**Energy Storage and Release in Circuits**

**Abstract**

In order to understand electrostatic storage and discharge of energy in capacitive circuits, an experiment involving charging and discharging of an R-C circuit was carried out. The time for charging and discharging the circuit, referred to as the R-C time constant, was found to be proportional to the product of resistance and capacitance in the circuit. On the other hand, it was established that the frequency of supply determines the time of the input waveform and this affects the output waveform, that is, whether the circuit is able to reach a stable state during the charging or discharging cycle. A function generator powered the circuit at varied frequencies while the input and output signals were then viewed on an oscilloscope. The experiment was successful and the charging and discharge transient times at different supply frequencies, resistance, and capacitance closely approximated the theoretical values.

**Introduction**

A capacitor is a device that stores electric charge. It consists of a dielectric, which is a layer of a shielding medium that separates the two conducting surfaces. The capability of the capacitor to store energy in the dielectric is called capacitance, defined as “the amount of charge needed to create a unit charge between its plates” (Theraja and Theraja, 2004, p.214).

Thus; Capacitance=

Where,

C= capacitance in Farads

∈= Permittivity of dielectric

A= Area of plate overlay in square meters

d= Distance between the plates in meters

Capacitance is expressed in units referred to as Farads (F): hence, 1 Farad = 1 Coulomb/1 Volt. Generally, capacitors normally accumulate charge in the form of micro- (μF) to pico-Farads (pF). The potential energy accumulated in a capacitor can be demonstrated as:

U =CV^{2 }

The storage of energy in the form of an electric field between the plates allows a capacitor to accumulate and release electrical energy. A completely discharged capacitor has a zero terminal voltage and initially acts as a short circuit when joined to a circuit with a voltage source, drawing optimal current as it builds charge. The voltage rises with time to reach the applied voltage, but in the opposite direction. Subsequently, the current through the capacitor decreases proportionally. When the capacitor voltage equals full supply voltage, it stops drawing current and acts as an open circuit. The cycle is repeated if it is shorted or discharged through a resistance and recharged.

**Charging a Capacitor**

- First ensure that the power supply is actually turned off. Consequently, you will also be required to assemble or arrange the circuits first prior to supplying energy to the system.
- Then go ahead and build the first circuit by joining one of the 1000-μF capacitors direct to the energy supply using a collection of banana plug wires, joining with suitable poles (black to-black and red-to-red). In this case, red and black point out the positive and negative poles of the power supply, correspondingly.
- Connect the leads of the Differential Voltage probe to the banana plug wires (use the back inserts), while, considering the poles. Again, ensure that the voltage probe records the voltage transversely towards the capacitor (VC) as it continues to be charged by the energy supply.
- Open the Logger Pro and confirm that the Differential Voltage Probe is the only sensor that is connected or plugged into the LabPro interface. Then set the information assembly time to 15 seconds, with a sample frequency of 10 samples per second.
- Fix the dial on the energy supply setting to 20, which matches to about 5 volts.
- Start the data collection by hitting the ’Collect’ switch, and then switch on the energy supply.
- After a period of 5 seconds, switch off the energy supply, but keep on gathering voltage information to view how the capacitor maintains voltage.
- Finally, it is advisable to print a copy of the graph for reference.

*Figure 1: A Circuit illustration for charging a capacitor*

**Discharging a Capacitor**

- First, unplug the wire leads from the energy supply, and fasten a nail and an alligator clip to the end of every wire. This will boost the effect in the subsequent observation.
- Start gathering information, again for a duration of 15-second of continued observation.
- Restore electric charge to the capacitor by a fixing the nails to the poles on the energy supply, in contrast to connecting the wires directly.
- As soon as the capacitor is completely charged, detach the tips of the nails from the energy supply, and attach them together. This process can be repeated for several times for each data set, since it will only take some few seconds to complete this process.

A resistor capacitor circuit mainly denoted as RC circuit, or an RC filter is an electronic circuit comprised of resistors and capacitors energized by a power source. The 1st order RC circuit that is comprised of one resistor and a single capacitor, is one of the simple examples of a resistor capacitor circuit. Resistor capacitor circuits, like other kinds of circuits, are mainly used to “refine” an electrical impulse waveform, thereby, altering the relative quantities of low-frequency and high-frequency data in their output electric impulses relative to their input electrical impulses or signals.

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*Figure 2: Schematic of an R-C circuit*

Considering a series RC circuits connected to a capacitor, a battery and a resistor in series. The capacitor is originally uncharged, but it begins to charge when the button is closed (Physics, 2016). Originally the probable difference transversely towards the resistor is the battery emf, but that gradually drops as well as the current as the probable difference transversely towards the capacitor increases (Physics, 2016).

Using Kirchoff’s loop rule: (Physics, 2016)

ε – IR – Q/C = 0

As Q rises I reduces, but Q changes because there is a current I. As the current reduces Q changes more slowly (Physics, 2016).

I = dQ/dt, so the equation can be expressed as:

ε – R (dQ/dt) – Q/C = 0

Generally, this is an equation comprised of derivatives of a function that can be explained or solved for Q as a function of time (Physics, 2016). The answer (obtained in the text) is:

Q (t) = Qo [1 – e-t/τ]

Where,

Qo = C ε and the time constant τ = RC.

Ascertaining this expression to get the current as a variable of time leads to:

I (t) = (Qo/RC) e-t/τ = Io e-t/τ

Where,

Io = ε/R is the highest current conceivable in the circuit.

The time constant τ = RC regulates how fast the capacitor charges. If RC is minimal the capacitor charges fast; if RC is large the capacitor charges bit by bit or slower.

**An RC Circuit: Discharging**

Applying Kirchoff’s loop rule:

-IR – Q/C = 0

I = dQ/dt, so the equation can be expressed as:

R (dQ/dt) = -Q/C

This is a differential equation that can be explained or solved for Q as a function of time. The result is:

Q (t) = Qo e-t/τ

Where,

Qo is the original charge on the capacitor and the time constant t = RC.

Setting apart this expression to get the current as a function of time results to:

I (t) = – (Qo/RC) e-t/τ = -Io e-t/τ

Where,

Io = Qo/RC

Note that the minus symbol or sign shows that the charge flows in the opposing direction. So, the time constant τ = RC regulates how fast the capacitor releases energy. If RC is minimal the capacitor releases energy fast; if RC is large the capacitor releases energy slowly.

**Conclusion**

The experiment demonstrates that a capacitor stores electrostatic energy between its plates when placed in a circuit that has a potential difference. The stored charge is proportional to the applied voltage, and it charges and discharges in transients, whose time depend on the capacitor and the energy-dissipating element of the circuit. The product of resistance and capacitance in such a circuit is called the time constant of the circuit, or the time that would be required for an ideal linear discharge or charging. Both theoretical calculations and experimental observations prove that approximately five time constants are required to charge or discharge a circuit to a steady state value. Therefore, the input supply period should be consistent with these time constants in order to produce a full-scale waveform at the output.

**References:**

Theraja, B. L. (2004), *A textbook of electrical technology.* *Rev. ed*. New Delhi S. Chand

Physics, (2016), *An RC Circuit*. Retrieved on 24^{th} March 2016 from http://physics.bu.edu/~duffy/semester2/c11_RC.html