## How a multimeter measures capacitance

13.03.2020 10:57

I've recently bought a Keysight U1241C multimeter. One of the features it has is a capacitance measurement. Since this is my first multimeter that can do that I was curious what method it uses. I was also wondering what voltage is applied to the capacitor under test and whether the probe polarity matters (for example, when measuring electrolytic capacitors).

The Figure 2-16 in the User's Guide seems to imply that polarity is important. The red probe (V terminal) is marked as positive and the black probe (COM terminal) is marked as negative:

Image by Keysight Technologies

The description of the measurement method is limited to this note and doesn't say what voltages or frequencies are involved, but does give a rough idea of what is going on:

Image by Keysight Technologies

Connecting an oscilloscope to a capacitor while it is being measured by the multimeter reveals a triangle waveform. I made the following screenshot with a 47 μF electrolytic capacitor connected to the multimeter set to the 100 μF range. The oscilloscope was set to DC coupling, so the DC level is correctly shown as 0 V at the center of the screen:

Since current into a capacitor is proportional to the time derivative of the voltage, a triangle-shaped voltage means that there is a constant current flowing alternatively in and out of the capacitor. Connecting different capacitors revealed that the current and the amplitude of the voltage stay constant for each measurement range, while the period of the signal changes. So the multimeter applies a known current source I to the probes and measures time t it takes for the voltage to rise (or fall) for a present difference Upk-pk. From the measured rise (or fall) time it then calculates capacitance:

C = \frac{I\cdot t}{U_{pk-pk}}

These are the approximate current and voltages used by the multimeter for each range:

Range [μF] I [μA] Upk-pk [mV]
1 1.5 800
10 15 800
100 150 800
1000 340 200
10000 340 200

Note that 1000 μF and 10000 μF ranges seem identical in this respect. I'm guessing the only change is how the time is measured internally. Perhaps a different clock is used for the counter.

If a high range is selected while a small capacitor is connected, the voltage on the capacitor can reach much higher amplitudes. The highest I saw was about 2 V peak-to-peak when I had a 4.7 nF capacitor connected while the instrument was set to 100 μF range.

In conclusion, the polarity of the probes isn't important. The applied signal to the capacitor is symmetrical and the capacitor will be alternatively polarized in the positive and negative direction regardless of how it is connected to the multimeter. The voltages do seem low enough that they probably don't damage polarized electrolytic capacitors.

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## The tiny curve tracer project

08.03.2020 10:14

About a year ago I got interested in some unusual transistor characteristics. Since I didn't have any suitable instruments at hand I first measured things with an oscilloscope and some improvised circuits on a protoboard. These setups gradually became more elaborate and for some time now I had a dusty rat's nest of wires on my desk that more or less implemented a simple curve tracer. It soon turned out to be useful in other cases as well, so I thought it would be worth moving the circuit from the protoboard to an actual printed circuit board.

The construction is through-hole on a single layer PCB. I've decided on this slightly vintage style because I could just move the components from the protoboard to the PCB without having to buy their surface-mount equivalents. My Dad offered to etch the board for me using the toner transfer method and it turned out very nice with 20 mil design rules. He made the overlay print on the component side for me as well. I've not etched a board at home in years, ever since cheap on-line prototyping services became available.

It took me quite a while to decide on what kind of contacts to use for connecting the device-under-test (DUT). I've considered a transistor socket (too limited in the pin diameters), a Textool-type ZIF socket (seemed wasteful to use a 16-pin socket when I only needed 6 pins) and just ordinary screw terminals (inconvenient to use). In the end, I went with a WAGO push-button style, 3.5 mm pitch terminal block (type 250-206 to be exact).

This seems to work really well and the only slight problem I have with that is that when inserting a transistor you need to push three buttons at the same time. Since the springs are quite stiff, this requires a fair amount of force. If I would be doing a second revision of the board I would make some provisions for better supporting the PCB around the terminal since it tends to flex quite a lot.

My circuit obviously isn't on the level of something like a Tektronix 575. The signal that will be applied to the DUT, usually a sine or a triangle wave, comes from a signal generator via a BNC connector, shown on the left of the block diagram. I'm using my GW Instek AFG-2005. After the input there are two amplifier stages and a power transistor in an emitter-follower configuration. Total voltage gain from input to DUT terminal is 5. The actual current and voltage on the DUT are amplified and passed to the BNC outputs. I use two 50 Ω coaxial cables to connect the outputs to an oscilloscope in XY mode.

