THERE MUST BE 50 WAYS TO FIND YOUR VALUES: AN EXPLORATION OF CIRCUIT ANALYSIS TECHNIQUES FROM OHM S LAW TO EQUIVALENT CIRCUITS


 Magdalen McBride
 3 years ago
 Views:
Transcription
1 THERE MUST BE 50 WAYS TO FIND YOUR VALUES: AN EXPLORATION OF CIRCUIT ANALYSIS TECHNIQUES FROM OHM S LAW TO EQUIVALENT CIRCUITS Kristine McCarthy Josh Pratti Alexis RodriguezCarlson November 20, 2006
2 Table of Contents Introduction:... 3 Objectives:... 3 Discussion of Theory:... 4 Circuit Analysis Tools: Ohm s Law, Kirchhoff s Voltage Law, and Kirchhoff s Current Law:... 4 Circuit Analysis Tools: Nodal Analysis and Loop Analysis:... 6 Circuit Analysis Tools: Superposition and Source Transformation: Equivalent Circuits: Thevenin Circuits: Equivalent Circuits: Norton Circuits: Practical Circuit Analysis: A Wheatstone Bridge: Verifying Nodal Analysis with a Real Circuit: Analyzing A Single Circuit Multiple Ways: Loop Analysis: Superposition: Source Transformation: Comparisons of Results: Verifying Thevenin and Norton Equivalent Circuits: Analyzing the Master Circuit: Calculating V Th and R Th : Conclusion: List of Tables: List of Figures: Page 2 of 59
3 Introduction: The purpose of this paper is to give a brief explanation of several of the basic laws of electrical circuit analysis and of different types of circuit analysis tools. To try to make this information as accessible as possible, the paper is divided into several sections. First, there is a list of objectives, listing exactly which laws and tools will be explored in this paper. The next section, entitled Discussion of Theory, gives a brief overview of what those laws and tools are and how and when they can be used to analyze networks. Following that there is a short section on Wheatstone Bridges detailing how several of principles discussed in the Discussion of Theory Section can be used in a practical way in a measurement system, and confirming those uses by detailing the results of an experiment involving a Wheatstone Bridge. The rest of this paper is dedicated to the experimental use and verification of each of the laws and principles detailed in the list of objectives, and includes the analysis and manipulation of several circuits by both measurements and calculations. These examples show that, in each case, the analysis methods discussed are accurate and compatible with one another when used with a variety of circuits. Objectives: To employ and verify several different circuit analysis techniques, including: o Ohm s Law o Kirchhoff s Voltage Law (KCL) o Kirchhoff s Current Law (KVL) o Nodal Analysis o Loop Analysis Page 3 of 59
4 o Superposition o Source Transformation To design and use Thevenin and Norton Equivalent Circuits, including: o To determine a network s Thevenin and Norton equivalent circuits by both calculations and measurements o To determine the different values of the Thevenin Resistance (R Th ), the Thevenin Voltage (V Th ), the Norton Current (I N ), and the Norton Resistance (R N ) for different loads in the same circuit o To build the Thevenin circuits by using a potentiometer in series with a resistor to create R Th o To verify that the equations of equivalent by comparing voltage and current measurements of from the original circuit with those from the Thevenin circuits o To transform the Thevenin Circuits into Norton Circuits To use PSpice to model and simulate circuits Discussion of Theory: Circuit Analysis Tools: Ohm s Law, Kirchhoff s Voltage Law, and Kirchhoff s Current Law: Three of the most basic tools in circuit analysis are Ohm s Law, Kirchhoff s Voltage Law, and Kirchhoff s Current Law. At lease one of these laws is used in the analysis of any circuit. Ohm s Law states that voltage is equal to the current times the resistance in a circuit or a resistor. Practically, this means that if you have a circuit and you know any Page 4 of 59
5 two of those values, that the third can easily be found, and that if you know the value of the resistor and either its voltage drop or the current through it, the other value can be easily found. The most common way to see Ohm s Law stated is V = IR Equation 1 Kirchhoff s Voltage Law states that in a given loop, the sum of all voltages must equal zero. In Figure &, this means that the source voltage must be equal to the voltage drop across R 1. Figure 1: In the above circuit, V 1 = V R1 This will hold true for any complete loop around a circuit where you begin and end at the same element and do not pass through any element or node twice. Kirchhoff s Current Law states that at every node, the sum of all currents entering and exiting must equal zero. This law basically means is that charge cannot accumulate in a node. Basically, what goes in must come out. If you think of it in terms of a water pipe system for a house, and think of the water as current and a point at which the pipe splits as a node, the water coming into the split must be the same as the water coming out of the it. Page 5 of 59
6 Figure 2: In the above circuit, KCL says that i 1 = i 2 + i 3, and KVL says that V S = V R1 = V R2 Circuit Analysis Tools: Nodal Analysis and Loop Analysis: Nodal analysis and loop analysis are other powerful tools in circuit analysis. To perform nodal analysis, you first label the nodes in the circuit, as in Figure 3, where the nodes are labeled A, B, and C. The node voltages are defined as positive with respect to a common point, generally the ground point in the circuit. You then write the node equations according to the steps below, and solve them to find the voltage values at each node. Once you have determined the node voltages, you can use those values to find the branch currents and resistor voltage drops. Page 6 of 59
