On stability of capacitive loaded op-amps.

29 May 2025

Introduction

Operational amplifiers are often, and for good reason, the go-to building block of many analog functions. They are often used with capacitive loads, for two main reasons. One, when operational amplifiers are used to produce DC voltage supplies, in which cases the capacitive load is used to ensure DC voltage stays constant. Second, when operational amplifiers are used to drive some capacitive load, typically the gate of the transistor of an RF power amplifier, including the capacitors of the biasing network.

However, a common trap with such circuits is that operational amplifiers tend to be unstable when they are capacitively loaded, even for relatively low values.

Beware of the “high capacitive load drive capability” op-amps

Some op-amps models like the OPA192¹ advertise an “high capacitive load drive capability”, but the stated value is only 1 nF. Even if this is only a typical value, its conditions are rather fair: only 40% overshoot. The main problem is that it is only 1 nF.

The same graphs suggest that a 25 Ω improves greatly the overshoot, with a maximum value of 25 %, and allows to use an arbitrarily high capacitive load.

And once the isolation resistor is added, it lacks only two components to make something perfect. Why not read the following and add them ?

In the discussion, the OPA994² was suggested. Indeed, it is advertised as having an “unlimited capacitive load drive capability” with a “phase margin of 50° when driving a load of 10 μF and 1 MΩ”. It is indeed the case. However, on some part of the phase margin plots, the phase margin is as low as 20°, and, again, is works much better with an isolation resistor. It should be noted, however, that the OP994 with a 50 Ω isolation resistor has a 45 % worst case overshoot while the OPA192 has only a 20 % overshoot.

And once the isolation resistor is added, same comment than before.

The root cause

A feedback system must have less than 180° phase shift from its output to its - input at the unity gain frequency in order to the negative feedback to stay negative and to avoid it going positive and causing unstability. There are more complicated ways to tell this principle but the core concepts are here.

An ideal operational amplifier has already a -90° phase shift due to its integrator nature (the gain bandwidth product). Due to various unperfections, there is also a little more phae shifting. The open loop output impedance of the operational amplifier combined with the capacitive load easily makes almost -90° phase shift at unity gain if the GBW and the capacitor are high enough. All this combined, this can lead to high overshoots, an high closed-loop output impedance, or a caracterized oscillation.

Note what matter for stability is the open loop operational amplifier output impedance. The closed loop output impedance is much lower due to feedback, at least when the feedback operates properly.

Some details on the topic

The details of the topic won’t be explained here, because the most important is to be aware of the problem and of its solutions. Once the problem is known, details are easy to find on Google. Reference ³ is a great starter, even if I would suggest an other solution on the “RC filter directly at the input”, ⁴ gives more informations on threee useful solutions: R_ISO, R_ISO + DFB, R_ISO + DFB + RFx, ⁵ give more solutions and details, and ⁶ is a rather long presentation on the general subject of operational amplifier stability which beats the topic to death, ⁶ gives additional details and equations on the topic. This application note from Microchip⁷, although not complete on compensation networks, give useful equation, useful tips for ADC interfacing and give useful information on large signal effects.

The careful reader will probably notice a mistake in the “in-loop compensation circuit” of ⁵: Vin and GND are inverted, as well as the - and + inputs of the operational amplifiers. Despite this mistake, the remaining of the contents is highly valuable.

Important note on coaxial cables

A important case of “capacitive” loads are coaxial cables. Their distributed nature and associated characteristic impedance and propagation time makes it differ from purely capacitive loads in ways that should not ignored. Coaxial cables loaded by their characteristic impedance, most often 50 Ω, behave like like a pure resistance. Conversely, coaxial cables driven by with a source impedance equal to their characteristic impedance behave like a pure resistive source impedance.

When it is sure a coaxial cable would be loaded by its characteristic impedance, from a stability point of view, it is just a resistor. Nothing more is needed, provided the operational amplifier can feed the requested current.

When a coaxial cable is loaded by something looking like an open circuit, typically an high impedance digital input, it is best to source terminate it, by putting a resistor between the output of the operational amplifier, wired in the usual way, and the input of the coaxial cable. The coaxial cable will be merged with the source resistor, and all will behave like the source resistor loaded by the open circuit.

Such a scheme can also be used in cases where the load is most often an high impedance but not always, because the source resistor will ensure a minimum resistance seen by the operational amplifier and avoid stability problems.

In some cases, it can be useful to terminate the coaxial cable on both sides. This has the drawback to make the final voltage half the voltage at the output of the operational amplifier, but is a robust solution.

However, when the coaxial cable is loaded by some capacitor, they must together be handled like a capacitor. Nevertheless, due to the cable propagation time, some ringing can be expected if the rise and fall times are too small compared to it. This can be avoided by ensuring long enough rise and fall times or by adding some damping resistors.

