AoNArt of Nanoelectronics
My space
05 · Advanced

Microwave losses, bandwidth and practical constraints

Conductor/dielectric loss, skin effect, distributed attenuation, manufacturing tolerances, SOLT/TRL calibration and de-embedding.

260–340 minAdvanced
In this lesson
  • Derive the lossy transmission-line propagation constant and characteristic impedance.
  • Derive the weak-loss contributions of series resistance and shunt conductance.
  • Estimate skin depth and surface resistance with explicit material assumptions.
  • Relate dielectric energy participation and loss tangent to unloaded Q.
  • Compute line propagation loss while separating mismatch and insertion loss.
  • Derive frequency sensitivities to resonator length and effective permittivity.
  • Distinguish phase/group velocity, return loss and standing-wave ratio.
  • Specify VNA SOLT/TRL calibration and correct reference planes.
  • De-embed both fixtures using the correct ordered chain matrices.
  • Quantify resonance and measurement uncertainty, retaining complex Touchstone data.

01Why matching is not the same as efficiency

A passive microwave circuit can exhibit almost perfect input matching while dissipating substantial power. Input reflection tests how well incident power enters the measured network; it does not reveal whether accepted power reaches the intended load. Start with the power budget, distinguish conductor and dielectric heating, radiation and power delivered to other ports, and state whether attenuation is intrinsic to the device or the measurement fixture.

Treat network reference impedances and the VNA calibration planes explicitly. In a two-port device, transmitted, reflected and dissipated powers are related by passivity only when ports, wave normalizations and excitation are specified. For a matched-source excitation in a passive two-port, the missing fraction after reflection and measured transmission can include dissipation and radiation, not an unmeasured magical loss coefficient.

γ=(R′+jωL′)(G′+jωC′)\gamma=\sqrt{(R'+j\omega L')(G'+j\omega C')}Zc=R′+jωL′G′+jωC′Z_c=\sqrt{\frac{R'+j\omega L'}{G'+j\omega C'}}γ=α+jβ,V+(z)=V+(0)e−γz\gamma=\alpha+j\beta,\qquad V^+(z)=V^+(0)e^{-\gamma z}

02Distributed transmission-line resistance and conductance

The telegrapher equations contain series resistance and inductance per unit length and shunt conductance and capacitance per unit length. Solving their harmonic form gives complex propagation constant and characteristic impedance. A real attenuation constant describes exponential decay of traveling-wave voltage amplitude with distance, whereas phase constant describes accumulated phase. The two must not be conflated.

For weak loss relative to reactive terms, expand the complex square root to first order. The attenuation separates into approximate conductor and dielectric contributions determined by series resistance and shunt conductance. The expression is valid only for a reasonably uniform line and well-defined modal impedance; strong loss, higher-order modes and discontinuities require the full complex propagation solution.

R′dzL′dz1/(G′dz)C′dzInputOutputCOMMON RF RETURNdz
Series R′dz and L′dz in the signal trace; shunt resistor 1/(G′dz) and capacitance C′dz return to the common RF conductor. All four underlying parameters are distributed per unit length.
R′≪ωL′,G′≪ωC′R'\ll\omega L',\quad G'\ll\omega C'Z0≃L′C′,β≃ωL′C′Z_0\simeq\sqrt{\frac{L'}{C'}},\quad\beta\simeq\omega\sqrt{L'C'}α≃R′2Z0⏟αc+G′Z02⏟αd\alpha\simeq\underbrace{\frac{R'}{2Z_0}}_{\alpha_c}+\underbrace{\frac{G'Z_0}{2}}_{\alpha_d}

03Conductor skin effect and RF current crowding

A sinusoidally varying magnetic field drives currents toward the metal surface. For a nonmagnetic, sufficiently thick, smooth conductor under a good-conductor approximation, field and current density decay over the skin depth inversely proportional to the square root of frequency. The corresponding surface resistance rises roughly with the square root of frequency, rather than staying at its DC sheet-resistance value.

Conductor loss of an actual microstrip also depends on the trace width, field concentration near edges, thickness relative to skin depth, copper roughness, surface finish and the return-plane current distribution. A surface resistance expressed in ohms per square is not, by itself, the series resistance per meter: a cross-sectional field or current-distribution model is needed to perform that conversion.

