| 13 | Supercapacitors |
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| Main | A circuit consisting of four capacitors and four resistors is built. The circuit models the diffusion of charge in a supercapacitor. This supercapacitor is charged and discharged across an external resistor. More…
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| Description | DESCRIPTION OF EXPERIMENT |
| Experiment | A circuit consisting of four capacitors and four resistors is built. The circuit models the diffusion of charge in a supercapacitor. This supercapacitor is charged and discharged across an external resistor. Voltages across the innermost and outermost capacitors are measured as functions of time (Data). Each of the capacitors is discharged separately in additional experiments (to yield independent data on the individual capacitors).
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| Dimensions | Capacitance of capacitors 1 and 2: 470 µF (nominal) Capacitance of capacitors 3 and 4: 100 µF (nominal) Resistances of resistors from inside to outside: 10.0, 9.90, 9.82, 9.88 kOhm Resistance of load resistor: 97.90 kOhm Resistance of differential voltage probes: 4 and 12 MOhm |
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| Assignment | A POSSIBLE PATH THROUGH THE INVESTIGATION… |
| Basics | Investigate the experiment, make sure you understand the setup of the system and the initial conditions. Draw a circuit diagram representing the system. Plot the Data to get a feeling for the dynamical process. Compare curves to simple exponential functions. Estimate time constants of the processes. Create a word model for the system and its processes. Create a formal dynamical model (consult with the models of previous Investigations), import data, simulate the model and determine the parameters of the model by comparing simulation and experimental data. (Determine capacitances first from the individual experiments.)
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Particulars | Change parameters (including initial values) and perform simulations to learn about the behavior of this system. In particular, investigate the role of the values of capacitance and resistances in the outer part of the supercapacitor model. What values lead to a behavior that is similar to the discharging of a real super- capacitor? |
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| Model | MODEL EQUATIONS AND MORE… |
| | Laws of balance dQ_1(t)/dt = - IQ_1 - IQ_1_leak
INIT Q_1 = C_1*UC_init_1
dQ_2(t)/dt = IQ_1 - IQ_2
INIT Q_2 = C_2*UC_init_2
dQ_3(t)/dt = IQ_2 - IQ_3
INIT Q_3 = C_3*UC_init_3
dQ_4(t)/dt = IQ_3 - IQ_4 - IQ_4_leak
INIT Q_4 = C_4*UC_init_4
Flows IQ_1 = (U_1-U_2)/R_1
IQ_1_leak = U_1/R_leak_1
IQ_2 = (U_2-U_3)/R_2
IQ_3 = (U_3-U_4)/R_3
IQ_4 = (U_4-UB)/(R_4+R_ext)
Relations U_1 = Q_1/C_1
U_2 = Q_2/C_2
U_3 = Q_3/C_3
U_4 = Q_4/C_4
Parameters R_ext = IF (TIME < 502) THEN 0 ELSE 98e3
UB = IF (TIME < 21.0) THEN 0 ELSE IF (TIME <502) THEN 4.584 + 0.025 ELSE 0.07
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| Questions | SOME SIMPLE QUESTIONS… |
| 1 | Which of the two functions measured during charging and discharging of the model supercapacitor belongs to the inside and outside capacitor, respectively? (See upper diagram in Graph) |
2 | At the beginning of charging, one of the voltages rises very steeply (lower left diagram in Graph). This function clearly is not a simply exponential function of the type U_max·(1–exp(–t/tau)). Explain why the voltage measured here behaves as seen in the graph. |
3 | At the beginning of charging, the second voltage measured (blue curve in the lower left diagram in Graph) rises slowly. In fact, the initial rate of change of voltage is zero. Why is this so? |
4 | The blue curve in the lower left diagram in Graph is nearly an exponential function. How can you veryfy this? |
5 | Why does the voltage across the outermost capacitor (red curve in the lower right diagram in Graph) fall quickly for a short period after which it continues to decrease slowly? |
6 | The voltage that can be measured across a real supercapacitor is modeled by the voltage across the load resistor R_ext (see the Model Diagram). Why? How could you calculate this voltage? What will this quantity look like as a function of time? |
7 | What changes have to be made to change the initial fast drop of voltage during discharging of a supercapacitor (to make this drop faster/slower, larger/smaller)? |
8 | What is the capacitance of the system modeled as a single simple capacitor? How does this value compare to the sum of all capacitances in the circuit? |
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