7 Figure 3: Node Analysis example circuit To write the node equations, first set up a system of three equations each consisting of three unknowns (the node voltages) on the left and the current entering and exiting the node on the right. V A V B V C = V A V B V C = V A V B V C = To find the coefficients of each variable, add the conductance (inverse of resistance, see equation) or each resistor touching that node and enter that value in the main diagonal (Row 1, Column 1; Row 2, Column 2, and Row 3, Column 3) G = 1/R Equation 2 Page 7 of 59
8 G1 + G2V A V B V C = V A G2+G3+G4V B V C = V A V B G1+G3+G5V C = To fill in the remaining coefficients, enter the negative value of the conductance of the resistors touching 2 nodes. So, for Row 2, Column 1, we entered the negative value of the sum the conductance of the resistors touching nodes A and B. G1 + G2V A G2 V B G1 V C = G2 V A G2+G3+G4V B G3 V C = G1 V A G3 V B G1+G3+G5V C = Last, to fill in the right hand side of the equations enter the value of all current sources feeding or drawing from a particular node. In the case of Figure 3 it is very simple because there is only one current source, and it is entering Node A. The rest of the values will be zero. G1 + G2V A G2 V B G1 V C = IDC G2 V A G2+G3+G4V B G3 V C = 0 G1 V A G3 V B G1+G3+G5V C = 0 These three equations can be solved by a variety of means, including algebraic substitution. In this paper, we will use matrices. From the above equations, 3 matrices are created. Matrix A = [G 1 + G 2 G 2 G 1 ] [G 2 G 2 +G 3 +G 4 G 3 ] [G 1 G 3 G 1 +G 3 +G 5 ] Page 8 of 59
9 Matrix B = [V A ] [V B ] [V C ] Matrix C = [IDC] [0 ] [0 ] So, Matrix B contains all of the unknown voltage values. To solve for them, use the relationship: A * B = C Equation 3 And solve for Matrix B by multiplying Matrix C by the inverse of Matrix A, using MATLAB or a calculator: B = A 1 * C Equation 4 The resulting matrix will have the values of V A, V B, and V C. Using those values and Ohm s Law, all of the other values in Figure 3 can be calculated. Loop analysis is the process of creating equations with unknowns by walking around the loops of a circuit. A loop is any path where the current begins and ends at the same element and does not cross any node or element twice. For the purposes of loop analysis, it is also important that there are no loops inside any of the other loops. In Figure 4, there are 2 loops, labeled A and B. Page 9 of 59
10 Figure 4: Example Circuit with 2 loops In order to find the values of current and voltage in Figure 4 using loop analysis, we first calculate the currents in loops A (I A ) and B (I B ) by solving a system of equations by using matrices, just as we did in nodal analysis. The difference is that instead of using conductance values, we will use resistance values touching the loop for the main diagonal and the resistance values touching both loops for the other values, and the right hand side of the equation will hold the negative voltage values instead of current values. Since there are 2 unknowns, we will have a 2x2 matrix, and two 2x1 matrices. Matrix A = [R 2 +R 1 +R 3  R 3 ] [R 3 R 3 +R 4 ] Matrix B = [I A ] Page 10 of 59
11 [I B ] Matrix C = [ V 1 ] [V 2 ] Once again, Matrix B contains all of the unknown values. We solve for them, in the same way we solved for the unknown node voltages above by using the relationship: A * B = C Equation 5 And solve for Matrix B by multiplying Matrix C by the inverse of Matrix A, using MATLAB or a calculator: B = A 1 * C Equation 6 Once the values of I B and I A are known all of the other values can be found using Ohm s Law, as will be seen in detail later. Circuit Analysis Tools: Superposition and Source Transformation: Superposition is a method of circuit analysis used when there is more than one voltage and/or current source in a network and it is easier to evaluate them one at a time then both together. To do that, you choose one source to evaluate, and redraw the circuit as if the other circuits weren t there. When you redraw the circuit, any voltage sources you are not currently evaluating are shorted, and any unused current sources are replaced by an open. Page 11 of 59
12 Figure 5: Circuit from Figure 4 with V 2 replaced with short Figure 6: Circuit from Figure 4: Example Circuit with 2 loops with V 1 replaced with a short Page 12 of 59
13 Then, the voltage or current for specific nodes and branches can be found. This is done for each source, always eliminating the other sources. When the circuit has been evaluated with each source, the values of the node voltages and branch currents from each individual circuit are added together (see Figure 7) to determine the values on the original circuit containing all of the sources. Figure 7: Circuit from Figure 4 showing principal of superposition Equivalent Circuits: Thevenin Circuits: When working with a variable load, it can be timeconsuming and painstaking to calculate the changes which take place within a circuit as the load varies. For example, in the drawing below, R 5 is a variable load resistor. The current and voltage values shown are with R 5 = 5.0 kω. Page 13 of 59
14 Figure 8: Circuit with Variable Load = 5.0 kω When the value of R5 changes to 10 kω, the circuit looks like this: Figure 9: Circuit with Variable Load = 10 Ω Had this network not been modeled in PSpice, both Figure 8 and Figure 9 would need to be analyzed in order to calculate and compare the values of voltage and current Page 14 of 59