Some comments of the common solutions

Input filtering

Not directly related to output driving load, this is nevertheless an interesting case to mention. Often it is derisable, particularly to reduce sensitivity to RFI and EMI, to filter the input. A capacitor straight at the op-amp inputs, like in figure below, is likely to make it unstable, for the reasons described above.

However, I would suggest an other solution than the one proposed by Analog Devices: splitting the input resistor and to put the filtering capacitor in the middle.

Output snubbing network

A perceived drawback of solutions using an isolation resistor is the voltage drop in the isolation resistor and the inability of using the full output voltage range of the operational amplifier, particularly for RRO (rail-to-rail-output) amplifiers.

Whether an isolation resistor is used or not, voltage margin between the desired voltage and the voltage rails of the operational amplifiers is useful so the retroaction loop can work properly.

In the case this solution must be used, like in the schematic below from Analog Devices, the principle can be explained in the following simple way. R_S is low enough compared to C_L to make the output load looks like resistive enough to ensure a good phase margin at the unity gain frequency, while C_S is high enough compared to R_S to make R_S+C_S close enough to only R_S.

This solution stabilizes the circuit, but makes the load harder to drive, making this solution ok for DC or low frequency signals, but not so much for higher frequency cases.

A tricky point of this calculation is that the needed components values depends on the unity gain frequency, but it depends itself on the values of the snubber network which tend to reduce it.

Isolation resistor

An isolation resistor ensures the opamp sees at the unity gain a convenient load, either a low enough capacitance when the capacitive load is low enough compared to the isolation resistor, or the isolation resistor when the capacitive load is high enough. This ensures stability, but at a price: the opamp produced the wanted voltage with the feedback before the resistor, and the actual voltage at the load is RC filtered.

Isolation resistor + double feedback

This technique exists in two common variants shown in the 2 figures below from the excellent article⁴ from Texas Instruments.

The operating principle is the same in the two cases: provide an high frequency and a low frequency feedback paths. The high frequency path, right at the operational amplifier output, undelayed, provides stability, while the low frequency path, at the load, provided an exact low frequency response.

With some mathematics, it is possible to determine the optimum values which ensure both stability and performance.

Since I had not yet the time to do the mathematics, in waiting, please find a picture of these beautiful cats from Wikipedia:

Some words on buffers

Unity gain buffer ICs can help to solve this problem, either as a standalone solution, when their offsets are tolerable, or as an addition to the opamp, kind of replacing the isolation resistors.

Since I had not yet the time to write on the topic, in waiting, please find a picture of this beautiful cat from Wikipedia:

Case studies

Sariel Hodisan

From this LinkedIn post⁸ from Sariel Hodisan:

Lately, I designed a simple biasing circuit for bipolar voltage rails (±13.5V) to bias a very precise low-noise analog circuit. To minimize offset in the analog section, it’s mandatory for the ±13.5V rails to be as closely matched as possible. The load is just a few mA, and it’s mostly a DC circuit. No fast current draws.

[...]

I did need decoupling for the biased circuit and figured 1nF would be enough. According to the OPA192 datasheet, there’s around 40 % overshoot when loaded with a [2nF, see below] cap.

[...]

The ±13.5V rails oscillated terribly.

Luckily, I had provisions to add an isolation resistor (30 ohm), and that solved the problem, though at the cost of a slight voltage drop.

I edited the message to make the total capacitive load more clear: 2 nF total, including 1 nF straight at the opamp output and 1 nF at the load.

Although some details are specific to his project, like his need of low noise, generating DC voltages to supply something is a very common need.

This circuit should not have oscillated so hard, according to the datasheet, which predicts less than 45 % overshoot for a 2 nF load.

He solved the problem using a simple isolation resistor, at the cost of a small voltage drop, and I suggested him to add a double feedback to eliminate the drop. Since it is a DC need, the simple double feedback is sufficient.

Microwaves101 gate pulser

In the excellent microwaves101 website⁹, a page is written on the subject of pulsed RF sources. Although it advises drain pulsing (this page was written long before before GaN was mainstream), it also gives some advice about gate pulsing (bold from myself):

For a good gate pulsing circuit, we recommend the Analog Device's AD8036 "clamping amp", shown below. It lets you set up on and off gate bias voltages independently; for example, if you are using a PHEMT power amp you can set VG(on) to -0.9 volts, and VG(off) to -1.5 volts. You still need charge storage on the drain bias lines, and stability caps on gate and drain biases, as near as possible to the amp. Like all op-amps, the AD8036 can be configured as an inverting or non-inverting amplifier. Yes, we need to add some resistive feedback to the figure!

The feedback was forgotten in the figure, but not in the text.

However, a point which might need to be added is the stabilisation for capacitive loads. The AD8036 datasheet does not mention performance curves when capacitively loaded, but instead a recommended isolation resistor value:

Given this curve and the typical gate bias network capacitance values, I would recommend a 20 Ω isolation resistor. The RC bandwidth is already 31.8 MHz. If not sufficient, a R_ISO + DFB + RFx can be used to both remove the R_ISO offset and to have speed up.

