Single-Stub Match for 60 − j80 Ω

A load of $60 - j80\,\Omega$ on a $50\,\Omega$ line is matched by a single shunt stub. The stub sits $0.110\,\lambda$ from the load, or $0.259\,\lambda$ if you take the other intersection. At the nearer position the shorted stub is $0.095\,\lambda$ and the open stub is $0.345\,\lambda$, a quarter-wave longer, as an open and a short must be. On $1.6\,\mathrm{mm}$ FR-4 at $2.4\,\mathrm{GHz}$ the $50\,\Omega$ microstrip has $\varepsilon_{\mathrm{eff}} = 3.377$ and is $117.42\,\mathrm{mil}$ wide ($2.982\,\mathrm{mm}$). Those electrical lengths become $7.51\,\mathrm{mm}$ of through line, a $6.46\,\mathrm{mm}$ short, and a $23.45\,\mathrm{mm}$ open.

The wavelengths are Pozar’s example, computed here rather than copied from a printed page. The millimetres are that example on this library’s FR-4, at one frequency.

How does one stub match 60 − j80 Ω?

A length of line rotates the load on a circle of constant $|\Gamma|$. You stop the rotation where the admittance has the right real part, $1/50\,\mathrm{S}$, and some leftover susceptance. A shunt stub is a pure susceptance. Its length is chosen to cancel that leftover. The sum lands on the center of the chart.

The arc on the chart is the through line, $0.110\,\lambda$, $39.8^\circ$. The load is $60 - j80\,\Omega$, in the lower half. The stub then supplies the susceptance that the rotation did not. Because the stub is in shunt, the rotation is aimed at a conductance of $0.02\,\mathrm{S}$, not at a resistance of $50\,\Omega$. Series stubs exist too. They rotate toward a resistance of $50\,\Omega$ and cancel a leftover reactance. For a microstrip you can build without a series gap, the shunt is the one on the board.

The match is exact at $2.4\,\mathrm{GHz}$ for a lossless line. It is not broadband. A stub tuner is a narrow device on purpose. If you wanted the $38\%$ band of an L-section, this is the wrong tool. That comparison is Pi vs T vs L.

Why are there four shunt solutions?

The constant-$|\Gamma|$ circle crosses the match conductance twice inside one half-wavelength. Each crossing has an open stub and a shorted stub. Four shunt answers, and four more if you allow a series stub. All eight are perfect at the design frequency. They are not the same circuit.

The two distances are $0.110\,\lambda$ and $0.259\,\lambda$. The short at the far position is $0.405\,\lambda$, which is the near short plus a half wavelength of the way around the stub circle. The open at the far position is $0.155\,\lambda$. Nothing in the electrical solution prefers the short. The board does.

How long is that stub on FR-4?

$\varepsilon_{\mathrm{eff}}$ is not $\varepsilon_r$. On this library’s FR-4, $\varepsilon_r = 4.4$, a $50\,\Omega$ microstrip at $1\,\mathrm{GHz}$ on $1.6\,\mathrm{mm}$ dielectric has $\varepsilon_{\mathrm{eff}} = 3.345$ and is $3.008\,\mathrm{mm}$ wide. That width is 50 Ω microstrip on FR-4. At $2.4\,\mathrm{GHz}$ dispersion has moved $\varepsilon_{\mathrm{eff}}$ to $3.377$, and the width that is still $50.00\,\Omega$ has narrowed to $117.42\,\mathrm{mil}$, $2.982\,\mathrm{mm}$. Use that $\varepsilon_{\mathrm{eff}}$ for the stub. Using $4.4$ makes the stub too short. Using the $1\,\mathrm{GHz}$ number is closer, and still not this frequency.

The guided wavelength is $c / (f \sqrt{\varepsilon_{\mathrm{eff}}})$. At $2.4\,\mathrm{GHz}$ and $3.377$ that is $68.0\,\mathrm{mm}$. Then $0.110\,\lambda$ is $7.51\,\mathrm{mm}$ and $0.095\,\lambda$ is $6.46\,\mathrm{mm}$. The microstrip calculator is where $\varepsilon_{\mathrm{eff}}$ came from: Hammerstad–Jensen with the Kirschning–Jansen correction, 1 oz bare copper, no solder mask, height $1.6\,\mathrm{mm}$. The same $\varepsilon_{\mathrm{eff}}$ is what the matcher multiplies into the physical length when you set the line to $50\,\Omega$.

