mixed-mode S-parameter simulation enables quantitative evaluation of these competing considerations.

Sensitivity analysis identifies which physical parameters most strongly affect mixed-mode performance. Varying trace width, spacing, dielectric height, and via geometry in parametric sweeps reveals which dimensions require tight manufacturing control. Parameters that minimally affect mixed-mode S-parameters can have looser tolerances, reducing manufacturing cost.

Multi-objective optimization algorithms can automatically tune physical layout parameters to meet targets for Sdd11, Sdd21, Sdc21, and other mixed-mode parameters simultaneously. These algorithms explore the design space more thoroughly than manual iteration, potentially identifying non-obvious solutions that satisfy all constraints.