Microwaves101 gate pulser, alternative solution

That being said, we would propose here an alternative solution.

A problem with the AD8036¹⁰ is that it is an input clamping amplifier, with clamping only available on its +V_IN, making it “works only for noninverting or follower applications”. This needs a 0 to -5V input signal to have the requested V_H and V_L voltages. Not very convenient since most digital circuits operate on positive voltages.

Switching between two voltages levels would be conveniently done by a switch integrated circuit. Most common switches are slow, either because they are plain slow or because they include some “break before make” circuitry which take some transition times. The switches in the “buffered analog multiplexers” section of Analog Devices¹¹ provide faster time, probably because the techniques to deal with switching transitions are easier to implement in unidirectional multiplexed buffer than in unidirectional buffer, like current steering:

The [ADV3129¹²] multiplexer is organized as two input transconductance stages tied in parallel with a single output transimpedance stage followed by a unity-gain buffer. Internal voltage feedback sets the gain.

The following chips are interesting, with only single channel devices mentionned:

The AD8170 is a current feedback amplifier with a switchable input. These category of operational amplifiers are sensitive to the impedance seen at the feedback pin, to DFB schemes cannot be used directly with them. The datasheet recommands isolation resistor values and feedback resistance. Simple RC calculations shows that the performance is mainly determined by the RC constant of the isolation resistor and the capacitive load.

The AD8180 is an open loop buffer and can drive capacitive loads without isolation resistors. However, an input resistor is recommended. This is often the case for buffer amplifiers, and often forgotten. Performance is again determined by the open loop impedance of approximately 25 Ω.

The ADV3219 is a feedback amplifier with an internal feedback. Performance curves are given for low capacitive loads without isolation resistor. For higher capacitive loads, the datasheets recommands an isolation resistor of “a few tens of ohms”, but does not give more performance details. Nevertheless it can be assumes that it will be dominated by the RC constant of the output.

AD8170	AD8180
AD8180	ADV3219

Conclusion

Capacitor unstability when capacitively loaded is a classical trap, catching even rather experienced designers, particularly RF designers who are not always proficient in low frequency analog design. Although it may seem hard, this problem is well known, well documented and has readily available efficient solutions.

Appendix

Previous version of the page included various cat pictures as placeholders for contents to be written. They are left here for posterity:

Courtesy of Microwaves101 (https://www.microwaves101.com/unknown-editor-121-cat-wrangler-april-2023):

IQ modulator and quadrature coupler sign issues.

05 Nov 2024

Update from 2024-11-05: Steve from Microwaves 101 also tackled this topic (https://www.microwaves101.com/encyclopedias/branchline-coupler-port-definition), proposing a different port-naming convention for the branchline coupler. Steve names the direct and coupled outputs in reverse compared to the approach I use. His convention offers some interesting arguments: the output he names "forward" exhibits a higher bandwidth and flatter phase response.

Upon further analysis, it appears that the operating principles of branchline couplers differ significantly from those of coupled line couplers. Consequently, the terms "direct" and "coupled," commonly used for coupled line couplers, no longer carry the same meaning.

For now, I’ll retain the "geometric" convention I initially used, as it seems more common, although I’m not entirely convinced it’s more "correct." Examples that follow this convention can be found in this research, this diagram, this article, and this paper).

Many thanks to Steve for raising this interesting topic. Thanks also ChatGPT for improving my English.

Introduction

IQ modulators and quadrature couplers are often used together to perform a frequency translation while rejecting the unwanted sideband. However, the documentation often does not clarify whether the quadrature signal has a +90° or -90° phase shift, and whether the IQ modulator expects a quadrature input with a +90° or -90° phase shift.

The purpose of this page is to clarify this sign issue.

Upconverter side

Here, we assume an upconverter is used. The principle of an IQ upconverter is to perform a complex frequency translation as follows:

"RF" = Re[(I + j \cdot Q) \cdot e^(j \cdot \omega_("rf") \cdot t)]

In the case of a complex cosine signal in baseband with frequency &&\omega_b&&:

I + j \cdot Q = e^(j \cdot \omega_b \cdot t) = cos(\omega_b \cdot t) + j \cdot sin(\omega_b \cdot t)

By application of the trigonometric identity &&sin(x) = cos(x - pi / 2)&&:

I + j \cdot Q = cos(\omega_b \cdot t) + j \cdot cos(\omega_b \cdot t - \pi / 2)

This clearly shows that the Q input has a phase lag compared to the I input, which is a -90° phase.

Coupler side case 1: coupled line coupler

Let’s assume that the coupled line coupler QCH-451+ from Mini-Circuits is used. The datasheet indicates which port is the quadrature output but not whether its phase is +90° or -90°.

Using ChatGPT and Plotly, the following plots were easily generated:

It’s clear from the plots than the quadrature output has a +90° phase (lead) compared to the in-phase output.

