Speed ​​considerations when making level signal measurements

The relationship between time and frequency

There is no conceptual difficulty in understanding what happens when a steady-state DC signal is applied to a voltmeter. However, if the signal has a time-varying component, such as an AC signal superimposed on a DC signal, the meter will follow the varying signal and indicate the instantaneous amplitude of the input signal. As the frequency of the AC component increases, the response of a DC meter becomes insufficient until, at a certain frequency, the meter displays only the average value of the input voltage. The frequency at which the voltmeter's response to an AC signal drops to 70% is often called the "3dB point" (f3dB). The bandwidth of a DMM is roughly half the rate at which its displayed readings change (the number of readings per second). Except in the case of reconversion from digital to analog, the bandwidth of the meter's analog output is generally much wider.

Bandwidth describes the ability of an instrument to respond to time-varying signals over a certain frequency range. Another measure of the speed of an instrument's response is its ability to respond to a step function signal. A typical measure of this response is the instrument's rise time. Bandwidth or rise time can be used to describe how well an instrument responds to time-varying signals.

The rise time of an analog instrument (or analog output) is generally defined as the time required for the output signal to rise from 10% to 90% of its final value when the input signal rises immediately from zero to a fixed value. This relationship is shown in Figure 2-46. Figure 2-46a shows a step function assuming a zero rise time, while Figure 2-46b shows the instrument's response and the corresponding rise time. The rise time, frequency response, and RC constant of a single-pole system (1st order system) are related. The 3dB point is given by:

The relationship between the rise time ( ) and the RC time constant is as follows:

For example, the rise time of a circuit with a source resistance of 1TΩ and a capacitance of 100pF is approximately:

Using the above relationship between RC and f3dB, we can see that:

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Therefore, a source resistance of 1TΩ and a capacitance of 100pF will limit the bandwidth to:

When the rise time and the measurement period are of the same order of magnitude, the accuracy of the measurement is affected. If the time allowed before taking a reading is equal to the rise time, an error of about 10% will occur because the signal can only rise to 90% of its final value. To reduce the error, you must wait longer. To reduce the error to 1%, you must wait about twice the rise time. And to reduce the error to 0.1%, you must wait about three times the rise time (or nearly 7 time constants).

In cases where the required error is better than 0.1% (sometimes 1%), two-pole effects kick in. For example, due to dielectric absorption of the insulator and other second-order effects, it will typically take more than 4 times the rise time to reach 0.01% of the final value.

In general, the response of an analog instrument (or the analog output of most digital instruments) to a varying input signal is a function of its bandwidth, since frequency response is directly related to rise time. To ensure accurate measurements, sufficient settling time must be allowed after the input signal is applied so that the source, the instrument's connections, and the instrument itself settle to their stable conditions.

Effect of Input Capacitance on Rise Time and Noise

Voltage measurement

When measuring voltage across a high impedance source (Figure 2-47), the capacitor (CIN) across the voltmeter (VM) must be charged through RS. The output voltage as a function of time is:

VM = VS (1-et/RSCIN)

Where: VM = the voltmeter reading at t seconds

VS = step function voltage source

t = time in seconds after the step occurs

RS = Equivalent series resistance in ohms

CIN = Equivalent parallel capacitance in Farads (capacitance of the instrument plus capacitance of the cable)

This results in the familiar exponential curve shown in Figure 2-48. You must wait 4 to 5 time constants to get an accurate reading. With large resistors and capacitors, the rise time can be several minutes. Adding more shunt capacitance increases the rise time, but it reduces the effective bandwidth of the voltmeter, thereby filtering out noise generated by the source and interconnecting cables.

Shunt current measurement

When using a shunt ammeter (Figure 2-49), the effect of input capacitance on current measurements is similar to that of voltage measurements. A shunt ammeter can be thought of as a voltmeter with a resistor connected across its input terminals. The circuit shows that the input capacitor (CIN) must charge to ISRS volts at an exponential rate with a time constant of RSCIN. Note that CIN is the sum of the source, connecting cable, and meter capacitances.

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Feedback current measurement

The effect of input capacitance on an ammeter using negative feedback is different from that on a shunt-type ammeter. The circuit for this mode is shown in Figure 2-50.

If the amplifier gain A is large, then V0 = -IINRFB. In this case, CIN does not shunt RFB. The effect is small compared to the case of the shunt picoammeter. The reason for the increased speed is that the input impedance of the picoammeter is reduced due to the negative feedback. In other words, the voltage developed across CIN is only VS = -V0/A volts, while the voltage is V0 for the shunt picoammeter. Therefore, even if the capacitor connected in parallel with the input is large, its effect on the rise time is small.

The rise time of a feedback picoammeter is a function of the physical or parasitic capacitance in parallel with the feedback resistor (RFB). Electrometers, SMUs, and picoammeters can all use larger values ​​of source capacitance. It should be recognized that increasing the value of the input shunt capacitance (including the parallel effects of the source, cable, and input capacitance) will degrade the signal-to-noise ratio of the measurement.

Resistance measurement (constant current method)

Input capacitance affects resistance measurements in the same way (Figure 2-51). In this case, CIN must also be charged by a current (IR), so the same formula applies.

Electrometer Rise Time Summary

For most high impedance source measurements, it is important to minimize the capacitance in parallel with the meter input when considering rise time. As mentioned earlier, this also reduces noise gain. Broadly speaking, the source impedance should be large compared to the meter's feedback impedance.

The most effective way to reduce input capacitance is to connect the electrometer, SMU, or picoammeter to the signal source with the shortest possible shielded cable. When measuring voltage from a high impedance source or measuring high resistance, guarding techniques can minimize the effects of input capacitance because the inner shield of the triaxial cable or the shielded box surrounding the input is driven with an appropriate potential, thereby minimizing the effective capacitance.