A switch allows the input to be either AC or DC coupled. AC coupling allows me to quickly change the amplitude on the signal generator without having to simultaneously also adjust the offset voltage. In this case the circuit clamps the lowest voltage to 0 V. On the other hand, DC coupling allows me to put the DUT under some DC bias and only measure some small signal characteristic.

The curve tracer is powered by a bench power supply with around 30 V and doesn't need a negative supply. It is capable of providing up to about 25 V of voltage to the DUT and around 200 mA of current. 25 V isn't enough to reach collector-emitter breakdown voltages of common transistors, but it is plenty to investigate knee regions of diodes or breakdown in low-voltage Zener and TVS diodes. For example, this is the forward characteristic of a 1N4148 diode I measured:

It's also possible to measure base-emitter junction characteristic in bipolar transistors. Here is a BC546, a common small-signal NPN transistor, in forward and reverse directions. Breakdown in the junction is visible in the reverse direction at around 11 V:

There's a protection circuit that turns off the voltage to the DUT, and lights up a warning LED, if DUT current exceeds 200 mA. The protection gets automatically reset when the input voltage goes to 0, which usually means on the next period of the signal generator. This protection is more about saving the curve tracer than the DUT. It's still perfectly possible to obliterate a small transistor that can't handle 5 W of dissipation.

The power to the DUT however can be limited by connecting a load resistor in series, which is similar to how the old Tektronix instruments did it (they came with this nice resistor selection graph). I've left two contacts on the DUT terminal for the load resistor connection. On the photo above they are shorted with a wire jumper.

One of the things I aimed for was also support for high frequency signals since I wanted to observe some dynamic effects. Traditional curve tracers only use 100 Hz or 120 Hz. Above you can see some reference measurements of a 100 ohm carbon-film resistor I did at different frequencies. Ideally, all plots should show a line with a 10 mA/V slope and the same length. However at around 100 kHz the amplitudes start falling and voltage and current are no longer in sync, which causes the curve to open up into an ellipse.

The performance is mostly limited by the slew rate of the amplifiers, especially when observing fast edges, like in an avalanche breakdown. I'm currently using the excellent 4.5 MHz Renesas CA3240, which are one of the fastest operational amplifiers I could get in a DIP package. There's also a limitation on the DUT capacitance. Since I'm using an emitter-follower and not a push-pull output stage, my curve tracer can only tolerate about 150 pF of DUT capacitance at 100 kHz.

Tweaking the high frequency performance was the most challenging part of designing this. Matching the delay between voltage and current outputs involved a lot of experimenting by adding small capacitances into the feedback networks. Despite this effort however each measurement still involves some double-checking to be sure that I'm seeing a real effect and not an artifact of the instrument.

In conclusion, I'm reasonably happy how this turned out. Obviously, there are some limitations and in a second iteration of the design I would do some things differently. For example the decision to go with 10 mA/V current output wasn't the best. Decreasing the gain would reduce the slew-rate requirements without losing much precision. The clamp circuit also doesn't work very well at high frequencies due to the opamp going into saturation and could do with a redesign. In the end, it was a useful refresh of my knowledge about various details of imperfect amplifiers.

I've also found out that the XY mode on my Tektronix TDS2002B seem to be a bit crude. You can't have cursors and for some reason it's also impossible to save a screenshot. Also, while the scope does support current probes, the 10 mA/V range is missing so I can't have the proper scale displayed on screen. While it's a bit annoying, it's not too much extra work to save individual traces to CSV in YT mode and then re-plot them with matplotlib or something when I have a measurement I want to save.

Another thing is that there's currently no step generator or any other provision for setting the base current (or FET gate voltage). However I did reserve a contact on the DUT terminal for the base/gate pin and there's a header already wired for a possible future expansion board with that capability. I might make that add-on at one point in the future but currently I'm not missing it too much since I've mostly been using the curve tracer with two-terminal devices.

Finally, if you're interested in this sort of things and maybe designing or building your own curve tracer, I can recommend reading Paul's Building Another Curve Tracer post. He goes into much more detail about the design of his own instrument. Another very useful resource I found is the TekWiki which contains a staggering amount of information about old Tektronix instruments, including manuals and often also full schematics.

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