Besides clarifying the +/-90° sign issue, it is useful to check to what physical element correspond each port number. The hypothesis is that this coupler is made with an equivalent to coupled lines performed using an LC network, as pure coupled lines are obviously too long to fit in the component.

Analysis of the transmission curves enables to deduce the connexions:

• At low frequencies, total transmission from port 1 to port 3 and no transmission to the other ports. Thus both ports are connected to the same first transmission line and ports 2 and 3 are connected to the second line.

• Port 4 is not connected at low frequencies but is coupled in the band. So port 4 is the coupled port of the other line.

• Port 2 is not connecter at low frequencies and remains approximately isolated in the band. So port 2 is the coupled output of the second line.

This leads to the following schematic, compared to the datasheet information:

The symmetry of the schematic is coherent with the symmetry of the table given in the Mini-Circuits datasheet:

The +90 phase shift stays even at low frequencies. This is coherent with the similar behaviour of an high-pass CR network, which is part of the equivalent schematic at low frequencies.

Coupler side case 2: branch-line coupler

In this part, the ports are numbered like in Microwaves101 (https://www.microwaves101.com/encyclopedias/branchline-couplers), from which this schematic is taken:

This part was performed in two steps. First, using scikit-rf with a some help of ChatGPT, a Python script was created to produce preliminary plots and an s4p file. The second step was to reuse the plotting code from the previous section, with some help from ChatGPT to refactor the duplicate parts, to make the following plots:

Contrary to coupled lines, which have a very broadband quadrature effect, the branch-line coupler has a narrowband quadrature effect. This is still useful for narrowband applications.

It’s clear from the plots than the coupled output has a -90° phase (lag) compared to the direct output.

Conclusion

Quadrature couplers are important for making image reject mixers, but the actual phase sign is sometimes not clearly stated. According to the theory, IQ mixers needs a -90° phase (lag) for the Q input. For coupled lines couplers, the coupled output has a +90° (lead) phase compared to the uncoupled output. Conversely, for branchline couplers, the coupled output has a -90° (lag) phase compared to the uncoupled output. These points must be considered when selecting the wanted mixer side.

Microstrip formulas comparison.

07 Jul 2024

Introduction

The designer has several tools and formulas available to calculate the characteristic impedance of microstrip lines. Some are highly precise but rather complex like the Hammerstad and Jensen formulas^1,2,3, while others are rather simple but with questionable accuracy like the IPC-2141 formulas^4,5,6. While approximations can be useful for the first steps of a design, their accuracy must be evaluated before use. The authors of Qucs³ made some comparison, but this comparison don’t include the common IPC formulas^5,6. A comparison of the most common microstrip calculation formulas is shown here.

While Hammerstad and Jensen formulas¹ stay the gold standard, other formulas can be safely used but that IPC-2141 formulas^4,5,6 must be used with extreme caution.

Geometry of the problem

Fig. 1 – Geometry of a microstrip line⁴.

The geometrical parameters of the microstrip line studied are defined in Fig. 1⁴: w is the width of the microstrip line, h the height of the substrate, t the thickness of the strip. Non-geometrical parameters are the relative permittivity of the substrate. We’ll note for the characteristic impedance of free space.

To simplify the analysis, the strip conductivity is assumed to be infinite, the substrate losses to be null, the thickness t to be null and the frequency to be null. Note that the infinite conductivity hypothesis implies the absence of low-frequency dispersion^7,8. These assumptions are often used in RF and microwave design, and are of good accuracy for the impedance calculation.

Reference formula for Microstrip

Evaluating the accuracy of models needs a reference to compare to. Hammerstad and Jensen formulas are the most accurate closed-form formulas, and their accuracy is higher than manufacturing processes. They are commonly used in CAD software^3,9 in which the accuracy of the models of single microstrip lines without discontinuities is recognized. However, these formulas are so complex than their practical use for a comparison can lead to type errors during implementation, which can lead to inaccuracies. 3D EM simulations can be as accurate as needed, but their setup and calculation time is time consuming when accuracy is needed. Measurements are highly expensive due to the need of accuracy in both manufacturing and characterization of board materials. So, for this setup, a well-known software is used, TXLine from AWR. This software implements probably Hammerstad and Jensen formulas and its wide use almost guarantees that the implementation is bug free.

Formulas to be compared

The formulas to be compared are the following:

TXLine

TXLine is a small software, lightweight and free to use, from AWR, which allows to calculate the characteristic impedance of transmission lines like microstrip and striplines. A screenshot is shown Fig. 2.

Since TXLine has no direct option to handle the infinite conductivity, the zero thickness or the zero frequency, they were approximated as follows: conductivity 10^99\ "S"\cdot"m"^-1, thickness 0.1 µm or 0.001 µm (depending on cases), and frequency 1 Hz. The high conductivity can be considered as infinite for all practical purposes, and is high enough to avoid the impact of low-frequency dispersion on the calculation results ^7,8. The thickness also is low enough to approximate the zero-thickness condition.