15 for R 5 (V R5 and I R5 ). As you can see by comparing Figure 8 and Figure 9, every value except for the source voltage (V S ) is different. This means that in order to find the value of the V R5 and I R5, you must calculate at least the two loop currents. If you need that information for many values of R 5 this is a daunting task with room for many errors. If you have a situation where it doesn t matter what is going on in the rest of the circuit, it only matters that R 5 receives the same current and has the same voltage drop as it would if the rest of the circuit were attached, you can build an equivalent circuit. That is, a circuit that is equivalent to the original circuit behind R 5. Figure 10: Circuit behind R 5 from Figure 8 By creating an equivalent circuit consisting of only two elements, the calculations for V R5 and I R5 become much quicker. There are two types of equivalent circuits that we are concerned about in this paper: Thevenin and Norton Equivalent Circuits. Page 15 of 59
16 Figure 11: Generic Thevenin Circuit A Thevenin Circuit consists of one voltage source (V Th ) in series with a resistor (R Th ). These values are determined by transforming the circuit and performing several calculations. To find the value of R Th, the voltage source in the original circuit is replaced by a short and the nodes where the variable resistor will be are left open. Page 16 of 59
17 Figure 12: Circuit used to determine R Th Then, the total resistance in the circuit is calculated to determine the resistance as seen between the points A and B in Figure 12. To do this you start at the other end of the circuit and begin tallying R Th, which is equal to the total resistance in Figure 12. Upon examining Figure 12, it becomes apparent that R 1 is in parallel with R 2, and that they are in series with R 2. So, R Th is: R Th = R 2 + R1 * (R3 + R4) = 2 kω + 1 kω * 7 kω = kω Equation 7 R 1 + R 3 + R 4 1 kω + 7 kω To find V Th the voltage source is replaced into the network and the location of the variable resistor remains open. V Th is equal to the voltage drop between points A and B. Page 17 of 59
18 Figure 13: Circuit used to determine V Th To find V Th, first apply Kirchhoff's Voltage Law to the loop in Figure 13 in order to find the current (i). 10 V = (1i + 3i + 4i) kω Equation 8 = 8i kω i = 1.25 ma Because the space between point A and B is open, there is no current flowing through R 2. Therefore, the voltage drop from A to B is equal to the voltage drop across R 3 and R 4 (between the blue dots on Figure 13). To find that value we apply Ohm s law to R 1 in order to find V R1. V R1 = I * R1 = 1.25 ma * 1 kω = 1.25 V Equation 9 If V R1 is the voltage drop across R 1, than the voltage drop from point A to point B in Figure 13 is equal to V S V R1. So: V Th = V S V R1 = 10 V V = V Equation 10 Page 18 of 59
19 Modeled in PSpice, the Thevenin Circuit looks like this: Figure 14: Thevenin Equivalent Circuit for Figure 8 To verify that Figure 8 and Figure 14 are equivalent when the load resistor if R 5, compare the two side by side and note that I R5 and V R5 are the same in both circuits: Page 19 of 59
20 Figure 15: Thevenin Equivalent Circuit to Figure 8 with voltage and current values Figure 16: Circuit from Figure 8 Page 20 of 59
21 Equivalent Circuits: Norton Circuits: Another type of equivalent circuit is the Norton Circuit. The Norton Circuit uses a current source in parallel with a resistor in order to equate the circuit behind the load resistor. Figure 17: Generic Norton Circuit There are two ways to find the values of I N and R N. If you have already found the Norton Circuit, there is a relationship between I N and V Th, and the value of R N is equal to R Th stays the same. I N = V Th Equation 11 So, looking back to Figure 14: Thevenin Equivalent Circuit for Figure 8, the value of the I N is: R Th I N = V Th = 8.75 V = ma Equation 12 R Th 2875 Ω Page 21 of 59
22 The Norton Equivalent Circuit looks like this: Figure 18: The Norton Circuit Equivalent to the circuits in Figure 8 and Figure 14 Note that, as in Figure 8 and Figure 14, the voltage drop across the 5 kω resistor is V and the current through it is ma. So, from the view point of the 5k resistor, the circuits in Figure 8, Figure 14, and Figure 18 are all equivalent. To determine the values of I N and R N without first finding the Thevenin Circuit, start by replacing the load resistor with a short. I N is equal to the current through points A and B, and R N is equal to the equivalent resistance in the circuit, and can be found using the same methods used to find V Th and R Th. Page 22 of 59
23 follows: Figure 19: Circuit used to find I N and R N From Figure 19 the value of R N can be found by calculating the total resistance as R N = (R3 + R4) * R1 +R 2 = (4 kω + 3 kω) * 1 kω + 2 kω R 3 + R 4 + R 1 4 kω + 3 kω + 1 kω = 7 * * 10 3 = kω Equation 13 8 * 10 3 In order to find the total current, I T, the total resistance must be calculated from the viewpoint of the 10 V source, meaning that the 4 kω and 3 kω are in parallel with the 2 kω resistor, and that they are all in series with the 1 kω resistor. R T = (R 3 + R 4 ) * R 2 +R 1 = (4 kω + 3 kω) * 2 kω + 1 kω R 3 + R 4 + R 2 4 kω + 3 kω + 2 kω = (7 * 103) (2 * 103) + 2 * 10 3 = kω Equation 14 9 * 10 3 Page 23 of 59
