## An electrical theory of the alternating current spectrographic spark source

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https://hdl.handle.net/2144/6810##### Abstract

The alternating Current Spark Source has become the principal mode of excitation for industrial spectrographic analysis.
This paper approximates the very complicated exact problem of the electrical behavior of the Alternating Current Spark Source by reduction to problems for which direct solutions are possible. By means of this treatment it establishes the relations between the two dependent variables, discharge repetition rate, and r.m.s.r.f. current, and the measurable independent electrical parameters. The total list of variables is reduced to those which from the physics of the solution as presented provide a unique description of the electrical source.
The most general form of the Alternating Current Spark Source can be represented by the following electrical circuit. [Figure I: Alternating Current Spark source in an electrical circuit]
Where G is the effective discharge gap
L is the discharge circuit inductance
RRis the discharge circuit resistance
C is the secondary capacitance
RS is the resistance of the transformer secondary
X is the transformer
and RP is the resistance of the primary circuit.
In this treatment the source is approximated by two principal circuits which are assumed to function independently.
The first of these is called the charging circuit. It is taken to be an R-C circuit in which a transient current is developed by the introduction of an effective D.C. voltage from the transformer at time zero.
This circuit is shown schematically by the following diagram [Figure 2: A circuit with an effective D.C. voltage from transformer at time 0]
Here EDC is the effective D.C. voltage impressed on the circuit at time zero, RS is the effective secondary circuit resistance, including such resistance as is reflected into the secondary from the primary by the use of the turns squared ratio, and C is the secondary capacitance. Thus the transient equation for the instantaneous charge q in the condenser C is (dq/dt)+(q/RSC)=(EOC/RS) during the charging cycle.
The boundary condition is, q = 0 at t= 0.
The second circuit is an R-L-C circuit including the discharge gaps. A transient current is developed in this circuit by the effective reduction of the gaps to a low resistance during discharge, when the voltage of the capacitance reaches a given value.
The discharge circuit is represented schematically in the follorring diagram. [Figure 3: discharge circuit] where the parameters are the same as in the overall circuit shown above. It should be noted that R includes the effective resistance of the discharge gaps. Thus the transient equation for the instantaneous charge q in the condenser C is, (L(d^2*q)/(dt^2))+(RR(dq)/(dt))+(q/C)=0 during the discharge cycle.
The boundary conditions are:
at time t = 0, q =V0 and (dq/dt) = 0.
By the use of the Charging Circuit approximation the following things are shown:
1. The efficiency of the source in making energy available for the discharge can never exceed fifty per cent, and is in fact, n = ((1/(2)^.5)) ((V0/ES)/((1+(V0/ES))^.5) where n is the efficiency of the system, V0 is the gap breakdown voltage, and ES is the peak rated secondary voltage of the transformer.
2. The discharge repetition rate, n, for a 60 cycle input is given to a reasonable approximation by the relation n-1 = IP(1-(2/π)sin^-1(V0/ES))/ 196NErmsClog(((1+(V0/ES)^.5)/((1+(VO/ES)^.5)-(VO/ES*((2)^.5))
Where Ip is current in the transformer primary,
N is the turns ratio of the transformer,
Erms is the rated r.m.s. secondary voltage of the transformer, and
C is the secondary circuit capacitance.
Throughout the treatment of the charging circuit the approximation is made that the duration of the discharge, during which the voltage on the capacitor C drops to zero, is negligible. In other words, the instant the voltage on the capacitor C reaches the discharge voltage V0, this voltage drops to zero, and the charging cycle starts over again.
By the use of the discharge circuit approximation the following things are shown:
1. Peak r.f. current is shown to be
ipeak = -V0((C/L)^.5)e^((πR/4)((C/L)^.5))
Where V0 is again gap breakdown voltage
C is secondary circuit capacitance
L is discharge circuit inductance
and R is discharge circuit resistance.
A graphical representation of this relation is given in figure 11.
2. The mean r.f. current, which is related to overall light intensity output of the source is shown to be Irms^2=((60V0^2CN)/R)(1-(e^(-RT/L))for a 60 cycle input.
Where V0 is gap breakdown voltage
C is secondary circuit capacitance
L is discharge circuit inductance
R is discharge circuit resistance
n is discharge repetition rate
and T is the time from initial breakdown to total quenching of any discharge.
Throughout the treatment of the discharge circuit it is assumed that the current supplied directly from the transformer is negligible compared with the circulating r.f. transient current developed in the discharge circuit.
The following parameters are established as sufficient to describe an Alternating Current Spark Source.
Ip - the primary current
N - the transformer turns ratio
Es - the peak rated secondary voltage of the transformer
V0 - the discharge voltage
C - the secondary capacitance
n - the number of discharges per half cycle
R - the series resistance of the discharge circuit
L - the series inductance of the discharge circuit
T - the duration time of a single discharge.
The duration time T of the discharge is only important in those cases where the series inductance L is appreciable. If L is small, the mean r .f. current equation reduces to Irms^2 = (60(V0^2)Cn)/R)
When inductance is added to the source, the term (1-e^(-RT/L)) must be retained. It is shown that T is dependent in a complicated way upon the capacitance and the inductance, and a first approach is made to a graphical presentation of this relationship. This is shown in figure 15.
As indication of the validity and the value of this development, the following things are shown.
1. The energy dissipated in the discharge circuit, as computed from the above developed equation for Irms, is exactly equal to the energy made available to the discharge circuit, as expressed simply by (1/2)C(V0^2). This indicates that the theory has retained energy relationships correctly in the circuit.
2. The predicted values of Irms as obtained from the above relations correspond very satisfactorily with data obtained experimentally from an actual spark source.
3. The theory in the case of high inductance values produces values of discharge duration which are of the order of magnitude indicated by existing experimental evidence.
4. The theory developed above is capable of including and explaining the observed phenomenon of linearity in the decay of instantaneous r.f. current for small discharge circuit resistance.

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Thesis (Ph.D.)--Boston University