Microstrip cross-section and loss channelsAIRSIGNAL COPPERSUBSTRATE · εr, tanδGROUND PLANERF surface currentsthickness h
Illustrative cross-section: fields occupy air and substrate while return current flows on the plane. Effective permittivity and conductor loss require actual geometry.
δs=2ωμσ\delta_s=\sqrt{\frac{2}{\omega\mu\sigma}}R□=1σδs=πfμσR_{\square}=\frac{1}{\sigma\delta_s}=\sqrt{\frac{\pi f\mu}{\sigma}}δs≪t ⟹ thick-conductor surface-current limit\delta_s\ll t\ \Longrightarrow\ \text{thick-conductor surface-current limit}

04Dielectric loss tangent and energy participation

An imperfect dielectric has a complex permittivity and dissipates power when its electric polarization lags the applied field. For small dielectric loss, time-averaged dissipated power is proportional to angular frequency, the electric energy in that region and loss tangent. An electromagnetic mode may store part of its electric energy in air and part in the substrate, so the substrate fraction is an essential factor.

With the standard resonant-mode energy convention, define the dielectric participation fraction as the ratio of electric mode energy in the lossy region to total electric energy. The approximate dielectric-limited quality factor is the inverse product of this fraction and the loss tangent. Adding weak conductor, dielectric and radiation channels by reciprocal Q is valid when the modal energy normalization is common.

Pd≃2ω Ue,participatingtan⁡δ(peak-phasor time-averaged electric energy)P_d\simeq2\omega\,U_{e,\mathrm{participating}}\tan\delta\quad\text{(peak-phasor time-averaged electric energy)}Qd≃1petan⁡δ,pe=Ue,participatingUe,totalQ_d\simeq\frac{1}{p_e\tan\delta},\quad p_e=\frac{U_{e,\mathrm{participating}}}{U_{e,\mathrm{total}}}1Q0=1Qc+1Qd+1Qr\frac1{Q_0}=\frac1{Q_c}+\frac1{Q_d}+\frac1{Q_r}

05Attenuation, insertion loss and the danger of mismatch

For a uniform line whose characteristic impedance is matched to both measurement reference ports, a forward traveling wave decays as the exponential of minus attenuation times length. The S21 amplitude follows this decay, and logarithmic conversion gives the familiar 8.686 decibel factor per neper. The line attenuation should be specified in nepers per meter of field amplitude, not in an unspecified decibel convention.

Using a 50-ohm line with series resistance eight ohms per meter and shunt conductance four millisiemens per meter gives approximate attenuation 0.18 nepers per meter. Over ten centimeters the pure propagation contribution is about 0.156 decibel. When a launcher or load is mismatched, measured S21 includes standing-wave interference and cannot be equated blindly to a single exponential factor.

∣S21∣=e−αℓ(matched, uniform line)|S_{21}|=e^{-\alpha \ell}\quad\text{(matched, uniform line)}ILdB=8.686 αℓ\mathrm{IL}_{\mathrm{dB}}=8.686\,\alpha\ellR′=8 Ω/m, G′=0.004 S/m, Z0=50 Ω⟹α=0.18 Np/mR'=8\,\Omega/\mathrm{m},\ G'=0.004\,\mathrm{S/m},\ Z_0=50\,\Omega\Longrightarrow\alpha=0.18\,\mathrm{Np/m}

06Phase/group velocity, dispersion and fabrication tolerance

Phase velocity determines the electrical length of a sinusoidal carrier, whereas group velocity describes transport of a narrowband wave packet in the weakly dispersive regime. Their equality is not guaranteed for dispersive microstrip or other guided modes. Group delay of a two-port transmission is the negative frequency derivative of its unwrapped transmission phase, at documented reference planes.

For an ideal quarter- or half-wave mode, frequency scales inversely with physical resonator length and approximately with the inverse square root of effective permittivity. A first-order sensitivity equation relates fractional frequency shifts to length and permittivity shifts. Separate worst-case tolerance limits from root-sum-square estimates, which additionally require justified independent uncertainty distributions.

vp=ωβ,vg=(dβdω)−1v_p=\frac{\omega}{\beta},\quad v_g=\left(\frac{d\beta}{d\omega}\right)^{-1}λg=vpf,τg=−darg⁡S21dω\lambda_g=\frac{v_p}{f},\qquad \tau_g=-\frac{d\arg S_{21}}{d\omega}Δf0f0≃−Δℓℓ−12Δεeffεeff\frac{\Delta f_0}{f_0}\simeq-\frac{\Delta\ell}{\ell}-\frac12\frac{\Delta\varepsilon_{\mathrm{eff}}}{\varepsilon_{\mathrm{eff}}}

07Return loss, VSWR and mismatch loss are different metrics

For a real reference impedance, the complex reflection coefficient describes the incident-to-reflected wave ratio at a specified reference plane. Its magnitude leads to return loss and voltage standing-wave ratio, both of which characterize matching. A small reflected amplitude does not demonstrate that a network is lossless or that any downstream stage receives the accepted power.