24 R 3 R 4 To find I T, divide V S by R T I T = VS = 10 V = ma Equation 15 R T 2555 Ω Finally, to find I N, use the Current Division Law: I R2 = I N = R3 + R4 * I T = 3 kω + 4 kω * ma = ma Equation 16 R 2 + R 3 + R 4 2 kω + 3 kω + 4 kω These values of I N and R N are the same as those determined using IN = VTh = 8.75 V = ma Equation 12 and those calculated by PSpice. Practical Circuit Analysis: A Wheatstone Bridge: A Wheatstone Bridge, invented by Samuel Hunter Christie in 1933, is comprised of a set of four resistors as shown in Figure 20. R 1 R 2 DC 6 V A B Figure 20: A Wheatstone Bridge Both arms of the bridge circuit must be balanced and the voltage from A to B should be 0V. If this is not the case, then there is something affecting the system, such as one resistor being unequal to the others. This voltage differential is the value measured when Page 24 of 59
25 using a Wheatstone Bridge in data acquisition, such as in weight measurement systems. In these systems, one or more of the resistors is replaced by a strain gauge of equivalent resistance, as shown in Figure 21. Figure 21: A ¼ Wheatstone Bridge Circuit A strain gage is a sensor that converts force into a change in electrical resistance. When s the train gauge is put under tension or compression its resistance changes and voltage output changes accordingly. When the metal foil of the gauge is placed under tension, is becomes longer and thinner and its resistance increases. When the foil is placed under compression, it becomes shorter and broader and its resistance decreases. If all of the resistors in Figure 20 are of equal value, the equivalent resistance as seen by the source is the value of one resistor. R 1 and R 3 are in series as are R 2 and R 4 and are all equal to 120 Ω, so each arm shows a total resistance of 240 Ω. The resultant resistances are in parallel with each other, and so act like one 120 Ω resistor, which is the equivalent resistance as seen by the voltage source. (For a more complete discussion of resistors and their behavior, please see our October, 2006 paper entitled The Building Blocks of Circuit Analysis: The Resistor, Ohm s Law, and Conservation of Power.) Page 25 of 59
26 When one of the resistors is replaced by a strain gage, as in Figure 21, the change in that strain gage can be calculated by the change in the voltage differential between points A and B. For our experiment, each of the resistors R 1 R 4 was valued at 120 Ω. We measured each of the resistors to obtain the actual resistances for calculations as are shown in Table 1. Resistor # Resistance (Ω) , ,390 Table 1: Resistance Values for our Wheatstone Bridge And the resulting circuit was: Figure 22: Our Wheatstone Bridge Page 26 of 59
27 Our first step to find the expected voltage differential was to calculate the equivalent resistances in legs A and B of Figure 22. R A = R 1 + R 3 = = Ω Equation 17 R B = R 2 + R 4 = = Ω Equation 18 And so, the equivalent resistance (R Eq ) in the entire circuit is equal to: R Eq = R A * R B = Ω * Ω = Ω Equation 19 R A + R B Ω Ω Using Ohm s Law, we found the total current in the circuit: I T = V S = 6 V = ma Equation 20 R Eq Ω From there we calculated the current through the legs (I A and I B ) of the circuit, and the voltage differential, V A  V B. I A = R B * I T = Ω * ma = ma Equation 21 R A + R B Ω I B = I T I A = ma ma = ma Equation 22 We again use Ohm s Law in order to find the values of V R3 and V R4, which are equal to V A and V B : V A = V R1 = I A * R 3 = ma * Ω = 3.009V Equation 23 V B = V R2 = I B * R 4 = ma * Ω = V Equation 24 And, at last, we find voltage differential: V Diff =VA  V B = V V =.022 V Equation 25 To see how accurate our calculations were, we then actually measured the values calculated in IA = RB * IT = Ω * ma = ma Page 27 of 59
28 Equation 21 through VDiff =VA  VB = V V =.022 V Equation 25. The results are below: Measured Value Calculated Values % Error ((Measured  Calculated)/Measured * 100) I A (ma) % I B (ma) % V A (V) % V B (V) % V Diff (V) % Table 2: Measure vs. calculated values for Figure 22 We then calculated how much V Diff would change if a R5 from Table 2 was placed in parallel with R 4. Figure 23: Circuit from Figure 22 with R 5 in parallel with R 4 To do this, we first had to calculate the new equivalent resistance. First, we found the equivalent resistance for R 4 and R 5, R eq1 Page 28 of 59
29 R eq1 = R4 * R5 = Ω * kω = Ω Equation 26 R4 + R Ω kω We then repeated the steps used to find the original equivalent resistance of the circuit in Figure 22, IA = RB * IT = Ω * ma = ma Equation 21 through VDiff =VA  VB = V V =.022 V Equation 25. We then actually placed R 5 in parallel with R 4, as shown in Figure 23, and compared the measured value for V DIFF with our calculated value. We then did the same for R 1, as shown in Figure 24, Figure 24: Figure 23: Circuit from Figure 22 with R5 in parallel with R4 and R 6 in parallel with R 1 Again, we measured and compared the results, first finding the new equivalent resistance of R1 and R4, R eq3, and following the same steps Page 29 of 59
30 Measured Values Calculated Values % Error ((Measured  Calculated)/Measured * 100) V Diff (V) (R 5 in parallel w/ R 4 ) % V Diff (V) (R 6 in parallel w/ R 1 ) % Table 3: Measured differential voltage compared to calculated differential voltage for Figure 23 and Figure 24 From these results we can conclude that the simpler the circuit, the closer to zero the differential voltage will be. This holds true because if you look at the values for V DIFF, the more elements that are added, the larger the discrepancy. This may also be due to error in measurement of the resistor values and/or error with in other measurements. When more elements are added to a circuit, the chance for measuring and rounding errors is also increased. To get as accurate results as possible, it is important to measure carefully and to as many significant digits as possible. Generally, though, since all of our measured values were within 10% of our calculated values, we can attribute this error to rounding and feel confident that the methods of analysis used in this section are accurate. Page 30 of 59
