When an atmosphere or ionization system is to be evaluated, the charged plate is exposed to the atmosphere, charged to a selected voltage, and the time, t₁₀, for the voltage to drop to one-tenth of the starting voltage is measured. This time depends on the geometry of the CPM and the negative resistivity of the air if a positive voltage is chosen. Consequently, it is not possible to use a CPM measurement to predict directly how fast an arbitrary item will be neutralized, even when both the CPM and the item are exposed to the same atmosphere. However, in many cases, the neutralization time for an item can be deduced indirectly from CPM measurements.

Figure 1. Charged-plate monitor.

Theory of a CPM

In Part I of this article, it was shown that the charge on a positively charged body suspended freely in air will be neutralized exponentially with a time constant of

, (1)

where r^– is the resistivity of the air due to negative ions.¹

The time constant in an exponential decay is the time it takes the decaying quantity to drop to 1/e of the initial value, with e being the base of the natural logarithm (≈ 2.72). Consequently, for any system with the time constant t, there is the relationship

. (2)

The time constant t_o (known as the fundamental time constant) given by Equation 1 is the lowest time constant for any system in an atmosphere characterized by the parameters e_o and r.

Figure 2 shows a system consisting of a positively charged metal plate with the area A, separated from ground by a dielectric with the thickness d and the relative permittivity e_r. In Part I, it was suggested that the plate A, when exposed to an atmosphere with the fundamental time constant t⁺_o, would have a time constant t⁺ given by

, (3)

where

. (4)

 

To verify experimentally the relationship expressed in Equation 7, it is necessary to be able to establish an atmosphere with a known value for t_o. Although the fundamental time constant, t_o, cannot be measured directly, Equation 1 may be reformulated, writing r^– as

, (8)

where e is the electronic (and small-ion) charge, k^– is the mobility of negative ions, and n^– is the relevant negative-ion concentration.

Combining Equations 1, 4, and 5 gives the following results:

(9)

and

(10)

The constants in Equations 4, 5, 9, and 10 had the following values: r = 9 · 10^–2 m, d = 1.5 · 10^–2 m, e_o = 8.85 · 10^–12 F · m^–1, e = 1.6 · 10^–19 C, k^– ≈ 1.8 · 10^–4 m² · V^–1 · s^–1, and k⁺ ≈ 1.4 · 10^–4 m² · V^–1 · s^–1. Substituting these values into the relevant equations results in the following capacitance and decay-time values to be recorded on the CPM: C_d = 15 pF, C_a = 5 pF,

, (11)

and

. (12)

The setup shown in Figure 3 was used for the experiments. An ionizer was placed at a well-defined distance from an ion-density meter. Because an ion blower was used as the ionizer, the air intake of the ion-density meter was placed perpendicular to the airflow so that the airflow would not be disturbed through the meter. The CPM was placed on top of the ion-density meter. It is important that the field from the charged plate is insignificant at the air intake, and the placement on top of the ion-density meter turned out to be the best because the CPM also needed to be exposed to the ionized airflow.

Figure 3. Experimental setup.

A series of experiments were performed in which the positive- and negative-ion density and the corresponding decay times (t₁₀ values) were measured. The ion densities were varied by varying the distance a between the ionizer and the ion-density meter. For each distance, the average ion density and the decay times for positive and negative voltages were measured. The results are shown in Table I. Each of the decay times is the average of five measurements.

Distance a (m) n+ (1011 ions/m3) n– (1011 ions/m3) t+10 (seconds) t–10 (seconds) Ion blower 1 3.2–3.6 2.5–2.9 11.8 12.3 1.5 2.2–2.6 1.8–2.2 16.9 17.7 2 1.0–1.4 0.8–1.2 32.0 34.1 2.5 0.4–0.6 0.2–0.4 105 83.7 Ion bar 1 0.20–0.24 0.16–0.20 180 190

Table I. Ion concentrations and CPM decay times.

To compare the experimental results with the predictions of Equations 11 and 12, the products

and

were calculated with the following
results:

(13)

and

. (14)

It appears that the coefficients in the experimentally determined Equations 13 and 14 are 14–15% greater than the theoretical values of the corresponding Equations 12 and 11, respectively. This discrepancy must be caused by the approximation used in evaluating the air capacitance C_a (Equation 5), which overestimated the capacitance by about 19%. Therefore, Equation 5 should read

, (5a)

resulting in C_a = 4.2 pF for the CPM investigated. Correspondingly, Equations 6, 7, 9, and 10 become

, (6a)

, (7a)

, (9a)

and

. (10a)

Discussion of the Results

The equations developed in this article indicate that, by the use of a CPM with given geometric and dielectric characteristics, it is possible to predict the ion concentrations and resistivities of a given (ionized) atmosphere. However, a more-important implication is that it is, in many cases, also possible to estimate how fast an ordinary charged item (i.e., not just a CPM) will be neutralized in a given atmosphere.

For example, if Equation 7a is applied on the CPM used in this investigation (i.e., r = 9 ·10^–2 m and d = 1.5 ·10^–2 m), the result would be

. (7b)

Equation 7b indicates that, for a given atmosphere, the CPM will measure a time constant that is 4.57 times greater than the fundamental time constant for the atmosphere in question.

Now assume there is an item like that shown in Figure 2, with a charged metal plate A, an effective radius r_i = 0.1 m, a thickness d_i = 10^-2 m, and a relative permittivity e_r,i = 4.0. According to Equation 6a, the time constant should be

. (6b)

Combining Equation 6b with 7b results in

. (15)

It is therefore possible to use a CPM measurement to predict how fast a charged metal plate, separated from ground by a dielectric, will be neutralized.

Labeling the parameters of the CPM with c and of the item with i, the general equation relating a predicted neutralization time with the corresponding time measured by a CPM will be

. (16)

Equation 16 is valid for positive as well as negative voltages, corresponding to ions of opposite polarities, and for time constants as well as for t₁₀ times.

Insulators

As was explained in Part I, the neutralization of a charge located on the surface of a dielectric follows the same rules as that of a charge located on the surface of a conductive plate.¹ The equations in this article are therefore valid when r_i is the radius of the charged area, e_r,i is the relative permittivity, and d_i is the thickness of the dielectric.

Conclusion

It has been demonstrated that it is possible to use a CPM with given dimensions to determine fundamental atmospheric electric quantities such as ion density and resistivity. Furthermore, it has been shown that the neutralization time for commonly shaped charged conductors, as well as for insulators, can be deduced from measurements of the neutralization time with a CPM.

Acknowledgment

The cooperation of Ion Systems (Culemborg, The Netherlands) and Simco (Lochem, The Netherlands) for allowing the use of their ion blower and ion bar, respectively, is greatly appreciated.

Reference