At a single ideal lossless interface with reflection magnitude one tenth, return loss is twenty decibels and the standing-wave ratio is about 1.222. The reflected power fraction is one percent and the corresponding mismatch loss is about 0.04365 decibel. This mismatch loss is not the dissipative insertion loss of a lossy two-port, and the two cannot be added without a defined measurement model.

Γ=Zin−ZrefZin+Zref\Gamma=\frac{Z_{\mathrm{in}}-Z_{\mathrm{ref}}}{Z_{\mathrm{in}}+Z_{\mathrm{ref}}}RL=−20log⁡10∣Γ∣,VSWR=1+∣Γ∣1−∣Γ∣\mathrm{RL}=-20\log_{10}|\Gamma|,\quad\mathrm{VSWR}=\frac{1+|\Gamma|}{1-|\Gamma|}ML=−10log⁡10(1−∣Γ∣2)(single interface, lossless)\mathrm{ML}=-10\log_{10}(1-|\Gamma|^2)\quad\text{(single interface, lossless)}

08VNA reference planes, SOLT, TRL and de-embedding

A calibrated VNA reports complex waves at its chosen electrical reference planes. SOLT uses short, open, load and thru standards appropriate to the connector model; TRL uses thru, reflect and a known-length offset line and is often useful for planar structures. For TRL, the extra line must generate enough phase separation over the frequency band to avoid singular or ill-conditioned calibration.

When left and right fixtures surround the device under test, a compatible cascade or transfer-matrix representation multiplies their matrices in propagation order. Proper two-sided de-embedding inverts each fixture in the correct order. Subtracting transmission magnitudes in decibels or adding S-parameters directly is not generally equivalent to this operation when port mismatch and multiple reflections are present.

VNA calibration planes and fixture de-embeddingLeft fixtureT leftDevice under testT DUTRight fixtureT rightVNA · Port 1VNA · Port 2P1P2Tmeas = TL · TDUT · TR
The measured chain matrix is the ordered product of left fixture, DUT and right fixture. Extract DUT by inverse fixture matrices; state calibration planes explicitly.
Tmeas=TL TDUT TR\mathbf T_{\mathrm{meas}}=\mathbf T_L\,\mathbf T_{\mathrm{DUT}}\,\mathbf T_RTDUT=TL−1TmeasTR−1\mathbf T_{\mathrm{DUT}}=\mathbf T_L^{-1}\mathbf T_{\mathrm{meas}}\mathbf T_R^{-1}ΔϕTRL=βΔℓ,Δℓ≃λg4 at calibration center\Delta\phi_{\mathrm{TRL}}=\beta\Delta\ell,\quad\Delta\ell\simeq\frac{\lambda_g}{4}\ \text{at calibration center}

09Putting the losses into an actual RF resonator

Real resonators possess conductor, dielectric, radiation and external coupling channels. The unloaded quality factor is set by internal dissipative channels; external Q values measure energy extracted by the feed ports, and their reciprocal sum determines the loaded quality. The bandwidth estimated by frequency divided by loaded Q assumes an isolated Lorentzian mode with approximately constant coupling.

For a sample calculation choose conductor-limited Q of 250, dielectric participation 0.70, loss tangent 0.004 and radiation Q of 2000. The dielectric contribution is about 357 and the combined unloaded Q about 137. Connecting two ports of external Q 100 each reduces loaded Q to about 36.63, broadening a 2.45-gigahertz resonance to about 66.89 megahertz. These values are pedagogical assumptions, not substrate specifications.

