Introduction
Coaxial cable is commonly used to carry radio frequency signals and is most commonly encountered in TV and radio wiring. It is typically used because it is extremely well shielded from interference from external electromagnetic fields, because it doesn’t radiate when carrying high frequencies and because it has well-defined impedance to high frequency signals.
This blog post describes its use at lower frequencies, particularly its use in experiments where a capacitance must be measured remotely with some length of cabling between the capacitor and sensing circuitry. This is particularly relevant to cryogenic experiments where capacitances often must be measured at low temperature inside a cryostat but the electronics must be kept some distance away, at room-temperature. In this case, we are using the coax purely for its shielding properties, and we don’t care about the other properties which make it ideal for RF use.
No detailed knowledge of electronics is required, so many of the electronics justifications will seem overly simplistic and many researchers will be able to skip straight to our next blog post; Fitting coax cabling into a cryostat: practical challenges and considerations.
What is Coaxial Cable?
Coaxial cable simply consists of an inner wire, the core, surrounded by a second wire, the braid, which entirely encloses it but is electrically isolated from it. Various types are available with single or multistrand cores and the outer may consist of a solid tube or in most cases a conductive sheath made of braided wires (hence the outer is often referred to as the braid) or foil tape. The ones with solid tube, known as rigid or semi-rigid coax, are designed for high frequency use, but as we’re only interested in shielding, the flexible types are sufficient and rather easier to work with.
Challenges in measuring capacitance
Naively it might be expected that the capacitance could be simply measured by connecting your capacitor plates up to your cryostats wiring and measuring the capacitance externally with a simple hand-held LCR meter. The reason why this approach falls down when a long wire is used, is that the wire itself will have its own capacitance. In most research situations, the capacitance being measured will be small and this parasitic capacitance of the wires will be larger than the signal. The LCR meter will simply be measuring the capacitance of the wires to their environment – junk! This is shown in the simple circuit below.
Figure 1. Though we’re only interested in measuring Cx the wires attached to each capacitor plate have introduced their own parasitic capacitance, completely obscuring Cx.
Using coaxial cables correctly allows the parasitic capacitance to be rejected. To the uninitiated, it can be surprising that to reject parasitic capacitance in the wires, we switch to coaxial cabling which has one of the largest capacitances of any cable – coax commonly has a capacitance of 100 pF/m, many times more than a typical unshielded wire.
The reason why the coax cabling is typically used is so the parasitic capacitance can be controlled and rejected. With a typical wire, the parasitic capacitance is caused by the wire acting as one of the plates in a capacitor, the insulation on that wire acting as the dielectric, and the other bits of random conductive items that the wire happens to go near acting as the other plate. The issue is that the random conductive items are an unknown size and distance away and hence an unknown capacitance. If the wire or anything else moves, the capacitance will change. They are also at an unknown potential, if they are other wires carrying other signals, then those signals will couple into the capacitance measurement.
Figure 2. A simple capacitive voltage divider.
In the absence of a parasitic capacitance, a typical way to accurately measure a capacitance is make a half bridge circuit or potential divider. This allows the determination of an unknown capacitance by comparing it to a high precision calibrated capacitor.
If Cref is known, it is trivial to determine Cx because Cx/Cref=V1/V. As before discussed, if long wires are used this just isn’t a practical approach because the wire to both sides of Cx will have their own parasitic capacitances which are impossible to distinguish from CX
The auto-balancing bridge method of determining capacitance
Rather than measure V1, as in the previous circuit, an alternative circuit could be to use an amplifier to hold the centre tap at ground. Because this point is held at ground but not directly connected to ground it is typically called a virtual ground.
In the circuit to the right, Cx can be determined by knowing the output from the ADC, V1. For example, if Cref=Cx then V1 will equal to the excitation voltage, V. More generally V1/V=Cref/Cx. The reason why this is a more useful approach is that holding the centre tap at a virtual ground using an amplifier offers a promising way to reject the parasitic capacitance if used in conjunction with the coax cables. Let us draw the circuit again with the parasitic capacitance explicitly drawn. CP1 is the parasitic capacitance of the driven ‘HI’ cable, and CP2 the parasitic capacitance of the detected ‘LO’ cable.
Figure 3. A simple auto-balancing bridge circuit
Below is the simplest circuit that can be used to detect and small capacitance Cx, while rejecting larger parasitic capacitances. Most bridges and LCR meters will use more sophisticated circuits which will be able to measure complex impedance and provide better signal to noise.
Figure 4. An auto-balancing bridge circuit showing explicitit parasitic capacitances being rejected.
The circuit is driven through the HI cable with an oscillating voltage V. The parasitic capacitance of the HI cable, CP1 is irrelevant in this case – it will affect the amount of current drawn but little else as HI is merely for excitation, not detection, and is driven by a low impedance source. This means that the only parasitic capacitance that needs to be worried about is CP2, the parasitic capacitance of the LO cable. If coax cables are used the braid can be attached to ground, meaning that both the parasitic capacitances will also be to ground. This is very convenient because as in the previous circuit, the centre tap is held at a virtual ground – so that CP2 will be the capacitance ground to ground which will have no effect on the circuit. Consequently V1/V=Cref/Cx is just as relevant – both CP1 and CP2 are irrelevant to the measurement. This neat trick would not be possible without coax on LO because the parasitic capacitance would not necessarily share a common ground with the amplifier and could not be perfectly rejected. This type of capacitance measuring circuit is called an auto balancing bridge and slightly more sophisticated versions are commonly used in high precision LCR metres.
Hopefully that has explained the importance of coaxial cables in measuring capacitance in inaccessible environments in a research context. The next blog post will explain what should be considered when fitting a coaxial cable into a cryostat for low-temperature use.