31 Verifying Nodal Analysis with a Real Circuit: Figure 25: Circuit on which to perform nodal analysis In order to verify nodal analysis, we built the circuit in Figure 25. We then wrote the node equations using the system outlined in the section entitled Circuit Analysis Tools: Nodal Analysis and Loop Analysis:. This resulted in the following 3 matrices: Matrix A = [ ] [ ] [ ] Matrix B = [V 1 ] [V 2 ] [V 3 ] Page 31 of 59
32 Matrix C = [I T ] [ ] [0 ] = [ 0 ] [0 ] [ 0 ] A * B = C Equation 27 B = A 1 * C = [8.9293] = [V 1 ] [5.3422] = [V 2 ] [4.4791] = [V 3 ] The comparison of the calculated node voltages to the measured node voltages shows: Node Measured Voltage (V) Calculated Voltage (V) Percent Error % % % Table 4: Comparison of measured and calculated node voltages Branch Current (ma) I I I I I I Power Table 5: Branch Currents and power The equations that follow are the equation derived by using nodal analysis on the circuit. These are the branch currents and are derived by simply looking at the circuit and applying nodal analysis. Also included is the equation to derive the power delivered to the system. To calculate this you must first take the voltage or the source and multiply that with the current through R 1 and this will give you the power in Watts. Page 32 of 59
33 I 1 15 V R 1 = Equation 28 1 I 2 V V R 1 2 = Equation 29 2 V 2 I 3 = Equation 30 R3 I 4 V V 2 3 = Equation 31 R 4 I 5 V V 1 3 = Equation 32 R 5 V 3 I 6 = Equation 33 R6 P = Vs * I 1 Equation 34 As you can see, nodal analysis is an extremely accurate way to analyze circuits. The fact that we got such a small percentage of error for each calculation in Table 4 shows that the method of analysis is working and the only error in the system is wire resistance and measurement rounding. Page 33 of 59
34 Analyzing A Single Circuit Multiple Ways: Loop Analysis: i 3 i 2 i 5 i 1 i 6 i 4 Figure 26: Circuit analyzed by loop Analysis Using the same methods discussed in the section entitled Circuit Analysis Tools: Nodal Analysis and Loop Analysis, the following matrices were built. Matrix A = [ ] [ ] [ ] Matrix B = [I A ] [I B ] [I C ] Page 34 of 59
35 Matrix C = [6.025] [15.047] [ 0 ] A * B = C Equation 35 C * A 1 = B Equation 36 B = loop currents = [ μa ] [ μa] [ μa ] Once the values of I A, I B, and I C were known, we could find the branch currents: i 1 = I A = μa Equation 37 i 2 = I A I C = μa μa = μa Equation 38 i 3 = I C = μa Equation 39 i 4 = I A I B = μa μa = μa Equation 40 i 5 = I B = μa Equation 41 i 6 = I B I C = μa μa = Equation 42 And with the branch currents, we could find voltage drops across each resistor: V R1 = i 3 * R 1 = μa * 9.93 kω =.4802 V Equation 43 V R2 = i 2 * R 2 = μa * kω = V Equation 44 V R3 = i 2 * R 3 = μa * 4.65 kω = V Equation 45 V R4 = i 5 * R 4 = μa *.976 kω =.787 V Equation 46 V R5 = i 1 * R 5 = μa * kω =.0267 V Equation 47 Page 35 of 59
36 V R6 = i 4 * R 6 = μa * 6.82 kω = V Equation 48 V R7 = i 6 * R 7 = * kω = V Equation 49 And that enabled us to find the node voltages: V 1 = V 9.022V  V R5 = V V = V Equation 50 V 2 = V 1  V R2 = V = V Equation 51 V 3 = V 2  V R3 = V V = V Equation 52 V 4 = V 3  V R4 = V V = V Equation 53 V 5 = V 9.022V = V Equation 54 V 6 = V 3  V R6 = V V = V Equation 55 Superposition: To see if the results would be equivalent, we then analyzed the same circuit using the principal of superposition. We began by removing the 9 V source and replacing it with a short. Page 36 of 59
37 Figure 27: Circuit from Figure 26 with 9 V source removed We then used loop analysis to find the loop currents: Matrix D: [ ] [ ] [ ] Matrix E [I D ] [I E ] [I F ] Matrix F [ ] [ ] [ 0 ] D * E = F Equation 56 E = D 1 * F Equation 57 E = loop currents = [ μa] [512.8 μa ] [ μa] We then repeated the process with the 15 V source replaced with a short: Page 37 of 59
38 Figure 28: Circuit from Figure 26 with 15 V source removed Matrix G = [ ] [ ] [ ] Matrix H = [I G ] [I H ] [I J ] Matrix J = [9.022] [ 0 ] [ 0 ] Page 38 of 59
39 G * H = J Equation 58 H = G 1 * J Equation 59 H = loop currents= [835.6 μa] [342.0 μa] [383.8 μa] Once we knew that values of the loop currents for the circuits with each source, the total loop currents for the whole circuit could be found. Figure 29: Circuit from Figure 26 I A = I D + I G = μa μa = μa Equation 60 I B = I E + I H = μa μa = μa Equation 61 I C = I F + I J = μa μa = μa Equation 62 Page 39 of 59
40 Source Transformation: The last way we analyzed this circuit was using source transformation. We found the values of the current sources as described in the section entitled Circuit Analysis Tools: Superposition and Source Transformation, and the results of those calculations can be seen in Figure 30. Figure 30: Circuit from Figure 26 with sources transformed We then used nodal analysis as described in the section Circuit Analysis Tools: Nodal Analysis and Loop Analysis. Three matrices were created. Matrix A = [ ] [ ] [ ] [ ] Page 40 of 59
41 Matrix B = [V A ] [V B ] [V C ] [V D ] Matrix C = [ ] [ 0 ] [ ] [ 0 ] A * B = C Equation 63 B = A 1 * C Equation 64 B = node voltages = [8.995 V] [ 9.14 V ] [9.303 V] [8.513 V] Comparisons of Results: Table 6: Comparisons of results from loop analysis, superposition, and source transformation compares the measured results to each of the calculated results for individual voltages and currents. Note that even the highest percent difference between the measured and calculated values is less than 4%. Page 41 of 59