Qd=10.70(0.004)≃357.14Q_d=\frac1{0.70(0.004)}\simeq357.14Q0=(1250+1357.14+12000)−1≃136.99Q_0=\left(\frac1{250}+\frac1{357.14}+\frac1{2000}\right)^{-1}\simeq136.99QL=(Q0−1+100−1+100−1)−1≃36.63Q_L=\left(Q_0^{-1}+100^{-1}+100^{-1}\right)^{-1}\simeq36.63

10Verification workflow and an honest uncertainty budget

A reproducible microwave design records substrate thickness and dielectric properties, trace width and copper thickness, connector/launch geometry, reference planes, calibration standard data, measured temperature, input power and sweep resolution. First verify the same device with a distributed or electromagnetic model and realistic tolerances. Then measure complex S11 and S21 and compare the complete frequency dependence, not only a matching minimum.

For a physical implementation distinguish instrumental uncertainty from fabrication variability, calibration drift and model error. On a 40-millimeter resonator with a 0.10-millimeter length limit and effective permittivity 2.80 with limit 0.08, a conservative first-order worst-case bound predicts about 1.68 percent frequency deviation. RSS is smaller only under justified independence assumptions. Save raw Touchstone data for audit, and do not invent a substrate permittivity or roughness value where it is unknown.

IL=−20log⁡10∣S21∣(specified reference impedances)\mathrm{IL}=-20\log_{10}|S_{21}|\quad\text{(specified reference impedances)}Δf3dB≃f0QL(single isolated Lorentzian)\Delta f_{3\mathrm{dB}}\simeq\frac{f_0}{Q_L}\quad\text{(single isolated Lorentzian)}uRSS(f0)=f0(uℓℓ)2+(uε2εeff)2(independent linear inputs)u_{\mathrm{RSS}}(f_0)=f_0\sqrt{\left(\frac{u_\ell}{\ell}\right)^2+\left(\frac{u_\varepsilon}{2\varepsilon_{\mathrm{eff}}}\right)^2}\quad\text{(independent linear inputs)}

Ten rigorous problems and corrected solutions

  1. Starting from the telegrapher equations, show how the weak-loss attenuation contains separate R′ and G′ terms.
  2. For R′ = 8 Ω/m, G′ = 0.004 S/m and Z0 = 50 Ω, find the conductor and dielectric attenuation contributions.
  3. Estimate copper skin depth and sheet surface resistance at 2.45 GHz, assuming nonmagnetic bulk copper with conductivity 5.8×10^7 S/m.
  4. For dielectric participation 0.70 and loss tangent 0.004, calculate dielectric Q and combine it with conductor Q 250 and radiation Q 2000.
  5. A perfectly matched 10 cm line has attenuation 0.18 Np/m. Find amplitude S21 and insertion loss without adding a fictitious reflection.
  6. Given a 40 mm resonator, ±0.10 mm length limit, εeff = 2.80 ± 0.08 and f0 = 2.45 GHz, estimate conservative frequency shift.
  7. At a lossless interface with reflection magnitude 0.10, distinguish return loss, VSWR, reflected power and mismatch loss.
  8. Design a quarter-guided-wavelength TRL line offset at 2.45 GHz for phase velocity 2×10^8 m/s and discuss calibration bandwidth.
  9. Combine Qc=250, dielectric participation 0.70, tanδ=0.004, Qr=2000 and two external Q values of 100 to predict loaded width.
  10. Explain why two-port de-embedding needs matrix inversion of fixtures rather than subtraction of scalar S21 data in decibels.
Show all worked solutions

1. Starting from the telegrapher equations, show how the weak-loss attenuation contains separate R′ and G′ terms.

Use the harmonic telegrapher solutions for a uniform line: their complex propagation constant is the square root of a product containing a series impedance per unit length and shunt admittance per unit length. Factor out their reactive terms, then expand the remaining square root to first order in the small resistive and conductive perturbations.

The real first-order coefficient is half the ratio of series resistance to characteristic impedance plus half the shunt conductance times that impedance. Both terms carry units of inverse length. This is a weak-loss approximation; the exact complex square root must replace it if either R′ or G′ is not small relative to its reactive counterpart.

γ2=(R′+jωL′)(G′+jωC′)\gamma^2=(R'+j\omega L')(G'+j\omega C')α≃R′2Z0+G′Z02\alpha\simeq\frac{R'}{2Z_0}+\frac{G'Z_0}{2}

2. For R′ = 8 Ω/m, G′ = 0.004 S/m and Z0 = 50 Ω, find the conductor and dielectric attenuation contributions.

The conductor contribution equals eight divided by twice fifty, yielding 0.08 nepers per meter. The shunt dielectric contribution equals four millisiemens per meter times fifty divided by two, yielding 0.10 nepers per meter. Both contributions are positive and add linearly at first order.

