Using Superposition Theorem, 3V source is removed to calculate the contribution of the 8 V supply. Similarly, 8V source is removed to calculate currents due to 3V supply. a) Voltage across the 18 ohm resistor is 4.6872 volts and current across is 0.2604 amperes; b)Current in the 8V generator is 1.667 amperes; c) Current in the 3V generator is 0.656 amperes. (John, 2003)
Question 1.3 a)
Transformers are electrical devices that play a very vital and key role in electrical power distribution and transmission systems. They are static devices, that is, they do not have any moving components. Transformers utilize one of the most basic forces in nature, the electromagnetic force to convert alternating electrical energy of one power rating to another power rating, but do not change the frequency of the primary electrical energy. Voltage transformers are used to lower or raise the magnitude of the incoming voltage and accordingly current increases or decreases. Electromagnetic induction is used to perform this conversion, more specifically mutual induction. In a simple voltage transformer, two coils are present that have common magnetic flux in between them but are electrically isolated. This path of magnetic flux has some magnetic reluctance that actually helps in power conversion. The core constitutes of laminated sheets around which the incoming voltage coils, also called the primary windings, and the outgoing voltage coils, also known as, secondary windings are wound. The laminations of the core are connected in the form of strips which are not continuous and are overlapped. Both the coils are highly inductive in nature and also have high mutual inductance. Due to mutual induction, an electro motive force is created that creates a voltage as the primary coil is connected to an alternating source. Almost all of the primary coil’s induced voltage is transferred to the secondary coil as both have high mutual induction. Value of electromotive force produced due to mutual induction is given by Faraday’s Law of Electromagnetic induction e=N.dφ/dt where N is the number of turns in the coil , φ ids the magnetic flux in Webers, e is the induced voltage and t is time in seconds. Magnetic flux flowing through a ferromagnetic material is similar to current through a conductor. that is, a motivating force is necessary. If an electrical circuit is connected to the secondary coil, then a current will flow in it and thus it can be said that electrical energy is transferred through electromagnetic induction from the primary coil and secondary coil. Almost all transformers have taps in either their primary or secondary windings to change the number of turns of coils in the windings, and thereby change the level of secondary and primary voltage in the respective coils. (Electrical4u.com, 2016)
If E1 is the primary voltage and I1 the primary current and E2 ,I2 the secondary voltage and current, with N1 & N2 being the primary and secondary coil turns, then E1/E2=N1/N2=I2/I1 ;
Stepping up of voltage leads to stepping down of current and vice versa. Number of turns can be changed with a tap changer. Power rating is considered almost constant, ignoring the minor losses, E1*I1=E2*I2
As they handle huge amounts of electrical power, the amount of heat produced in transformers is always very high; more the amount of voltage handled, more is the heat produced due to copper losses, core losses and other sources of energy loss. Enhancing of the cooling rate of a transformer increases its power handling capacity. Cooling procedures should be able to prevent local formation of extreme heat at any portion of the device and also overall increase in the device’s temperature. This is why the active parts of a transformer are immersed in special cooling oils that have very good insulation properties. It should be ensured that this oil is free of any air bubbles, gases, particulate matter or any kind of contaminants. Radiators and ducts are constructed to enable circulation and sufficient cooling of the oil. There are various processes of transformer cooling, such as, oil filled self cooling where the heat of the oil is radiated to the air; forced air and oil cooled where cooling fans are utilized; conservator cooling systems; oil filled inert gas cooling systems. Transformers can be classified according their construction, cooling processes, purpose, supply type etc. Based on construction, transformers are primarily of two types, Core type where the windings are wound in the shape of a cylinder with proper insulation around a laminated core through which the flux flows and shell type, where the windings are mounted in a layered format. According to their uses, transformers can be Step Up and Step Down. On the basis of the type of power supply, there are Single Phase and Three Phase transformers. Power transformers are the ones used in high tension voltage transmission networks; Distribution transformers are used in distribution networks and have lower rating than power transformers; Instrument transformers are used for experiments and other system protection purposes and they too can be either convert potential or current. Based on cooling types, we have oil filled self cooled type, oil filled water cooled type and air blast type. (Circuitstoday.com, 2011)
Question1.3) b)
E1= 200V, N1=1200, E2=10V Therefore, N2=60 and, in the second case, primary current I1 is 0.125 amperes
Question 1.4)
250V phase angle 0 deg 50 Hz
79.58 mh
18 ohms
64.96 µf
Inductive reactance of the coil is XL=2∏fL= 24.98 ohms; Impedance = √(R2+XL2)=30.79 ohms. Thus, d) current in coil is 250/30.79=8.11 amperes and branch phase angle is tan-1(24.98/18) = 54.22 deg lag. Capacitive reactance is XC=1/(2∏fC)=49.26 ohms. Thus, e) current in capacitor =5.075 amperes with 90 deg lead; b)total supply current is 4.76 amperes and c) circuit phase angle 5.96 deg lag; a) total circuit impedance is 250/4.76=52.52 ohms. (Basic Electronics Tutorials, 2013)
Question 3.2) i)
Fourier series
A Fourier series is a type of infinite series in mathematicswhich involves functions of trigonometry. Fourier series are used in applied mathematics, physics and electronics, in order to express periodic functions comprising communication signal waveforms.Many of the phenomena studied in engineering and science are periodic in nature for example, the current and voltage in an alternating current circuit. These periodic functions can be analyzed into their constituent components (fundamentals and harmonics) by a process called Fourier analysis (Asher, 2013).
The first thing that comes under notice is representation of Fourier series for a function that is periodic in nature. A function f(x) is said to be Periodic when f(x + T) = f(x) for some value T. The minimum value of T is called the Fundamental Period. A notable feature of the periodic function is that if two functions are added or multiplied that has the same period, then the resultant sum or product will be of the same period (Combined estimator Fourier series and spline truncated in multivariable nonparametric regression, 2015).