42 Voltage (V) Node, Element, or Loop % difference from measured % difference from measured % difference from measured Resistance Value (kω) Measured Loop Analysis Super Position Source Transformation Node % % Node % % Node % % Node % % Node % % Node % % R % % % R % % % R % % % R % % % R % % R % % R % % % Loop A Loop B Loop C Node, Element, or Loop Resistance Value (kω) Loop Analysis % difference from measured Current (μa) Super Position % difference from measured Source Transformation % difference from measured Measured Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 R % % % R % % % R % % % R % % % R % % R % % R % % % Loop A % % Loop B % % Loop C % % Table 6: Comparisons of results from loop analysis, superposition, and source transformation analysis Page 42 of 59
43 Verifying Thevenin and Norton Equivalent Circuits: Analyzing the Master Circuit: Our goal in this part of the experiment was to analyze specified circuits to determine their Thevenin and Norton Equivalent Circuits, then to build the Thevenin circuits to confirm that they are indeed equivalent. Figure 31: Master Circuit We worked with one main circuit (Figure 31). To make sure that we got the most accurate results possible, we measured the actual resistance of each resistor and the actual voltage output from the power source. The results of those measurements can be seen in Figure 31 and Table 7. Resistor # Resistor Value (kω) Page 43 of 59
44 Resistor # Resistor Value (kω) Table 7: Measured Resistance Values Figure 32: Master Circuit Once we knew the values of the resistors and the actual voltage output by the power source, we calculated the voltage drops across each resistor. In order to do that, we first had to calculate the total resistance (R T ) and the total current (I T ) in the circuit, and the branch currents (I 2 and I 3 ) (see Figure 32). Finding total resistance: R T = R 1 + (R 3 + R 4 ) (R 2 + R 5 ) = 3.9 kω + (2.137 kω kω) (4.622 kω kω) R 3 +R 4 +R 2 + R kω kω kω kω = 3.9 kω kω * kω = kω kω Equation 65 Page 44 of 59
45 Finding total current: I T = V S = V = * 103 A = ma Equation 66 R T kω Calculating the branch currents: I 2 = (R 2 + R 5 ) * I T = kω * ma = * 104 A = μa R 3 +R 4 +R 5 + R kω Equation 67 I 3 = I T I 2 = ma μa = μa Equation 68 Calculating voltage across each resistor: V R1 = I T * R 1 = ma * 3.9 kω = V Equation 69 V R2 = I 3 * R 2 = μa * kω = V Equation 70 V R3 = I 2 * R 3 = μa * kω = V Equation 71 V R4 = I 2 * R 4 = μa * kω = V Equation 72 V R5 = I 3 * R 5 = μa *.982 kω =.8653 V Equation 73 Calculating the Node Voltages: V A = V1 = V Equation 74 V B = V A V R1 = V V = V Equation 75 V C = V B V R2 = V V =.8660 V Equation 76 V D = V B V R3 = V V = V Equation 77 Page 45 of 59
46 We then built the circuit according to Figure 32 and measured the voltage across each resistor and compared them to our calculated values. The results were all within 1% of each other, confirming our calculations. Resistor # Resistor Value (kω) V M (V) V C (V) % Error % % % % % Table 8: Measured and calculated voltage drops across each resistor Figure 33: PSpice Model of Master Circuit with voltage and current values Calculating V Th and R Th : Armed with all of the information gathered in Equations 47 thru 52, we were ready to begin calculating the values of V Th and R Th for our equivalent circuits. We used Page 46 of 59
47 the same process to obtain these numbers as was explained in the section entitled Discussion of Theory: Equivalent Circuits: Thevenin Circuits. We began by assuming that R 5 was the load. We first drew the circuit with an open across R 5 and a short across the voltage source in order to calculate the total resistance from the viewpoint of R 5, which is equal to R Th. Figure 34: Finding R Th with R 5 as the load R Th = (3.9) (2.137 kω kω) kω = kω Equation kω kω Then, we drew the circuit leaving R 5 open and reinserting the voltage source in order to find V Th. Page 47 of 59
48 Figure 35: Finding V Th with R 5 as the load The first step in finding V Th is to find the total current (I 1 ) in the one closed loop that exists in Figure 35. To do this we use Kirchhoff s Voltage Law, starting with the voltage source (V S ). V S + (R 1 + R 3 + R 4 ) * I1 = V + (3.9 kω kω kω) * I 1 = kω * I 1 = V I 1 = ma Equation 79 After finding the value of I 1, the value of the current drop across R 1 can be found by using Ohm s Law. V R1 = I 1 * R 1 = ma * 3.9 kω = V Equation 80 Page 48 of 59
49 Since there is an open after R 2, no current will flow through it. Therefore, we know that voltage drop across R 3 and R 4 will be equal to V Th. To find the voltage drop across R 3 and R 4 we subtract V R1 from the total voltage. We know that the voltage at that node will be equal to the voltage drop across R3 and R4 because on the other side of those resistors is the ground, which is always equal to 0 V. V Th = V R3R4 = V S  V R1 = V V = V Equation 81 We also found the values of V Th and R Th by building and Figure 34 and Figure 35 and actually measuring the values (V ThM and R ThM ). The results of both methods can be seen in Table 9, along with the percent difference to one another. Load Resistor R ThC R ThM % Difference V ThC V ThM % Difference R % % Table 9: Measured and Calculated Values of V Th and R Th with R 5 as the load The measured and calculated values were sufficiently close to convince us that our calculations were correct, and that the differences could be explained by rounding errors. However, to get the most accurate results possible, we used the measured values when building the Thevenin Circuit. Once the values of V Th and R Th were known we built the Thevenin Equivalent Circuit in PSpice and let it calculate the voltage drop across and current through R 5 to make sure that, in theory at least, the two circuits were equivalent. Checking it in PSpice was a quick and easy way to make sure that the circuit we designed was correct before we built it. The result can be seen in Figure 36. Page 49 of 59
