Yes - you can measure battery voltage directly using the DrDAQ USB datalogger.
However, the oscilloscope channel will measure voltages up to 10V directly and the 3 x RJ11 analog input channels (on the right hand side, labelled Ext 1 ... 3) will measure up to (only) 2.5 Volts directly. You will be able to use a "voltage divider" using two resistors to measure higher voltages than these. See below.
The specification sheet provides information on voltage levels and general capability.
You do not need an oscilloscope probe. You can either use the oscilloscope channel with a BNC connector wired to whatever wires you want or you can use any of the 3 x RJ11 connectors (aka FCC68 4/4) using an RJ11 plug. RJ11 is used for telephone cables and many other applications and you can obtain one for little or nothing.
The "pinout" of the RJ11 connectors is not shown on the specification sheet (reference above) and there is also power out on these. Due to the cost of the board you will no doubt want to be a little wary when "playing" You may have a full manual that shown pinouts. If not then
Turn PCB over. Are two pins of each RJ11 connector joined together If so, these will be ground.
The RJ11 will probably use two pins for signal_in + ground and two pins for power_out + ground. The grounds will probably be joined.
With a voltmeter measure between each pair or pins to try to locate power and ground. Probably 0/5 or 0/3.3 or perhaps 0/3 or ? Pin combinations are 12 13 14 23 24 34 so you only need 4 measurements max. If you gt -5 or -3.3 etc swap leads to get +5 +3.3 etc. Then positive lead is + psu and -ve lead is ground.
Signal in is two leads that are not psu. Grounds are probably common at connector or elsewhere on PCB. It is better to use ground actually made for Vin if not commined at connector but should work well enough using either if there are two grounds.
They MAY have some other unction on the last pin eg signal_in, +V, ground, ???.
Asking thenm for pinout if not in manual would be a good idea [tm]
Using the oscilloscope input is the safest start. You will very very probably NIT hurt it with up to 10 V_DC applied (or quite a bit more you'd hope. Scope input has 4 ranges which are software switched - hopefully using supplies cope software will allow you to choose range.
Using resistors to make a voltage divider.
Two resistors can be used to scale battery or other voltages down to the level acceptable to the DrDAQ board. In the diagram below the battery to be measured is Vin and the output to the DrDAQ is Vout. The voltage scaling down ratio is
R1 + R2 could be a "potentiometer" (aka pot, variable resistor ...) if you want to play. A 10K pot would (probably) be a suitable value. Note that using a pot or divider of your choive you can apply more voltage to the input than its maximum rating. I'd hope that DrDAQ were competent enough to protect their inputs to well above rated value BUT you need to see what they and their data sheet says.
If you make R1 = 10k then if R2 = 2.2k the scaling range is 3.3:1. If you make R1 = 1k and R2 = 6.8k then the scaling factor is 7.8:1 etc.
I'd recommend making R1 = 1k to start playing then later try R1 = 10k. It MAY be you can use R1 = 100k but that will probably introduce inaccuracies.
The above diagram is from here Worth a read for you I think.
I do not know how you accss the EXT channels using the supplied software. Using the BNC scope input it should be aasy to use the supplied software. The Ext inuts may be easily accessible with the supplied softwae, or not. If not then using the scope BNC input would be a good starting point. The resistor scaling applies, but the R1=100k is liable to work more accurately than with Ext inputs.
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