For example, cos(2πt), sin(2πt) both have periods of 1. The sums of these two functions are: cos(2π(t + 1)) + sin(2π(t + 1)) = cos(2πt + 2π) + sin(2πt + 2π) = cos(2πt) + sin(2πt)
It is observed that the resultant is of the same period as that of the initial trigonometric functions is of period 1.
The representation of the Fourier series of a function f ( x ) is as follows(Weisz, 2000): f(x) = a0 / 2 + m=1∞( amcos mπx / L + bm sin mπx / L )
Application of Fourier series:
The Fourier series has varied range of applications that ranges from experimental problems to difficult mathematical techniques for analysis. As this series is represented in terms of sine and cosine, hence it is easy to perform integration and differentiation of these functions which gives a simplified analysis for instance, saw waves that are common output for most of the experiments. The Fourier series comes to the survival to the functions that are discontinuous through approaches that are of simplified analysis type. This theory of Fourier series can also be applied in order to perform solutions to the functions belonging to the orthogonal system.
In case of non – sinusoidal periodic waveform, the wave which reaches the highest amplitude is known as the peak while those having a lower amplitude are known as side lobes. This is clearly shown in the following diagram.
It is required to calculate the Fourier series of the following triangular waveform, f ( t )
This is an even triangle wave for cosine series.
For the above figure, a0 = 0. For n > 0, an=4V 1-(-1)n / (πn)2 when n is odd =0 when n is even
In the given case, f (t) = cosx
Therefore the fourier series for this f (x) is, f(x) = a0 / 2 + m=1∞( ancos nπx / T + bn sin nπx / T ) bn = 0 a0 = 0
When n is odd, f (x) = 0
When n is even, f (x) = n=1∞( 4V 1-(-1)n / (πn)2)cos nπx / T
Question 3.2) ii)
Utilization of the Laplace transform as part of the transient circuit analysis gives an idea or a prediction of circuit response. The Laplace transform provides a general idea of output behavior.
Follow these basic steps to analyse a circuit using Laplace techniques (External version of Laplace method, 2014):
1. The differential condition is developed in the time-domain with the help of Kirchhoff's laws and other elemental conditions.
2. The Laplace transform theory is applied to the differential condition in order to put the condition in the domain of time
3. Algebraically explanations are given for the arrangement, or reaction change.
4. Apply the backwards Laplace change to create the answer for the first differential condition portrayed in the time-domain. To get comfortable with this process, you simply need to practice applying it to different types of circuits such as an RC (resistor-capacitor) circuit, an RL (resistor-inductor) circuit, and an RLC (resistor-inductor-capacitor) circuit(Ziemer and Tranter, 2002).
Transient state occurs in any electrical circuit when any of the circuit variables are in the process of change. (Electrical4u.com, 2016) Energy storing elements of a circuit such as inductors and capacitors portray transients until and unless they reach a steady state. These transient conditions are caused by any rapid or instantaneous change of the circuit voltage or current or when any element is added or removed from the circuit or due to the application or removal of the power source. Laplace Transform is a very useful mathematical tool that helps one to convert equations derived in the time domain to the frequency domain. (Dr. Gamal, 2011) A function f(t), t>=0, is transformed into its s-domain or frequency domain equivalent using the formula, F(s)=ʟ{f(T)}=∫0ᴔ e-stf(t)dt
As evident, Laplace transform is used to change differential equations in the time domain to algebraic equations in the frequency domain.
Question 4.2
Given a circuit with a resistance and inductance in series (I=0 when t=0), if a step input is applied, then current in the circuit is i=V/R(1-e-Rt/L) . And the expression for the exponential decay that occurs to the circuit current once the switch is opened is i=Ie-Rt/L or V/Re-Et/L . The current exponentially decays to zero. I is the initial value of the current. (Basic Electronics Tutorials, 2013)
References
Basic Electronics Tutorials. (2013). Parallel RLC Circuit and RLC Parallel Circuit Analysis. [online] Available at: http://www.electronics-tutorials.ws/accircuits/parallel-circuit.html [Accessed 7 May 2016].
Basic Electronics Tutorials. (2013). Series RLC Circuit and RLC Series Circuit Analysis. [online] Available at: http://www.electronics-tutorials.ws/accircuits/series-circuit.html [Accessed 7 May 2016].
Circuitstoday.com. (2011). Transformer -Working principle,Construction,Types of Transformers. [online] Available at: http://www.circuitstoday.com/transformer [Accessed 7 May 2016].
Dr. Gamal, S. (2011). Applications of Laplace transform to circuit analysis.
Electrical4u.com. (2016). Laplace Transforms | Table Method Examples History of Laplace Transform | Electrical4u. [online] Available at: http://www.electrical4u.com/laplace-transformation/ [Accessed 7 May 2016].
Electrical4u.com. (2016). What is transformer? Definition and Working Principle of Transformer | Electrical4u. [online] Available at: http://www.electrical4u.com/what-is-transformer-definition-working-principle-of-transformer/ [Accessed 7 May 2016].
John, B. (2003). Electrical Circuit Theory and Technology.
Asher, M. (2013). Fourier series And Fourier Transform. IOSR Journal of Mathematics, 4(6), pp.73-76.
Combined estimator Fourier series and spline truncated in multivariable nonparametric regression. (2015). ams.
External version of laplace method. (2014). imf.
Weisz, F. (2000). Riesz Means Of D-Dimensional Fourier Transforms And Fourier Series. Analysis, 20(2).
Ziemer, R. and Tranter, W. (2002). The Laplace transform. Harlow: Prentice Hall