Total amplitude attenuation is therefore 0.18 nepers per meter. Do not confuse these values with eight ohms per meter of DC resistance or with voltage transmission in decibels. Conversion to decibels also requires specifying the physical length over which the attenuation is accumulated.

αc=82(50)=0.08 Np/m\alpha_c=\frac{8}{2(50)}=0.08\,\mathrm{Np/m}αd=0.004(50)2=0.10 Np/m\alpha_d=\frac{0.004(50)}{2}=0.10\,\mathrm{Np/m}

3. Estimate copper skin depth and sheet surface resistance at 2.45 GHz, assuming nonmagnetic bulk copper with conductivity 5.8×10^7 S/m.

At the stated frequency, insert angular frequency, the vacuum permeability and bulk conductivity into the good-conductor skin-depth formula. The result is approximately 1.335 micrometers. This is a magnetic diffusion depth, not the full microscopic electron mean free path or the actual patterned copper thickness.

The surface resistance is the reciprocal of conductivity times skin depth, approximately 12.91 milliohms per square. This approximation presumes an adequately thick smooth conductor and negligible magnetic dispersion. It does not directly give the series resistance per meter of a microstrip, which depends on trace and return-current geometry.

δs=22πfμ0σ\delta_s=\sqrt{\frac{2}{2\pi f\mu_0\sigma}}R□=1σδsR_{\square}=\frac{1}{\sigma\delta_s}

4. For dielectric participation 0.70 and loss tangent 0.004, calculate dielectric Q and combine it with conductor Q 250 and radiation Q 2000.

Multiply the dielectric participation factor by the loss tangent to obtain the effective dielectric inverse quality, 0.0028. Its reciprocal is approximately 357.14. Physically the energy participation cannot be omitted because some electric energy may reside in air rather than the lossy substrate.

The conductor, dielectric and radiation inverse-Q values add: 0.004 plus 0.0028 plus 0.0005, equaling 0.0073. Inverting gives an intrinsic combined quality of approximately 136.986. The external coupling quality is not included here, because this is the unloaded internal-loss result.

Qd=10.70(0.004)=357.14Q_d=\frac{1}{0.70(0.004)}=357.14Q0=(250−1+357.14−1+2000−1)−1≃136.99Q_0=(250^{-1}+357.14^{-1}+2000^{-1})^{-1}\simeq136.99

5. A perfectly matched 10 cm line has attenuation 0.18 Np/m. Find amplitude S21 and insertion loss without adding a fictitious reflection.

Multiply attenuation coefficient by physical length: 0.18 per meter times 0.10 meters equals 0.018 nepers. Under matched measurement reference planes the traveling-wave transmission amplitude is exp(-0.018), approximately 0.98216, so the transmitted power ratio is the square of that amplitude.

The insertion loss magnitude is minus twenty times the base-ten logarithm of the amplitude, or 0.15635 dB. This is propagation-only loss in the explicitly matched uniform-line model. Adding a generic mismatch penalty would double count or invent an effect not present in the assumptions.

αℓ=0.18(0.10)=0.018\alpha\ell=0.18(0.10)=0.018∣S21∣=e−0.018,IL=8.686(0.018) dB|S_{21}|=e^{-0.018},\quad\mathrm{IL}=8.686(0.018)\,\mathrm{dB}

6. Given a 40 mm resonator, ±0.10 mm length limit, εeff = 2.80 ± 0.08 and f0 = 2.45 GHz, estimate conservative frequency shift.

Begin with resonance scaling inversely with length and inversely with the square root of effective permittivity. Differentiate logarithmically at the nominal values and take the absolute magnitudes of the two independent sensitivity coefficients. Their worst-case contributions are 0.10/40 and 0.08/(2×2.80).

Adding these upper bounds gives 0.0167857, or about 1.679 percent; multiplying by 2.45 GHz gives approximately 41.125 MHz. This is a conservative first-order tolerance bound, not a statistical standard uncertainty. Combining quadratically would require justified independent random inputs and would yield a different result.