Hammerstad and Jensen

According to Hammerstad and Jensen formulas^1,2,3 the calculation is made in two steps. First, the effective dielectric constant is calculated:

\begin{array}{c} u(w, h) = \frac{w}{h} \\ a(u) = 1 + \frac{1}{49} \cdot \ln \left[ \frac{u^4 + \left( \frac{u}{52} \right)^2}{u^4 + 0.432} \right] + \frac{1}{18.7} \cdot \ln \left[ 1 + \left( \frac{u}{18.1} \right) \right] \\ b(\varepsilon_r) = 0.564 \cdot \left( \frac{\varepsilon_r - 0.9}{\varepsilon_r + 3} \right)^{0.053} \\\ \varepsilon_{r, \text{eff}}(u, \varepsilon_r) = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2} \cdot \left( 1 + \frac{10}{u} \right)^{-a(u) \cdot b(\varepsilon_r)} \\ \end{array}

Thereafter, the following formulas are used to calculate the characteristic impedance of the microstrip line, assuming the line is in an homogeneous medium of dielectric constant :

\begin{array}{c} f(u) = 6 + (2 \cdot \pi - 6) \cdot \exp \left[ - \left( \frac{30.666}{u} \right)^{0.7528} \right] \\ Z_L(u) = \frac{Z_{F0}}{2 \cdot \pi \cdot \sqrt{\varepsilon_{r, \text{eff}}}} \cdot \ln \left[ \frac{f(u)}{u} + \sqrt{1 + \left( \frac{2}{u} \right)^2} \right] \end{array}

Note that in reference¹, the formulas are not very clear about whether or should be used, but this point was double-checked in the Excel spreadsheet formulas.

These formulas are used in these calculators^10,11.

Wheeler 1965

Wheeler’s formulas are sometimes encountered in technical literature^3,12. They are rather accurate for most uses, as will be seen in following sections. Their main problem is the undesirable impedance step. Note that, contrary to Hammerstad and Jensen formulas, the effective dielectric constant is not used in the characteristic impedance formulas.

Z_L(w, h, \varepsilon_r) = \begin{cases} \frac{Z_{F0}}{2\sqrt{\varepsilon_r}} \cdot \left( \frac{w}{2h} + \frac{1}{\pi} \cdot \ln 4 + \frac{\varepsilon_r + 1}{2\pi\varepsilon_r} \cdot \ln \left[ \frac{\pi \cdot e}{2} \cdot \left( \frac{w}{2h} + 0.94 \right) \right] + \frac{\varepsilon_r - 1}{2\pi\varepsilon_r^2} \cdot \ln \left( \frac{e \cdot \pi^2}{16} \right) \right) & \text{if} \quad \frac{w}{h} \leq 3.3 \\ \\ \frac{Z_{F0}}{\pi \sqrt{2(\varepsilon_r + 1)}} \cdot \left[ \ln \left( \frac{4h}{w} \right) + \sqrt{\left( \frac{4h}{w} \right)^2 + 2} - \frac{1}{2} \cdot \frac{\varepsilon_r - 1}{\varepsilon_r + 1} \cdot \left( \ln \frac{\pi}{2} + \frac{1}{\varepsilon_r} \cdot \ln \frac{4}{\pi} \right) \right] & \text{if} \quad \frac{w}{h} > 3.3 \end{cases}

We have not found any calculator using these formulas.

Wheeler 1977

Wheeler 1977 formulas are seen more often than Wheeler 1965 formulas in literature^4,13,14,15. They are attributed sometimes to Wadell because he made a good summary in his book¹³. Note that these formulas are incorrectly typed a reference⁴, where a pi^2 was incorrectly replaced by a sqrt(pi), and an extra square root is present, as pointed by an other reference¹⁶.

\begin{array}{c} A = \frac{14 + \frac{8}{\varepsilon_r} \cdot \frac{4 \cdot h}{w'}}{11} \\ B = \sqrt{A^2 + \frac{1 + \frac{1}{\varepsilon_r} \cdot \pi^2}{2}} \\ Z_L = \frac{Z_{0F}}{2 \cdot \sqrt{2} \cdot \pi \cdot \sqrt{\varepsilon_r + 1}} \cdot \ln \left[ 1 + \frac{4 \cdot h}{w} \cdot (A + B) \right] \end{array}

They are used in several microwave calculators^{17,18,19,20,21}. Calculator²¹ has a mistake in the handling of epsilon_(r,"eff"), which can be diagnosed by calculating impedances with epsilon_r=1.

No explicit formula is given for epsilon_(r,"eff"). The reason is that it can be calculated with: epsilon_(r,"eff")=[(Z_0(h,w,epsilon_r=1))/(Z_0(h,w,epsilon_r=epsilon_r))]^2. This makes these formulas both simple and complex at the same time. In a programming language, it is easy to define a function and to use it two times to calculate . In an old school Excel sheet this would lead to use twice plus one many columns, which more than Hammerstad & Jansen formulas.