50 Figure 36: PSpice Model of Thevenin Circuit with R 5 as the load Once we had confirmation from PSpice that the circuit we designed was indeed an equivalent circuit, we built the actual equivalent circuit using on a bread board, using a potentiometer in series with a resistor to create R Th. Then, we measured the actual values of V R5 and I R5 to see if the circuits were indeed equivalent in real life. After that, we calculated the value of I N for a Norton equivalent circuit by using the values of V Th and R Th. I N = V Th = V = ma Equation 82 R Th kω Page 50 of 59
51 Figure 37: PSpice Model of Norton Circuit with R 5 as the load This process was repeated four more times, each time with a different resistor acting as the load resistor. The PSpice models of the Thevenin and Norton Circuits for resistors R 2, R 3, and R 4 are below, as well as comparisons of the measured and calculated values of V Th and R Th for each resistor (see Table &). There is no circuit with R 1 as the load because there is no circuit behind it, since the short where R 1 exists stops any current from traveling. Load Resistor # R ThC R ThM % Difference V ThC V ThM % Difference % % % % % % % % Table 10: Measured and Calculated Values of V Th and R Th with R 2 thru R 5 as the load Page 51 of 59
52 Figure 38: PSpice Model of Thevenin Circuit with R 2 as load Figure 39: PSpice Model of Norton Circuit with R 2 as load Page 52 of 59
53 Figure 40: PSpice Model of Thevenin Circuit with R 3 as load Figure 41: PSpice Model of Norton Circuit with R 3 as load Page 53 of 59
54 Figure 42: PSpice Model of Thevenin Circuit with R 4 as load Figure 43: PSpice Model of Norton Circuit with R 4 as load Page 54 of 59
55 To verify that the Thevenin Circuits were equivalent to the Master Circuit (Figure 32), we measured the voltage drops across the load resistors in the Thevenin Circuits (Figure 38 thru Figure 43) to see if they were the same as in the Master Circuit. As you can see in Table 11, all of the voltage drops across the chosen load resistor in the Thevenin Circuits were less than.5% from the voltage drop in the Master Circuit. Although we did not actually build the Norton Circuits, PSpice verifies that the values are also quite close. Load Resistor # Resistor Value (kω) Voltage In Thevenin Circuit V M (V) Voltage In Master Circuit V M (V) % Difference % % % % Table 11: Measured Values V Th for R 2 thru R 5 as the load in the Master Circuit and the Thevenin Circuits To get a better idea of how accurate PSpice is in evaluating circuits, we built a table comparing the measured, calculated, and simulated (V P ) voltage drops across the resistors in the Master Circuit. Load Resistor # Resistor Value (kω) V M (V) V C (V) V P (V) % Diff of V P to V M % Diff of V P to V C % 0.36% % 0.02% % 0.00% % 0.00% % 0.01% Table 12: Comparisons of PSpice Voltage drops in the Master Circuit across each resistor to the measured and calculated values Page 55 of 59
56 As seen in Table 12, the values from PSpice were within.6% of the measured values and almost all within.03% of the calculated values. It makes sense that PSpice would be closed to the calculated values than the measured value, since any rounding that occurred in the measurement instruments or the reading of that instrument would feed both sets of calculations. Conclusion: The general conclusion to be drawn from these many examples is that all of the circuit analysis techniques we used have been verified and can be said to be accurate. Throughout this paper, the highest percent difference between a measured and a calculated value was less than 10%, and that figure includes any error that can be attributed from rounding, calculation, or measurement. Specifically what we have found is that it is extremely important to take measurement as accurately as possible and to keep as many significant figures as you possibly can. A small discrepancy in a measurement that is used in many calculations will lead to many discrepancies. For example, when looking at the section A Wheatstone Bridge:, an incorrect measurement of just one of the resistors in Table 1 would have lead to incorrect values for every calculated value of the differential voltage. As it is, most if not all of the error in Table 3 can be attributed to the rounding of measurement values, since even the digital multimeter we use is only accurate to 5 significant digits. While that is a high level of accuracy, ultimately all of the small errors add up to a much larger discrepancy. While all of the other values in Table 3 have a less than 1% error, the differential voltage, which is calculated by using all of the other values, has an 8.33% error. Page 56 of 59