Δf0f0≃−Δℓℓ−12Δεeffεeff\frac{\Delta f_0}{f_0}\simeq-\frac{\Delta\ell}{\ell}-\frac12\frac{\Delta\varepsilon_{\mathrm{eff}}}{\varepsilon_{\mathrm{eff}}}∣Δf0∣max⁡≃2.45 GHz(0.1040+0.082(2.80))|\Delta f_0|_{\max}\simeq2.45\,\mathrm{GHz}\left(\frac{0.10}{40}+\frac{0.08}{2(2.80)}\right)

7. At a lossless interface with reflection magnitude 0.10, distinguish return loss, VSWR, reflected power and mismatch loss.

For a reflection magnitude one tenth, the amplitude-based return loss is minus twenty times the logarithm of 0.1, which is 20 dB. VSWR is the ratio of 1.1 to 0.9, approximately 1.2222. Reflected power is the squared amplitude, one percent of incident power.

An ideal lossless single interface therefore accepts 99 percent of incident power, and its mismatch loss is minus ten times the logarithm of 0.99, or approximately 0.04365 dB. This is not thermal dissipation. A lossy two-port can show the same return loss but a radically different transmitted power.

∣Γ∣=0.10⟹RL=20 dB|\Gamma|=0.10\Longrightarrow\mathrm{RL}=20\,\mathrm{dB}VSWR=1.10.9≃1.2222,ML=−10log⁡10(0.99)\mathrm{VSWR}=\frac{1.1}{0.9}\simeq1.2222,\quad\mathrm{ML}=-10\log_{10}(0.99)

8. Design a quarter-guided-wavelength TRL line offset at 2.45 GHz for phase velocity 2×10^8 m/s and discuss calibration bandwidth.

For a uniform guided line with constant phase velocity, one quarter of the guided wavelength at 2.45 GHz equals velocity divided by four times frequency. Substituting two hundred million meters per second gives an ideal offset-line electrical length of 20.408 millimeters relative to the thru.

The corresponding phase separation is 90 degrees at calibration center. At other frequencies it scales with frequency only under the constant-phase-velocity assumption. A real TRL kit needs stable reflect and thru standards and must avoid phase separations approaching integer half-turns or zero over its calibration band.

Δℓ=2.00×1084(2.45×109)=20.408 mm\Delta\ell=\frac{2.00\times10^8}{4(2.45\times10^9)}=20.408\,\mathrm{mm}Δϕ(f)=2πfΔℓvp\Delta\phi(f)=2\pi f\frac{\Delta\ell}{v_p}

9. Combine Qc=250, dielectric participation 0.70, tanδ=0.004, Qr=2000 and two external Q values of 100 to predict loaded width.

The dielectric-limited quality is 357.14 and the unloaded quality from conductor, dielectric and radiation powers is approximately 136.986. To obtain loaded quality, add the inverse quality for each external port, so the total inverse quality is 0.0073 plus 0.01 plus 0.01, equal to 0.0273.

The loaded quality factor is approximately 36.6300. For an isolated Lorentzian centered at 2.45 GHz, its full half-power width is about 66.885 MHz. This estimate presumes comparable energy normalization and a dominant isolated mode; broad coupled filters cannot always be described by one Lorentzian.

QL=(136.986−1+100−1+100−1)−1Q_L=\left(136.986^{-1}+100^{-1}+100^{-1}\right)^{-1}Δf3dB≃2.45 GHzQL\Delta f_{3\mathrm{dB}}\simeq\frac{2.45\,\mathrm{GHz}}{Q_L}

10. Explain why two-port de-embedding needs matrix inversion of fixtures rather than subtraction of scalar S21 data in decibels.

When a left fixture, device and right fixture are cascaded, their chain matrices multiply in physical order. As a result the measured matrix contains mismatch-induced forward and backward wave interactions between fixtures and device, not merely the sums of three attenuation values.

If the individual fixtures are independently characterized and the matrices are invertible, recover the device matrix by left-multiplying by the inverse left fixture and right-multiplying by the inverse right fixture. This operation requires consistent reference impedances and non-singular conversions. Scalar subtraction of S21 decibels is valid only under restrictive special matching assumptions.

TDUT=TL−1TmeasTR−1\mathbf T_{\mathrm{DUT}}=\mathbf T_L^{-1}\mathbf T_{\mathrm{meas}}\mathbf T_R^{-1}S21,DUT≠S21,meas−S21,L−S21,R(in general)S_{21,\mathrm{DUT}}\ne S_{21,\mathrm{meas}}-S_{21,L}-S_{21,R}\quad\text{(in general)}

RF-06 ends with a verified end-to-end loss-aware design and measurement workflow.

Volume IITable of contents