THE EQUIVALENT SERIES RESISTANCE IN ELECTROLYTIC CAPACITORS

Electrolytic capacitors can be represented by an equivalent circuit consisting of a series combination of R, L and C elements. A typical impedencefrequency characteristic in shown in Figure 1. The impedance becomes inductive at frequencies which are for most applications beyond the region of capacitor operations. The equivalent series resistance R of the capacitor is responsible for heat generation and temperature rise in the capacitor. It also has a damping effect in fast charge/discharge applications. Therefore, it is desirable to reduce the equivalent series resistance (E.S.R.) to a minimum.


INTRODUCTI ON
Electrolytic capacitors can be represented by an equivalent circuit consisting of a series combination of R, L and C elements. A typical impedencefrequency characteristic in shown in Figure 1. The impedance becomes inductive at frequencies which are for most applications beyond the region of capacitor operations.
The equivalent series resistance R of the capacitor is responsible for heat generation and temperature rise in the capacitor. It also has a damping effect in fast charge/discharge applications. Therefore, it is desirable to reduce the equivalent series resistance (E.S.R.) to a minimum.

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The construction and performance of electrolytic capacitors have been reviewed extensively. [1][2][3][4][5] The most common type of construction consists of an etched aluminium anode and high grain cathode foils separated by electrolyte impregnated low density tissues. The dielectric is provided by the anodization of the anode, and its thickness is determined by the anodization voltage and other anodization parameters, such as temperature and the constituents of the anodization solution. 6 The total impedance of a capacitor has been evaluated by treating the capacitor as a distributed network. This has been applied to plain foil capacitors '/and to the evaluation of the impedance of tunnels in etched foils. 3 More elaborate study has been made of various types of electrolytic capacitors including sintered-anode, etched foil, vacuum deposited and plain foil capacitors. 8 The use of two transmission line treatment has recently been employed in the analysis of an etched foil electrolytic capacitor. 4 The exact transmission line model is extremely complex for etched foil capacitors. However, many approximations can be introduced without changing the final results drastically. The following approximations can be made" 1) Inductive effects due to winding and tunnels can be ignored. In practice, inductance is mainly due to lugs and leads. 4 2) The capacitance of impregnated tissues is insignificant compared with the capacitance of the anode.
3) The capacitance and resistance of the cathode foil can be lumped together with those of the anode foil. This is especially true for low voltage capacitors and high gain cathode foil.

4) Interaction between the turns of the wound
capacitor can be ignored. The effect of winding is accounted for merely by doubling the admittance of the unrolled plates. This results from the utilization of both sides of each foil when rolled.
With the above simplifications the equivalent circuit of a unit length of an unrolled capacitor can be represented by the circuit shown in Figure 2. The analysis of this circuit shows that below two critical frequencies fc and fc the capacitance C and resistance R of the capacitor are given by: R -Rt + %Z, Rp + %Rf +Rox + Ri (2) where C o is the lumped capacitance of a single runnel and the summation is for all tunnels on both sides of Rp is the resistance of dectrolyte in a tunnel.
Rt is the resistance of electrolyte impregnated tissues separating the anode and cathode foils.
Rf is the resistance of anode and cathode foils.
Ri is the resistance due to lugs and connections.
Rex is the resistance due to dielectric losses. This is given by: where tan 6 o is the loss of the oxide. The two critical frequencies are given by fc, 2zrRpCp (4) fc2 2rCRf (5) Above these frequencies the resistance and capacitance of the transmission line begin to fall with frequency according to f-v2 law.4 The oxide loss is almost frequency independent and thus the resistance Rex is expected to fall linearly with frequency. It is therefore expected that at high frequencies the E.S.R. is mainly due to the resistance of tissues and small contributions from lugs and connections.

EXPERIMENTAL
A large number of capacitors were made under production conditions. Plain and etched foils were used together with four types of electrolytes and varying anode-cathode separation. A batch of five capacitors was made for each specific type, and the average of each batch was taken.
In all cases the anode plate was formed at 115 V.
The capacitance of the plain anode foil was 0.78 mF/ m 2 while that of the etched anode was 12.5 mF/m2.
In small capacitors, e.g. 1 btF, the series resistance due to oxide is approximately 32  where L is the length of the plate. In the test capacitors pf was found to be 1.6 x 10 -a 2/square, k 12.5 mF/m 2 and plate length was 0.25 m. These figures give f in the region of 150 kHz. For most practical cases fc is much lower than this value as the foil capacitance is larger and so is the length of the plates.
It is evident from above that the contributiom of electrolyte in tunnels and the foil resistance are present in the region of 100 Hz, while at 150 kHz and above these are expected to become negligible.

