Equivalent circuits

Figme 8.16 Equivalent circuit for cathodic protection of a buried pipeline [24]. 

Protective potential = Anodic potential Cathodic potential = Soil resistance = Coahng resistance = External resistance 

Ohm s law predicts the corrosion current for the coated and bare steel pipelines before a cathodic current is apphed to the cathodic protection system [24]. Hence, at equilibrium 

If the protective anodic and cathodic potentials are equal, then Ohm s law yields Ea = Ec and 

Combining eqs. (829) and (8.36) and solving for la, and inserting the resultant expression into (8.37) yields 

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Figure Bl.28.8. Equivalent circuit for a tliree-electrode electrochemical cell. WE, CE and RE represent the working, counter and reference electrodes is the solution resistance, the uncompensated resistance, R the charge-transfer resistance, R the resistance of the reference electrode, the double-layer capacitance and the parasitic loss to tire ground.

W. J. Eleming, Zirconia Ouygen Sensor—-An Equivalent Circuit Model, SAE 800020, Society of Automotive Engineers, Warrendale, Pa., 1980. 

Other Useful Information Obtained by Probes Both EIS and electrochemical noise probes can be used to determine information about the reactions that affect corrosion. Equivalent circuit analysis, when properly applied by an experienced engineer, can often give insight into the specifics of the corrosion reactions. Information such as corrosion product layer buildup, or inhibitor effectiveness, or coating breakdown can be obtained directly from analysis of the data from EIS or indirectly from electrochemical noise data. In most cases, this is merely making use of methodology developed in the corrosion laboratory. 

This is a very useful nomogram to determine the performance of a motor with the help of only no-load and short-circuit test results. In slip-ring motors, it also helps to determine the external resistance required in the rotor circuit to control the speed of the motor and achieve the desired operating performance. Slip-ring motors are discussed in Chapter 5. The concept behind this nomogram is that the locus of the rotor and the stator currents is a circle. Consider the equivalent circuit of an induction motor as shown in Figure 1.15, where... 

Figure 1.15 Simple equivalent circuit diagram of a motor...

Figure 2.4(a) Equivalent circuit diagram of a single squirrel cage motor... 

Figure 12.4(b) Equivalent circuit diagram with an unbalanced supply voltage (at slip = 2 - S)... 

To illustrate the important features of a VT, let us analyse its equivalent circuit diagram. Refer to a simple diagram as in Figure 15.1 which is drawn along similar lines to those for a motor (Section I.IO, Figure I.I.5). For ease of analysis, the ratio of primary and secondary turns has been considered as l l. Then from the circuit diagram, the following can be derived ... 

Figure 15.1 Equivalent circuit diagram for a voltage transformer...

R Damping resistor to prevent ferro-resonance effects Figure 15.7 Equivalent circuit diagram of a CVT... 

I2 - Secondary current referred to the primary side Figure 15.17 Equivalent circuit diagram of a current transformer... 

On the load side, the interrupter is connected through a cable to the equivalent circuit, with the required quantities of lumped resistance and reactance, to represent the motor to be tested, under a locked rotor condition. The circuit would also represent an interruption immediately after a start, to check for the most onerous operating condition for the interrupter to generate the highest surges as discussed earlier. 

Then applying Thevenin s " theorem, the equivalent circuit can be represented as show n in Figure 18.18(b). On application of a voltage surge, the arrester will start... 

Figure 18.18 Equivalent circuit applying Thevenin s theorem...

Figure 21.9 Equivalent circuit of a CBCT protection circuit...

Figure 23.22 Equivalent circuit for n number of capacitor bank already switched on the circuit and another (C ) being switched...

Figure 23.25 Equivalent circuit for a 60 kVAr bank in star made of 3 x 20 kVAr units...

Consider the case when all five units are already energized and the sixth is switched. The equivalent circuit can now be represented as shown in Figure 23.26. 

For ease of calculation, let us consider the impedance of the transformer as its leakage reactance, ignoring resistance and draw an equivalent circuit diagram as in Figures 24.25 (b) and (c) Assuming the length of the primary distribution line to be 15 km, the total line parameters will become... 

Fig. 23-4 Equivalent circuit diagram for a pipeline subjected to interference.

The equivalent circuit diagram in Fig. 23-4 shows a differential fundamental section and the closed-circuit impedance at the end of the close proximity region. 

It follows from the equivalent circuit diagram for a loop pipe/soil of the differential fundamental section ... 

The heat run will he equivalent load method. Efficiency at full, and V2 load and power factor at full, /4 and V2 load and breakdown torque will be determined by equivalent circuit calculation (IEEE 112,... 

Fig. 19.36 Basic circuit for a poiemiostat. (a) Basic circuit for a potentiostat and electrochemical cell, (b) Equivalent circuit, (c) Circuit of a basic potentiostat. A.E. is the auxiliary electrode, R.E. the reference electrode and W.E. the working electrode (6 and c are from Polen-tiostat and its Applications by J. A. von Fraunhofer and C. H. Banks, Butlerworths (1972))...

According to this model, the SEI is made of ordered or disordered crystals that are thermodynamically stable with respect to lithium. The grain boundaries (parallel to the current lines) of these crystals make a significant contribution to the conduction of ions in the SEI [1, 2], It was suggested that the equivalent circuit for the SEI consists of three parallel RC circuits in series combination (Fig. 12). Later, Thevenin and Muller [29] suggested several modifications to the SEI model ... 

Figure 13. Schematic presentation of a small segment of polyheteromicrophase SEI (a) and its equivalent circuit (b) A, native oxide film B, LiF or LiCl C, non conducting polymer D, Li2CO, or LiCO, R GB, grain boundary. RA,/ B,RD, ionic resistance of microphase A, B, D. Rc >Rqb charge-transfer resistances at the grain boundary of A to B or A to D, respectively. CA, CB, CD SEI capacitance for each of the particles A to D.

The equivalent circuit of a section of this SEI is presented in Fig. 13(b). It was recently found [123, 124] that at temperatures lower than 90 °C, the grain-boundary resistance of composite polymer electrolytes and composite solid electrolytes based on Lil-A Ojis many times larger than their ionic resistance. At 30 °C / GB is several orders of magnitude larger than (the ionic resistance) and for 100 pm-thick CPE foils or Lil-A Oj pellets it reaches [125] 105-106Qcm2 (depending on CPE composition). 

A value of Rqb for an SEI lOnm thick can be estimated from its values for CPE and CSE by assuming that these solid electrolytes consist of nanometer-sized particles. Thus the expected value for / GB at 30 °C for a lOnm SEI is in the range 10-lOOQcm2, i.e., it cannot be neglected. In some cases it may be larger than the ionic (bulk) resistance of the SEI. This calculation leads us to the conclusion that 7 GB and CGB must be included in the equivalent circuits of the SEI, for both metallic lithium and for LixC6 electrodes. The equivalent circuit for a mosaic-type... 

Figurel4. Equivalent circuit for two sublayer polyheteromicrophase SEI (for notation, see text) [125],...

Normally, the impedance plots are fitted to an often-complex equivalent circuit. Mathematically, this means searching for a global solution in R". However, problems arise if a complicated equivalent circuit is found which does not allow physical interpretation. Therefore, it is preferable to run a wide variety of experiments with different samples rather than trying to fit in detail the results of a single measurement in order to analyze the resulting impedance plots. 

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