Hammerstad 1975 formulas

Often seen in websites²² or in lectures on microwave techniques^23,24, Hammerstad’s 1975 formulas are the following^{22,25,26,27,28}. They are often attributed to Bahl, who made an improvement to their strip thickness correction^26,27, not investigated in this article, but the zero-thickness formulas originates from Hammerstad²⁵.

\begin{eqnarray} \varepsilon_{r, \text{eff}} &=& \begin{cases} \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2} \cdot \left( \left(1 + \frac{12 \cdot h}{w}\right)^{-1/2} + 0.04 \cdot \left(1 - \frac{w}{h}\right)^2 \right) & \text{if} \quad \frac{w}{h} \leq 1 \\ \\ \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2} \cdot \left( 1 + \frac{12 \cdot h}{w} \right)^{-1/2} & \text{if} \quad \frac{w}{h} > 1 \end{cases} \\ Z_L &=& \begin{cases} \frac{Z_0}{2 \cdot \pi \cdot \sqrt{\varepsilon_{r, \text{eff}}}} \cdot \ln \left( \frac{8 \cdot h}{w} + \frac{w}{4 \cdot h} \right) & \text{if} \quad \frac{w}{h} \leq 1 \\ \\ \frac{Z_{F0}}{\sqrt{\varepsilon_{r, \text{eff}}}} \cdot \left[ \frac{w}{h} + 1.393 + 0.667 \cdot \ln \left( \frac{w}{h} + 1.444 \right) \right]^{-1} & \text{if} \quad \frac{w}{h} \geq 1 \end{cases} \end{eqnarray}

These formulas are used in these calculators^29,30,31. Note that this calculator³⁰ also includes frequency thickness correction and dispersion.

Schneider

Not often seen, Schneider formulas^3,32 are as following:

\varepsilon_{r, \text{eff}} = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2} \cdot \frac{1}{\sqrt{1 + 10 \cdot \frac{h}{w}}} Z_L = \frac{Z_{F0}}{\sqrt{\varepsilon_{r, \text{eff}}}} \cdot \begin{cases} \frac{1}{2 \cdot \pi} \cdot \ln \left( \frac{8 \cdot h}{w} + \frac{w}{4 \cdot h} \right) & \text{if} \quad \frac{w}{h} \leq 1 \\ \\ \frac{1}{\frac{w}{h} + 2.42 - 0.44 \cdot \frac{h}{w} + \left(1 - \frac{h}{w}\right)^6} & \text{if} \quad \frac{w}{h} > 1 \end{cases}

We have not found any calculator using this formula.

IPC-2141 formulas

The IPC-2141 formulas^4,5,6,28 are very popular. However, despite their popularity, they should be used with extreme caution, as will be demonstrated during our comparison, because their range of validity is extremely narrow, and they have a bad asymptotic behavior.

Although they are widely quoted, their narrow range of validity is much less often quoted²⁸: 0.1<w/h<2.0 and 1<epsilon_r<15. This range becomes even narrower because it enables to synthesize 50 Ω lines only for 3.9<epsilon_r<15.

Contrary to other formulas (Hammerstad and Jensen, Wheeler 1965, Wheeler 1977, Schneider and Hammerstad 1975), IPC-2141 formulas have a totally nonsense asymptotic behavior. All other formulas were found to have an accuracy better than 1.5 % for epsilon_r=4.5 and w/h ratios ranging from 0.001 to 1000. On the contrary, IPC-2141 formulas give a nonsense negative impedance result for w/h>7.5. This is a large line for most applications, but it still happens in some designs, and give an idea of the problem.

Fig. 1 compares the asymptotic behavior of IPC-2141 formulas with the good formulas. While IPC-2141 have a very bad asymptotic behavior for large lines, all good formulas have a good asymptotic behavior on the very wide range 0.001<w/h<1000, well outside of their guaranteed validity range.

Despite the problems of IPC-2141 formulas, they are used in several online calculators^33,34,35,36,[^37]^,37,38,39. Some calculators^33,34,35 give warnings when using IPC-2141 formulas outside of their validity range like shown in Fig. 3. On the contrary, some other calculators^36,[^37]^,37,38,39 give neither a warning nor a validity range, including a calculator on a renowned website³⁷. Worse, some calculators^37,38,39 even give nonsense negative impedance when fed with proper values without any warning.

Fig. 4 – Screenshot of a microstrip line impedance calculator³³ raising a warning when trying to calculate impedances outside IPC-2141 validity range.

It should be mentioned that a calculator³³ not only gives the validity range of the IPC-2141 formula and warns when trying to enter parameters outside of this range, but it also gives accuracy data.

Comparison results

From the Fig. 5 graph, a clear outsider in inaccuracy is the IPC-2141 formula, reaching up to 44 % inaccuracy! This formula will be commented later. Fig. 6 graph, without the IPC-2141 formula on a reduced scale, shows the relative error of the remaining formulas.