Loss on that $50\,\Omega$ strip is $0.224\,\mathrm{dB/in}$ at $2.4\,\mathrm{GHz}$ in the transmission-line model. The $7.51\,\mathrm{mm}$ run is about $0.066\,\mathrm{dB}$. The matcher’s line is lossless, $\alpha = 0$. For this stub the omission is smaller than the etch tolerance. It is not smaller at $10\,\mathrm{GHz}$, and it is not a reason to ignore mask and finish on a long run. Those are effective permittivity and delay.

Should the stub be open or short?

At the near position the short is $6.46\,\mathrm{mm}$ and the open is $23.45\,\mathrm{mm}$. I build the short. It is the smaller resonator, it radiates less, and the via at the end is an ordinary ground via. The open is a quarter-wave longer and it is an antenna whether you meant it to be or not. Take the open when a via is the thing you cannot have, and keep it away from the antenna you are actually trying to match.

The far solutions are $17.64\,\mathrm{mm}$ of through line plus a longer stub. Same match at $2.4\,\mathrm{GHz}$, more board, more loss, a slightly different bandwidth once the line is dispersive. There is no reason to pick them on this board unless the near position lands in a keep-out.

Draw the through line and the stub at the same width, $2.982\,\mathrm{mm}$, because both are $50\,\Omega$ in this solution. A different stub $Z_0$ is legal and changes the length. This note does not. The via fence, if the ground at the short is a coplanar pour rather than a backside plane, is a different calculator. The impedance here is microstrip.

What this length is not

It is not a double stub, and it is not a quarter-wave transformer. A real $100\,\Omega$ load wants $70.71\,\Omega$ for a quarter wave; $60 - j80\,\Omega$ is not that problem. It is not $\varepsilon_r = 4.4$ used as $\varepsilon_{\mathrm{eff}}$. It is not the length at $1\,\mathrm{GHz}$: scale by $2.4/1$ only if $\varepsilon_{\mathrm{eff}}$ is the same, and it is not. It is not a lossy line, a solder-mask correction, or an ENIG correction. The model is a TEM line of the $\varepsilon_{\mathrm{eff}}$ you typed, plus a lumped open or short. If the coupon’s delay disagrees, believe the coupon and recompute $d$ and $l$ from the $\varepsilon_{\mathrm{eff}}$ you measured.

References

• D. M. Pozar, Microwave Engineering, the single-stub example in the impedance-matching chapter. The wavelengths here are computed for $Z_L = 60 - j80\,\Omega$ and $Z_0 = 50\,\Omega$. They are not a transcription of a printed page.
• P. H. Smith, “Transmission Line Calculator,” Electronics, vol. 12, January 1939. The rotation with line length is the original use of the chart.
• E. Hammerstad and Ø. Jensen, “Accurate Models for Microstrip Computer-Aided Design,” IEEE MTT-S, 1980, and M. Kirschning and R. H. Jansen, Electronics Letters, 1982. $\varepsilon_{\mathrm{eff}} = 3.377$ is that pair of models on this library’s FR-4 at $2.4\,\mathrm{GHz}$.

Related

• 50 Ω microstrip width on 1.6 mm FR-4
• Effective permittivity and trace delay
• How to read a Smith chart
• Shunt shorted stub at 2.4 GHz

FAQ

How do you single-stub match 60 − j80 ohms?

On a 50 Ω line the shunt stub sits 0.110 λ from the load, or 0.259 λ at the other intersection. At the nearer position the shorted stub is 0.095 λ and the open stub is 0.345 λ. All four are a perfect match at the design frequency. The series-stub solutions are a second set of four.

How long is Pozar’s stub on FR-4?

At 2.4 GHz on 1.6 mm FR-4, 1 oz bare copper and no solder mask, the 50 Ω microstrip is 2.982 mm wide and εeff is 3.377. The nearer short is then 7.51 mm from the load and 6.46 mm long. The open stub at that same distance is 23.45 mm. The matcher’s line is lossless. The microstrip model’s loss on that width is 0.224 dB/in.

Should a matching stub be open or shorted?

At the position nearer the load, the short is 6.46 mm and the open is 23.45 mm on this 2.4 GHz FR-4 line. The short is the smaller resonator and needs a ground via. The open is a quarter-wave longer and radiates. Both are exact at 2.4 GHz.

Is effective permittivity the same as the substrate εr?

No. This library’s FR-4 has εr 4.4. The 50 Ω microstrip on 1.6 mm at 2.4 GHz has εeff 3.377, up from 3.345 at 1 GHz. Using 4.4 as εeff makes the stub too short. The guided wavelength at 2.4 GHz is 68.0 mm, not the free-space 125 mm.