Vacuum capacitors are used for high end applications that require high voltage withstand and low loss.

Though they are called capacitors, and simple analyses treat them as a capacitance with some small equivalent series resistance (ESR), there is more to it.

Above is a view (courtesy of N4MQ) looking into one side of a vacuum capacitor. It consists of an outer cup, and a series of 5 inner cups progressively smaller in diameter. The other side of the capacitor has a similar structure but the cups site in the middle of the spaces between cups in the first side.

Above is a cross section diagram of a fixed vacuum capacitor from an early patent, it shows the concentric cup structures.

Importantly, these cups are usually electrically bonded circumferentially and completely to the outer cup. Because of skin effect, most current at radio frequencies flows in a very thin layer near the surface of the material, and that means that the current that flows onto the inner cup has flowed along the outside, then inside of each of the outer cups in series.

As a result of the isolation provided by well developed skin effect, what we have with two nearly fully meshed electrodes is a cascade of vacuum dielectric copper transmission lines of successively reducing diameters.

## Modelling of transmission line ‘capacitors’

Whilst treating a very short section of open circuit (O/C) transmission line as a capacitance proportional to length might be an adequate approximation for some purposes, more insight is had by treating it as a transmission line (ie with distributed inductance, capacitance, resistance, and leakage conductance).

### Some examples to explore the underlying effects

Lets consider a coaxial line fabricated from copper conductors inside OD 30mm, outside OD 32mm, and terminated in an O/C.

First case, length=50mm. Input Z 2.827e-4-j3.693e+2Ω, Xc=369.3Ω and ESR=0.0002827Ω, Q=Xc/ESR=1,306,000.

What would you expect if it was twice as long?

Zin for a 100mm O/C section has input Z 5.654e-4-j1.846e+2Ω. Whilst Xc is halved, ESR is double so Q is a quarter of the first case.

What would you expect if two 50mm sections were connected in parallel?

Zin for a 100mm O/C section has input Z 1.416e-4-j1.840e+2Ω. Whilst Xc is halved, ESR is halved so Q is the same as the first case.

What would you expect if it was ten times as long?

Zin for a 500mm O/C section has input Z 2.831e-3-j3.679e+1Ω. Whilst Xc is just over a tenth, ESR is just more than ten times so Q is less than a hundredth of the first case. Evan at this short length, we observe that Xc is not simply inversely proportional to length, nor ESR simply proportional to length. The departure from proportionality increases as the length approaches an electrical quarter wave.

The lessons here are:

- Xc is not simply inversely proportional to length, and therefore equivalent capacitance is not simply proportional to length;
- Q of the ‘capacitor’ degrades with increased line length; and
- parallel TL sections may have higher Q that series sections.

### Analysis of the structure pictured above

Lets make some assumptions about the size of the structure pictured above, lets assume:

- frequency is 10MHz;
- cups mesh by 50mm length;
- outer cup is 90mm diameter;
- inner cup is 30mm diameter;
- there are 10 transmission lines formed by the meshed plates; and
- the space between TL surfaces is 2mm.

On that basis, the innermost TL section has inner OD 30mm, outer ID 32mm, and 50mm length. We can calculate the input impedance of that section assuming copper loss and ignoring dielectric loss. We obtain 2.827e-4-j3.693e+2Ω. Expanding that, Xc=369.3Ω, ESR=0.0002827Ω, and Q=1.306e6.

The next section has inner OD 33mm, outer ID 35mm, and 40mm, and is terminated by 2.827e-4-j3.693e+2 Ω. We obtain 5.654e-4-j1.846e+2Ω. Expanding that, Xc=184.6Ω, ESR=0.0005654Ω, and Q=326.5e3. Xc is just under half, ESR is double, and Q is about a quarter.

The next section has inner OD 33mm, outer ID 35mm, and 40mm, and is terminated by 5.654e-4-j1.846e+2Ω. We obtain 7.437e-4-j1.180e+2Ω. Expanding that, Xc=118.0Ω, ESR=0.0007437Ω, and Q=158.7e3. Xc is lower, ESR is higher, and Q is lower.

You could continue the analysis of the remaining 7 sections, Q will continue to fall at a decreasing rate. Though there are further ESR components in the connections to the external terminals, one can see how the series cascaded sections progressively deliver lower and lower Q, and why Q of these type of caps is more commonly specified of the order of 10,000 rather than 1,000,000.

## The obvious question

Since the Q of individual line sections may be very very high, why does the structure commonly connect them in series rather than parallel?