The H&J formulas are probably the formulas used in TXLine software since the calculated difference between them is lesser than 0.06 % ! Note that this calculated error is always positive, because TXLine took a line thickness which is non-zero, although very thin, contrary to our calculation. Since we are comparing the H&J formulas to themselves, this benchmark does not prove that they are the most accurate. However, since they are believed to be the most accurate in all the recent literature, we’ll stick to that conclusion. They still have the inconvenient of their complexity, which can lead to potential typing mistakes.

Hammerstad 1975 formulas are at the second place for accuracy: error less than 0,38 % on tested values. However, they still have the inconvenience of their complexity, potentially leading to typing mistake, and of the different expressions for different subdomains.

Wheeler 1965 formulas are at the third place for accuracy: error less than 0.59 % on tested values. However, the calculation step makes them troublesome to use in some cases. And, for use in Excel spreadsheets, it makes mandatory to duplicate the formulas. The 0,06 % accuracy gained from Wheeler 1977 is not worth it.

Wheeler 1977 formulas, although not very known, are rather simple and rather accurate: the error is lesser than 0.66 % on tested values. The absence of impedance steps in these formulas make them interesting.

Schneider formulas are less complex than Hammerstad 1975 formulas but less accurate. Their error is less than 1.6 %. Their interest is mainly historical.

The real strange point is the accuracy of the IPC-2141 formulas. Their relative error reach 44 % on the tested values! A closer look reveals that most of the error happens when the normalized width u is higher than 2. When plotted on a narrower range, like in Fig. 7, the relative error is much lower: less than 2.1 %. This is precisely the range of values of the 50 Ω lines.

Review of calculators

The following table sums up some microstrip calculators and the formulas which they use. Only microstrip calculators for which the formula was told or could be inferred from JavaScript source code were included.

Website	Formula	Val. warn	Comments
www.microwaves101.com10	H&J
mcalc.sourceforge.net11	H&J
www.eeweb.com17	Wheeler 1977
cepd.com18	Wheeler 1977
www.finetune.co.jp19	Wheeler 1977
leleivre.com20	Wheeler 1977
chemandy.com21	Wheeler 1977		Confusion between εr and εr,eff.
www.pasternack.com29	Hammerstad 1975
www.emclab.cei.uec.ac.jp30	Hammerstad 1975
chemandy.com31	Hammerstad 1975
emclab.mst.edu33	IPC-2141	Yes
referencedesigner.com34	IPC-2121	Yes	Formulas not told in document, but seen in JavaScript source code.
technick.net35	IPC-2141	Yes
a8blog.com36	IPC-2141	No	No warnings, but at least does not print the calculated negative impedance.
www.everythingrf.com37	IPC-2141	No
chemandy.com38	IPC-2141	No
ncalculators.com39	IPC-2121	No

Conclusion

While H&J formulas are the gold standard for calculations, several other formulas give a good trade-off between accuracy and simplicity which are sufficient for most applications. Wheeler 1977 is the clear winner of this tradeoff with 0.66 % error when epsilon_(r,"eff") is not needed. However, on old school Excel sheets, when epsilon_(r,"eff") is needed, the need to duplicate the calculation makes it not anymore convenient than H&J formulas. In this case, Hammerstad 1975 is the winner of this tradeoff with 0.38 % error.

IPC-2141 formulas have severe issues and must be used with extreme caution.

MLCC voltage dependence.

24 Apr 2024

This content was originally published on Microwaves 101 (https://www.microwaves101.com/encyclopedias/capacitor-voltage-effects). Many thanks to Steve for improvements on the original version. Have a look on his website for more interesting content.

This page was suggested by Hadrien, who has had recent experience in MLC capacitor variations with voltage. Did you know your capacitor nominal value can drop 80% when you apply a DC voltage to it? Worse, there does not seem to be any standards for voltage variations like there are for temperature variations.

This page is primarily discussing MLCCs, or multi-layer cerami capacitors, in Class 2. Class 2 uses exotic dielectrics such as barium titanate (BaTiO) with some other strange additives in order to get high dielectric constant in the thousande (to increase capaitance density). Barium titanate is a ferro-electric material, which is the source of voltage/capacitance misery.

From here down, thanks to Hadrien!

Case study

Since this page was originally written for Microwaves101, the case study to illustrate this problem was taken from it. Suppose we want to redesign the good old breadboard RF pulsed (https://www.microwaves101.com/encyclopedias/breadboard-rf-modulator) source from 2006 with more recent components (first version of this page written in 2019). Start by the capacitors. Capacitors values which were at this time most often made with electrolytic capacitors are nowadays more often made with MLCC (multiple layers caramic) capacitors.

To replace the 4.7 µF capacitor, a simple search gives this component:

https://fr.rs-online.com/web/p/condensateurs-ceramique-multicouche/8467296/ (4.7 µF)

Seems rather nice, and smaller than the original. Wrong! Let’s check the detailed characteristics:

https://www.murata.com/en-global/products/productdetail.aspx?partno=GRM188R61A475KE15%23

Not good! If we need the 4.7 µF capacitance value, we’ll be soon in trouble.

