How does the terminal velocity of the metal ball in cm s-1 moving vertically downwards through glycerin depend on the radius (in cm) of the metal balls, determined by measuring the time taken for the ball to fall?

Life has become so dependent on laptops, i-pad and mobile now, especially during this lockdown! Every day since morning all the plug points of the house have an electric component hanging which is called a charger. The chargers are of different shape and size! Being intrigued by the same I wondered “what if I directly connect my iPad or the phone to the plug point with wires without the charger?” “What is present inside the charger which makes it so useful? “In order to search for an answer to my question, I started searching the net and read across a number of journals and watched some youtube videos. I was fascinated with the thought that our supply current is ac but the phone charges on dc. I was more intrigued to know “how is this conversion possible?”  In order to quench my thirst, I read more articles and watched more videos? Thus, I came across the new terms like diode, capacitor, dielectric which are all dealing with electricity, which is a major component in IB syllabus in the chapter of electricity dealing with capacitance and dielectrics and in power transmission where we will learn about diodes and transformers and how they are used. Hence, I decided to take up a project on this interesting topic as to how an ac is converted to dc and thus what would be the best method of such conversion? My ever-enquiring mind further asked “Can I vary the ratio of ac to dc conversion rate?” Thus, I came across the concept of ripple factor which finally led me to my research question.

This project of mine might help in better understanding of dielectrics and help make better adaptors using the permittivity contempt to increase the capacitance and thus reduce the ripple factor.

In a way, a capacitor is a little like a battery. Although they work in completely different ways, capacitors and batteries both store electrical energy. A capacitor is much simpler than a battery, as it can't produce new electrons -- it only stores them. Inside the capacitor, the terminals connect to two metal plates separated by a non-conducting substance, or dielectric.

Permittivity is a constant of proportionality that relates the electric field in a material to the electric displacement in that material. It characterizes the tendency of the atomic charge in an insulating material to distort in the presence of an electric field. The larger the tendency for charge distortion (also called electric polarization), the larger the value of the permittivity. The permittivity of an insulating, or dielectric, material is commonly symbolized by the Greek letter epsilon, ε; the permittivity of a vacuum, or free space, is symbolized ε₀; and their ratio ε/ε₀, called the dielectric constant is symbolized by the Greek letter κ. The permittivity ε and the dielectric constant κ in the cgs system are identical; both of them are dimensionless numbers.

The bridge rectifier is made up of four diodes namely D₁, D₂, D₃ and D₄. The input signal is applied across the two terminals A and B while the DC output is obtained across the load resistor connected between the terminals C and D.

The pulsating DC output obtained across the load resistor R_L contains small ripples. To reduce these ripples, we use a filter at the output. The filter normally used in the bridge rectifier is a capacitor filter. In the below circuit diagram, the capacitor filter is connected across the load resistor R_L..

When an input AC signal is applied, during the positive half cycle both diodes D₁ and D₃ are forward biased. At the same time, diodes D₂ and D₄ are reverse biased. On the other hand, during the negative half cycle, diodes D₂ and D₄ are forward biased. At the same time, diodes D₁ and D₃ are reverse biased. Thus, the bridge rectifier allows both positive and negative half cycles of the input AC signal.

The DC output produced by the bridge rectifier is not a pure DC but a pulsating DC. This pulsating DC contains both AC and DC components. The AC components fluctuate with respect to time while the DC components remain constant with respect to time. So, the AC components present in the pulsating DC is an unwanted signal. The capacitor filter present at the output removes the unwanted AC components. Thus, a pure DC is obtained at the load resistor R_L.

In the parallel-plate capacitor in the above simulation if the potential difference between the plates is changed then

Q ∝ ΔV

⇒ Q = C Δ V

where C is called the capacitance of the capacitor.

The capacitance depends on the geometry of the capacitor, namely the distance d between the plates [Q ∝ 1/d] to the area A of the plate  [Q ∝ A] and on the material between the plates. Hence Capacitance is given by: C = ε₀ A/d where ε₀ is the permittivity of free space and has a value of 8.85 x 10^-12  C²\Nm². In most capacitors, an electrically insulating material called a dielectric is inserted between the plates. This decreases the electric field inside the capacitor. This happens because the molecules inside the dielectric get polarized in the field and they align themselves in a way that sets up a field in the opposite direction. The ratio of the field without the dielectric, E_o, and the field with the dielectric, E, is known as the dielectric constant, k. That is: k = E_o/E where is dimensionless and always greater than or equal to one. For a given potential difference across the plates, a capacitor with a dielectric can store more charge than the one without. The capacitance of a capacitor with a dielectric is given by: C = k ε₀ A/d

Ripple factor is a measure of the AC component present in a rectified AC signal. More the Ripple factor, the less efficient the rectification methodology is. An ideal rectification circuit or an ideal AC to DC converter would have 0 as the magnitude of ripple factor as there should not be any AC component in the output after rectification. Ripple Factor is the ratio of input AC current and the output DC current.The amount of ripple in power supplies is often indicated by the ripple factor:

For practical purpose, it is defined by: γ = \(\frac{V_{AC}}{V_{DC}}\)

Theoretically, its form becomes:  γ =  \(\sqrt{\frac{V^2_{rms}}{V^2_{DC}}-1}\)

Where – V ripple (rms) is the rms value of ripple voltage at the output and V_dc is the absolute value of the power supply output dc value.

