The Open University of Sri Lanka Department of Electrical and Computer Engineering Bachelor of Technology Honors in Engineering – Level 3 ECX 3231 – Electrical Circuits and Measurements Academic Year 2015/2016 Lab – 02 Session I – Network Synthesis Aim –To study the Synthesis of Passive Networks Objectives – 1. Understand LC network realization methods. 2. Understand the physical meaning of poles and zeros of a function. 3. Formulate a network function for a given requirement. 4. Learn to use pole – zero maps. Introduction – Network synthesis can be viewed as the inverse operation of the network analysis. In network analysis, the response is predicted for a known network by applying various electric circuit theorems. On the other hand, in Network synthesis, a network is obtained for a desired response. The driving point impedance Z(s) is a mathematical representation for the input impedance of a network in the frequency domain using Laplace transform ( s-domain) or Fourier transform. The expression is expanded and this expansion is transformed in to a network of electrical elements. There are number of realization methods which can be used to obtain the driving point impedance. Foster 2nd form is considered in this lab session. Frequencies at where the impedance becomes infinite are called Poles and frequencies where the impedance becomes zero are called Zeros. If the Poles and Zeros of a desired network are given, the impedance function can be formed as follows. Zeros – Z1, Z2, Z3, ….., Zn Poles – P1, P2, P3, ….., Pn 𝑍(𝑠) = (𝑠 − 𝑍1 )(𝑠 − 𝑍2 ) … … … . (𝑠 − 𝑍𝑛 ) (𝑠 − 𝑃1 )(𝑠 − 𝑃2 ) … … … . (𝑠 − 𝑃𝑛 ) Foster 2nd form: The Foster 2nd form of LC network realization is obtained by, I. II. Partial fraction expansion of admittance function Y(s) Identifying each term in the summation as admittances. Y(s) = Y1(s) + Y2(s) + ……. + Yn(s) Page 1 of 6 III. Obtain the values for each element. If, 𝑌𝑘 (𝑠) = 1 𝐴𝑠 + 1 𝐵𝑠 Yk(s) is a series configuration of an inductor and a capacitor as shown in Figure 1.1, where the inductance and the capacitance respectively equal to A and B. A Z(s) B Figure 1.1 Laboratory Tasks 1. Use the pole-zero combination given by your instructor and obtain the driving point impedance function Z(s) Important – Sinusoidal Frequency is always a complex conjugate pair. If an impedance function has, a Pole at A rads-1 and a Zero at B rads-1, the impedance function can be formed as follows, 𝑍(𝑠) = (𝑠 − 𝐵𝑖)(𝑠 + 𝐵𝑖) (𝑠 − 𝐴𝑖)(𝑠 + 𝐴𝑖) 2. Find the values for L, C elements using the Foster 2nd method and build the circuit. 3. Use a small resistor in series with the network as shown in the Figure 1.2 to observe a current waveform in phase with the voltage waveform. 4. Use the signal generator to supply appropriate sinusoidal signal and observe both current and voltage waveforms. 5. Find the frequencies at where the current is minimum (maximum impedance) and maximum (minimum impedance). Note that, there will be no phase difference between the current waveform and the voltage waveform at these frequencies. 6. Compare them with the pole and zero values which were given to you. 7. Change the circuit elements with the components given by the instructor and repeat step 5 8. Construct the pole – zero maps for all experiments separately using the frequencies you have found. Page 2 of 6 Z(s) R Figure 1.2 9. Try to use MATLAB® to find the partial fractions of the impedance function. 10. Try to plot the pole-zero graph of the impedance function using MATLAB®. Analysis of Results Compare the frequencies of poles and zeros with the given values. Discussion Discuss the importance of network realization regarding your field of specialization. Find real examples. Describe the meaning of a pole and a zero in your own words. Explain the importance of pole – zero maps in your own words. Session II – Use the X – Y mode of the oscilloscope for Electrical measurements Aim - Use the X-Y mode of the oscilloscope to plot characteristics of a system. Objectives – 1. Understand the use of X-Y mode of the oscilloscope 2. Understand the magnetic characteristics of a material. Introduction – Magnetic materials have non-linear magnetization characteristics. A great deal of information can be learned about magnetic properties by studying its hysteresis loop (B-H loop). The hysteresis loop shows the relationship between the induced magnetic flux density (B) and the magnetizing force (H). Once a material has been magnetized and the magnetizing force is removed, demagnetization does not follow the original magnetization path. Refer Figure 2.1. XY mode of the oscilloscope can be used to plot one varying voltage versus another. Usually the channel 1 of the oscilloscope is used for the independent variable (X) and the channel 2 for the dependent variable (Y). Page 3 of 6 Flux density (B) Magnetizing force (H) Figure 2.1 R Figure 2.2 i1 i2 l R1 N1 N2 R2 e2 C VC VR Figure 2.3 Equations magnetizing force (𝐻) = N1 = number of turns of coil 1 𝑁1 . 𝑖1 𝑙 l = length of magnetic circuit of coil 1 i1 = current through the coil 1 Voltage across R1 = 𝑉𝑅 = 𝑖1 . 𝑅1 Page 4 of 6 Induced e. m. f of coil 2 = 𝑒2 = 𝑁2 𝑑𝜙 𝑑𝑡 N2 = number of turns of coil 2 Magnetic flux = 𝜙 = 𝐴𝑆 𝐵 AS = Cross section area of the magnetic specimen B = Magnetic flux density Assuming R2 >> XC 𝑖2 = 𝑒2 ⁄𝑅 2 For the capacitor 𝑖2 = 𝐶 𝐻= 𝑑𝑉𝑐 𝑑𝑡 𝑁1 𝑉 𝑙𝑅1 𝑅 𝐻 ∝ 𝑉𝑅 𝐵= 𝑅2 𝐶 𝑉 𝑁2 𝐴𝑆 𝐶 𝐵 ∝ 𝑉𝐶 Laboratory Tasks Step 1 1. Build the circuit in shown in Figure 2.2 2. Use a small resistor for R. 3. Use channel 1 of the oscilloscope as the independent variable (Voltage across the diode) and channel 2 of the oscilloscope as the dependent variable (Current through the diode). 4. Select the suitable points where you should apply ground, channel 1 and channel 2 inputs. 5. Observe V-I characteristics of the diode using X-Y mode of the oscilloscope. Page 5 of 6 Step 2 1. Using the given equations find voltages in the circuit shown in Figure 2.3 which are proportional to B and H. 2. Use the signal generator to supply an appropriate sinusoidal signal to the circuit. 3. Select appropriate inputs for channel 1 and channel 2 of the oscilloscope. 4. Observe the hysteresis loop by using the X-Y mode in oscilloscope. Analysis of Results Compare the characteristic curve of the diode with the result you have obtained using X –Y mode Discussion Draw the hysteresis loop and find the area covered by the loop. Find the hysteresis loss caused by the core, using the area of the hysteresis loop. You can use a graph paper to find the area. Refer the course material (Book 2) for the calculation. Discuss the importance of the use of XY mode and its limitations. Write a comprehensive note on Measurement techniques you have learned in this lab session. State any assumptions made in this experiment. You will be evaluated during the viva mostly based on this note. (Please limit the note to a one side of an A4 page.) Page 6 of 6