Let’s search a bigger capacitor. We search the biggest value among MLCC capacitors having 10 V rating. https://fr.rs-online.com/web/c/passifs/condensateurs/condensateurs-ceramique-multicouche/?applied-dimensions=4294244793,4291386526&pn=1 We find some nice models:

Let’s plot together their bias characteristics^1,2,3,4,5,6:

The values of the capacitors at 5 V range from 28.7 to 45.9 µF. Still low for 100 µF capacitors but all would be more than sufficient to replace the 4.7 µF capacitor of the original design.

Personnally, to keep some safety margin from the maximum voltage, I don’t like using a capacitor at a voltage where its capacitance drops more than 50 % of its nominal value. So, if I had to redesign this board, I would continue searching capacitors, even when the Murata and Kemet capacitors are close matches.

Conclusion of case study

As a conclusion, the “100 µF” capacitors of our study, when a DC voltage is applied to them, have an actual capacitance value much lower than their nominal value. This capacitance value would be still sufficient for our application because we oversized the capacitor. However, to keep a sufficient margin from voltage breakdown, I find careful to not use a capacitor at a voltage where its capacitance drop is more than 50 %. In designs where the capacitors are undersized, the capacitance drop effect on applied voltage can be a bad surprise and lead to severe troubles. Such problems are pretty hard to debug if not aware because the cap meters instruments most often measure at 0 V.

An other interesting point is what happens when we combine both biasing and temperature effects. Let’s see an other curve provided by TDK:

The temperature dependence is better when biased. This leads to the conclusion that temperature coefficient is most often less troublesome than voltage coefficient for MLCC capacitors.

Calculation of characteristic impedance from S parameters.

08 Jan 2024

Introduction

Suppose one wants to measure or simulate the characteristic impedance of something similar to a transmission line. This can be for example a 75 Ω coax cable or a via structure on HFSS. The characteristic impedance can be calculated form its measured or simulated S parameters references to 50 Ω.

Suppose also that the structure is “symmetric enough” to have the same characteristic impedance on each side (see https://en.wikipedia.org/wiki/Image_impedance)…

Recommended method using ABCD parameters

According to https://en.wikipedia.org/wiki/Image_impedance, and assuming the symmetry hypothesis which allows to simply discard the second result, the characteristic impedance can be calculated as:

Z_0 = sqrt((A B)/(C D))

The ABCD parameters can be obtained from S parameters with https://www.microwaves101.com/encyclopedias/network-parameters:

{: ( A = ((1 + S_(11))(1 - S_(22)) + S_(12)S_(21))/(2S_(21)) , B = 50 Omega ((1 + S_(11))(1 + S_(22)) - S_(12)S_(21))/(2S_(21)) ), ( C = 1/(50 Omega) ((1 - S_(11))(1 - S_(22)) - S_(12)S_(21))/(2S_(21)) , D = ((1 - S_(11))(1 + S_(22)) + S_(12)S_(21))/(2S_(21)) ) :}

Calculation is made as follows:

Z_0 = 50 Omega sqrt( ( ((1 + S_(11))(1 - S_(22)) + S_(12)S_(21)) ((1 + S_(11))(1 + S_(22)) - S_(12)S_(21)) ) / ( ((1 - S_(11))(1 - S_(22)) - S_(12)S_(21)) ((1 - S_(11))(1 + S_(22)) + S_(12)S_(21)) )

These formulas can be conveniently entered into an Excel spreadsheet.

Alternative non recommended method using transfer S parameters

Transfer S parameters can also be used for this calculation. However, this method is NOT recommended because the calculations are cumbersome.

Expressing the characteristic impedance as a reflection coefficient from 50 Ω, and recalling that by definition its invariant through the system, the following can be written:

[[b_1],[a_1]] = [[T_11,T_12],[T_21,T_12]] \ [[a_2],[b_2]] Gamma = b_1 / a_1 = (T_11 Gamma b_2 + T_12 b_2)/(T_21 Gamma b_2 + T_12 b_2) = (T_11 Gamma + T_12)/(T_21 Gamma + T_12)

Rearranging:

Gamma (T_21 Gamma + T_12) = (T_11 Gamma + T_12) T_21 Gamma^2 + (T_12 - T_11) Gamma - T_12 = 0

Solving by the usual methods:

Delta = (T_(12)-T_(11))^2+4*T_(21)*T_(12) Gamma = (T_(11) - T_(12) +- sqrt((T_(12)-T_(11))^2+4*T_(21)*T_(12)))/(2*T_(21))

So, the procedure, can be outlined as follows:

1. Convert S parameters to T parameters using the previous formulas or scikit-rf (https://scikit-rf.readthedocs.io/en/latest/api/generated/skrf.network.s2t.html)

2. Calculate the Gamma of the characteristic impedance references to 5 Ω.

3. From the Gamma, calculate the characteristic impedance.