Contribution of Connections and Foil Resistance
The contribution of connections (lugs, leads and terminals) to the total resistance is usually negligible, especially in small capacitors. For capacitors with axial leads and with lugs rivetted to the deck and stitched to plates the resistance R was found to be in the region of 5-10 milliohms. The resistance of anode and cathode foils were measured using straight forward d.c. technique. The total resistivity of anode and cathode was found to be 1.6 x 10 -a 2/square (per unit thickness of foil). For plate of size 0.025 rn x 0.25 m the resistance Rf is 16 milliohm. In the worst case where foil length is 100 times the width (this gives a capacitor with equal height and diameter) the resistance Rf is 0.16 Transmission line analysis shows that only one third of this is contributed to E.S.R.

Contribution of lmpregnated Tissues
The resistance of impregnated tissue with resistivity Pt is given by the relationship" t Rt Pt - (10) where t is the thickness of tissues and A is the area of the capacitor plate.
According to analysis, the resistance E.S.R. at high frequencies is predicted to be mainly due to tissues. The experimental vertification of this requires measurements of high frequency resistance for different electrolytes and tissue thickness. The electrolytes shown in Table I were used. Figure 3 shows the dependence of the high resistance on tissue thickness for several electrolytes. The frequency of measurements were chosen where the impedance was minimum (Figure 1). In all cases this frequency was above 100 kHz, which was well above the critical frequencies.  is the dependence of the high frequency resistance on electrolyte resistivity (Figure 4). The differences between the tissue resistances in etched foil and plain foil capacitors for the same electrolyte cannot be attributed to any one cause with certainty. It could be the result of variation in the resistivity of one type of electrolyte at different impregnation cycles. It could also be due to different winding tension in the capacitors, and the timetemperature history during and after impregnation. 9 The results in Figure 3 dearly indicate the direct proportionality between resistance and tissue thickness. This suggests that the high frequency resistance is due to impregnated tissue. Further evidence of this

Contribution of Tunnels
Since it is expected that at high frequencies the resistances of oxide, foil and tunnels become negligible it is possible to estinaate the contribution of tunnels from high frequency data R[ and those at 100 Hz. The tunnel resistance can be evaluated from: ERp =3 [R(oo nz)-Rox(oo nz)-R[] (11) The term Rox(oo nz) due to dielectric was calculated using Eq. (9) and assuming a dielectric loss of 2%. The contribution of tunnels is expected to be independent of tissue thickness and dependent of electrolyte resistivity. The results in Table II show the calculated values of ,Rp for various electrolytes and tissue thickness. These results clearly indicate their independence of tissue thickness. Moreover, the plot of tunnel contribution versus electrolyte resistivity for 120 rn tissue thickness, Figure 5, shows direct proportionality between the two to within -+10%.   (Table III). Nevertheless, they still seem to be independent of the tissue thickness. It is puzzling however, to find that these contributions are larger than those found in etched foil. It is very hard to give a definite explanation for this discrepancy. This can arise from variation in the dielectric loss from one foil to another. For example, a value of 2.1% for tan i o instead of 2% nearly halves the contributions given in Table III, while those in  Table II will be reduced by less than 3%. This can also be the cause of the larger fluctuations noted in Table III. 3.6 Dependence of Capacitor Resistance on Plate Area and Type ofElectrolyte Five batches of 100 V capacitors were produced with different plate areas, but using fixed tissue thickness and two types of electrolytes E 93 and E 98. The resistance of the capacitors was determined at 100 Hz and the contribution of oxide, lugs and foil were subtracted. The remainder was plotted as a function of the reciprocal of plate area, Figure 6. The results show a straight line relationship. The ratio of the slopes of the lines for E 93 and E 98 is 1.85. This is close to the ratio of the resistivities of the two electrolytes at 100 Hz, which was found to be nearly 2. These findings indicate that the calculated resistances are due to electrolyte in tunnels and impregnated tissues.    The average tunnel diameter in the anode used was 3.5 x 10-Tm at a formation voltage of 20 V. At 115 V the diameter of the tunnel and thickness of the dielectric can be evaluated. 4 These are 1.8 x 10 -7 m and 1.6 x 10 -7 m respectively.
The cathode is of the same type used by Morley and Campbell4. The resistance of electrolyte in the cathode tunnels is determined for electrolyte E 93 and found to be nearly 0.2 [2 in a capacitor. This is negligible compared to the total resistance due to anode and cathode tunnels ( Table II).
The total anode capacitance and tunnel resistance are known experimentally. These are used in Eqs. (6) and (7) to determine tunnel length and density. The average tunnel length is found to be 25 x 10 -6 m and the density is 1.4 x 1012 tunnels/m2" These are close to the values normally found 4,s

CONCLUSIONS
The transmission line analysis of electrolytic capacitors has been examined experimentally and the results indicate dose agreement with theoretical predictions. The high frequency resistance is shown to be mostly due to impregnated tissue. The low frequency resistance, on the other hand, is the sum of the contributions of all parameters. The relative contribution of each parameter can be evaluated for any capacitor and for small capacitors it is found that resistance is mostly due to the dielectric loss.