Though this is an old post there seems to be renewed interest in measurement of small currents.
Many approaches to measuring leakage currents have been provided in previous replies but direct measurement of the current has advantages. One of the most effective ways of measuring small currents is the transimpedance amplifier, with the basic circuit shown (Fig.1). For this the output is simply:
v(out) = i . RF or v(out) / i = RF (1)
Thus for any given current it is necessary to select a suitable feedback resistor RF. The ‘gain’ of the amplifier is thus RF and the unit is ohm.
However there are a number of important additional matters to be considered. For very low currents, of the order of say pico or femtoamp, the resistor then has to be very high: for a current of 1pA and a full scale output of say 10V then RF =10Gohm (10^10), and for 1fA, RF =10Tohm (10^13). Though such resistors are available they are generally rather expensive. The amplifier A must have a bias current low relative to the current i to be measured. There is necessarily an input capacity CI (deliberate or stray, Fig.2) which will tend to cause dynamic instability. To counteract this it is necessary to add a suitable feedback capacity CF (and there is in addition the self capacity of RF). CI should be kept to a minimum to minimize noise [3, 4]. The combination RF of and CF leads to a system bandwidth of:
f = 1/(2pi . RF . CF) Hz (2)
and though CF may be small RF can be large so the bandwidth will be low. It can be shown that adding the components R1 and C1 (Fig.3) [ref 1, 2] such that:
R1.C1= RF.CF (3)
will provide some compensation for the restriction of the bandwidth. The system will again be tending to instability, but splitting R1 as shown in Fig.4 (R2<< R1), but still with:
(RF+ R2)C1= RF.CF (4)
provides a damping term in the transfer function that allows adjustment of stability. The bandwidth is now:
f = 1 / (2 pi . R2 . C1) Hz (5)
a considerable improvement over (2). Connecting the component under test (DUT), say CT together with a bias voltage VT, as shown in Fig.5, now allows measurement of the leakage current. The voltage VT should be increased slowly so that the CT charging current does not overload the system.
Since high value resistors can be (very) expensive it is worth considering an alternative approach which eliminates this component. If RF is removed and CF increased then the input current will charge CF and for a fixed i the output v(out) will be a ramp. CF must be a good quality capacitor with low leakage. If Q is the charge on CF and t is time, then:
CF = Q / v(out) and since Q = i . t
= (i . t) / v(out)
or v(out) = (i . t)/CF and hence
dv(out)/dt = i / CF or i = CF. dv(out) / dt (6)
so measuring the rate of output change dv(out)/dt and knowing CF allows the determination of i. CF needs to chosen to suit the magnitude of i and some means of resetting by discharging CF provided.
Though in the past it has been necessary to use special opamps in TO5 cans (that use high resistivity glass seals and hence are rather expensive) it is now possible to use some low cost CMOS type opamps in high resistivity DIL epoxy packages that can provide very low bias currents e.g. LMC6041/61/81. The LMC6041datasheet quotes a bias current of about 2fA, though it has a very restricted bandwidth. In mounting the opamp the input pin should be bent out (or the IC mounted upside-down with pins in the air) and not connected to anything other than the input and the feedback resistor, thus avoiding any additional leakage. Remember that in using very high value resistors there are no insulators, only other high resistances.
Some additional information is available in references [3] and [4].
[1] Pelchowitch I., Zaalberg van Zelst J.J. (1952): A Wide-Band Electrometer Amplifier;
Rev. Sci. Instrum. 23, 73-75.
[2] Hamilton S (2007) An Analog Electronics Companion (Cambridge University Press
ISBN 0780521687805), section 5.12, p488.
[3] Burr-Brown: Photodiode monitoring with op amps; Burr-Brown Application Bulletin
AB-075. See
Analog, Embedded Processing, Semiconductor Company, Texas Instruments and search for sboa035.
[4] Burr-Brown: Noise analysis of FET transimpedance amplifiers; Burr-Brown
Application Bulletin AB-076. See
Analog, Embedded Processing, Semiconductor Company, Texas Instruments and search for sboa060.