Theoretically, for a bridge rectifier with capacitor filter the ripple factor depends on the capacitance and is given by, γ =  \(\frac{1}{4\sqrt{3}f\,C\,R_{L}}\)

where, f = frequency of the AC supply, C is the capacitance of the capacitor and R_L is the load resistance.

The cathode ray oscilloscope (CRO) is a type of electrical instrument which is used for showing the measurement and analysis of waveforms and other electronic and electrical phenomena. It is a very fast X-Y plotter that shows the input signal versus another signal or versus time. The CROs are used to analyse the waveforms, transients, phenomena, and other time-varying quantities from a very low-frequency range to the radio frequencies.

First the bridge rectifier circuit using four diodes with capacitive filter was prepared. A step-down transformer was used with household AC supply as input to the transformer and the bridge rectifier circuit was connected to the output of the transformer. The value of capacitance of the capacitive filter was varied throughout the experiment by varying the dielectric inside the capacitor and the AC voltage (V_rms) and (V_dc) as measured using a multimeter across the load. The ripple factor has been calculated for each trial and the variation of ripple factor with an increase in capacitance of the capacitive filter due to dielectric was studied.

In the paper by P. P. Sahu, M. Singh and A. Baishya, "A Novel Versatile Precision Full-Wave Rectifier," in IEEE Transactions on Instrumentation and Measurement, vol. 59, no. 10, pp. 2742-2746, Oct. 2010.

In this paper, it was proposed and realized a novel precision full-wave rectifier using an all-pass filter as a 90° phase shifter.

Prediction

It was predicted that the dielectric introduced in the capacitor will increase the capacitance of the capacitive filter in the bridge rectifier circuit which would reduce the ripple factor. Thus, a negative correlation is expected between the permittivity of the medium and the magnitude of Ripple factor of the Bridge Wave Rectifier circuit.

Justification

The assumption was based on the fact that, with an introduction of dielectrics in the capacitor with permittivity “κ” the capacitance of the filter circuit would increase, hence more AC current will be bypassed to the output in which the capacitor was connected because capacitance offers extremely low resistance towards AC current. Since reactance is inversely proportional to capacitance, it would decrease with increase in capacitance and hence, more AC current will flow through the filter thus decreasing the amount of AC current component in the load.

The schematic graph shown below depicts a relationship between the Ripple factor and the permittivity of the medium represented graphically as a straight line sloping downwards and indicating a negative correlation.

Dielectric of the capacitive filter

The permittivity of the dielectric of the capacitive filter inserted between the capacitor connected across the load of the bridge rectifier circuit was the independent variable of this exploration. The magnitude of capacitance was varied over a range of 10 μF from 1 μF to 10 μF at an interval of multiple of permittivity value. The capacitance of the capacitive filter was increased by inserting dielectric inside the plates of the capacitor within the dielectric constant range of 1.0 to 9.0, connected parallel to the load resistor (output). The dielectric constant was increased over the above-mentioned range to understand the variation in AC component of voltage across the output over a wide range of capacitance to strengthen the correlation obtained.

The values of permittivity of the medium used have been taken from literature sources and to use more reliable data, the values have been taken from three different secondary sources and a mean value has been considered.

In certain cases, a range was observed instead of a specific value and as the investigation demands a specific value of permittivity along the x axes in the graphical analysis, those materials were intentionally avoided. For example, pyrex glass has been used as a medium which is reported to have a permittivity of the medium in the range of 4.0 to 6.0.

The range of permittivity has been kept within 1.0 to 9.0 as the materials required to have the permittivity within this range are all solids and can be easily procured. Materials having dielectric above this range are usually in the liquid state and will thus have problems while executing the experiment.

The table below shows the permittivity of the medium used. The data has been collected from three sources-

• Source A- engineertoolbox.com – This is a website with physical data of electrical properties of various dielectric mediums.
• Source-B - Foundations for microstrip circuit design published in onlinelibrary.wiley.com
• Source-C- nvlpubs.nist.gov – a government approved official website of National Institute of Standards and Technology under US Department of Commerce.

Sample calculation

For asbestos,

Mean permittivity = \(\frac{Source\,-A\,+\,Source B\,+\,Source\,-\,C}{3}= \frac{4.00\,±\,0.05 ,+\,4.10\,±\,0.05\,+\,4.00\,±\,0.05}{3}\)= 4.03 ± 0.05

Absolute uncertainty = \(\frac{Max value\,-\,Min value}{2}= \frac{4.80\,-\,4.70}{2}\) = 4.03 ± 0.05

Ripple Factor

Ripple factor is the dependent variable of the exploration.

It is a measure of the AC component present in a rectified DC signal. More the Ripple factor, the less efficient the rectification methodology is. An ideal rectification circuit or an ideal AC to DC converter would have 0 as the magnitude of ripple factor as there should not be any AC component in the output after rectification. Ripple Factor is the ratio of output AC current and the output DC current. It is also represented by the following formula:

γ = \(\sqrt{\frac{v^2_{rms}}{V^2_{dc}} -1}\) or γ = \(\frac{V_{AC}}{V_{DC}}\)

γ = Ripple Factor

v_{rms = Output RMS Voltage}

V_dc = Output DC Voltage

In the experiment, the rms output voltage and the dc output voltage has been obtained using a multimeter and by using the above-mentioned expression the ripple factor has been calculated.