57 In the section Verifying Nodal Analysis with a Real Circuit: we got better results, with the highest percent error in our comparison of measured and calculated node voltages being.35% (see Table 4: Comparison of measured and calculated node voltages). The reason for these better results may well have something to do with the fact that the calculations for these values did not involve as many different calculated values as the differential voltages in the Wheatstone Bridge did. The margin of error we obtained for the section Analyzing A Single Circuit Multiple Ways: was truly impressive, with the largest difference of any calculated value to any measured value being 3.57% (see Table 6). These results were especially gratifying since all of the analysis techniques used gave more or less the same values, which verified our supposition that all of the different circuit analysis tools used were equally accurate. The last section on equivalent circuits substantiated the claim that Thevenin and Norton Equivalent Circuits were indeed equivalent. With the margins or error between the measured values of the currents through and voltages across the load resistors hovering remaining below 0.5% (see Table 11), there is no doubt that the circuits are equivalent to each other and there should be no hesitancy to use an equivalent circuit, should the need arise. Indeed, all of the analysis techniques explored in this paper were highly accurate and can be used with confidence. Page 57 of 59
58 List of Tables: Table 1: Resistance Values for our Wheatstone Bridge Table 2: Measure vs. calculated values for Figure Table 3: Measured differential voltage compared to calculated differential voltage for Figure 23 and Figure Table 4: Comparison of measured and calculated node voltages Table 5: Branch Currents and power Table 6: Comparisons of results from loop analysis, superposition, and source transformation analysis Table 7: Measured Resistance Values Table 8: Measured and calculated voltage drops across each resistor Table 9: Measured and Calculated Values of V Th and R Th with R 5 as the load Table 10: Measured and Calculated Values of V Th and R Th with R 2 thru R 5 as the load 51 Table 11: Measured Values V Th for R 2 thru R 5 as the load in the Master Circuit and the Thevenin Circuits Table 12: Comparisons of PSpice Voltage drops in the Master Circuit across each resistor to the measured and calculated values List of Figures: Figure 1: In the above circuit, V 1 = V R Figure 2: KCL says that i 1 = i 2 + i 3, and KVL says that V S = V R1 = V R Figure 3: Node Analysis example circuit... 7 Figure 4: Example Circuit with 2 loops Figure 5: Circuit from Figure 4 with V 2 replaced with short Figure 6: Circuit from Figure 4 with V 1 replaced with a short Figure 7: Circuit from Figure 4 showing principal of superposition Figure 8: Circuit with Variable Load = 5.0 kω Figure 9: Circuit with Variable Load = 10 Ω Figure 10: Circuit behind R 5 from Figure Figure 11: Generic Thevenin Circuit Figure 12: Circuit used to determine R Th Figure 13: Circuit used to determine V Th Figure 14: Thevenin Equivalent Circuit for Figure Figure 15: Thevenin Equivalent Circuit to Figure 8 with voltage and current values Figure 16: Circuit from Figure Figure 17: Generic Norton Circuit Figure 18: The Norton Circuit Equivalent to the circuits in Figure 8 and Figure Figure 19: Circuit used to find I N and R N Figure 20: A Wheatstone Bridge Page 58 of 59
59 Figure 21: A ¼ Wheatstone Bridge Circuit Figure 22: Our Wheatstone Bridge Figure 23: Circuit from Figure 22 with R 5 in parallel with R Figure 24: Circuit from Figure 22 with R 5 in parallel with R4 and R 6 in parallel with R Figure 25: Circuit on which to perform nodal analysis Figure 26: Circuit analyzed by loop Analysis Figure 27: Circuit from Figure 26 with 9 V source removed Figure 28: Circuit from Figure 26 with 15 V source removed Figure 29: Circuit from Figure Figure 30: Circuit from Figure 26 with sources transformed Figure 31: Master Circuit Figure 32: Master Circuit Figure 33: PSpice Model of Master Circuit with voltage and current values Figure 34: Finding R Th with R 5 as the load Figure 35: Finding V Th with R 5 as the load Figure 36: PSpice Model of Thevenin Circuit with R 5 as the load Figure 37: PSpice Model of Norton Circuit with R 5 as the load Figure 38: PSpice Model of Thevenin Circuit with R 2 as load Figure 39: PSpice Model of Norton Circuit with R 2 as load Figure 40: PSpice Model of Thevenin Circuit with R 3 as load Figure 41: PSpice Model of Norton Circuit with R 3 as load Figure 42: PSpice Model of Thevenin Circuit with R 4 as load Figure 43: PSpice Model of Norton Circuit with R 4 as load Page 59 of 59
Chapter 7. Chapter 7
Chapter 7 Combination circuits Most practical circuits have combinations of series and parallel components. You can frequently simplify analysis by combining series and parallel components. An important
More informationUNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS 1.0 Kirchoff s Law Kirchoff s Current Law (KCL) states at any junction in an electric circuit the total current flowing towards that junction is equal
More informationPOLYTECHNIC UNIVERSITY Electrical Engineering Department. EE SOPHOMORE LABORATORY Experiment 2 DC circuits and network theorems
POLYTECHNIC UNIVERSITY Electrical Engineering Department EE SOPHOMORE LABORATORY Experiment 2 DC circuits and network theorems Modified for Physics 18, Brooklyn College I. Overview of Experiment In this
More informationVoltage Dividers, Nodal, and Mesh Analysis
Engr228 Lab #2 Voltage Dividers, Nodal, and Mesh Analysis Name Partner(s) Grade /10 Introduction This lab exercise is designed to further your understanding of the use of the lab equipment and to verify
More informationNotes for course EE1.1 Circuit Analysis TOPIC 4 NODAL ANALYSIS
Notes for course EE1.1 Circuit Analysis 200405 TOPIC 4 NODAL ANALYSIS OBJECTIVES 1) To develop Nodal Analysis of Circuits without Voltage Sources 2) To develop Nodal Analysis of Circuits with Voltage
More informationChapter 5. Department of Mechanical Engineering
Source Transformation By KVL: V s =ir s + v By KCL: i s =i + v/r p is=v s /R s R s =R p V s /R s =i + v/r s i s =i + v/r p Two circuits have the same terminal voltage and current Source Transformation
More informationKirchhoff's Laws and Maximum Power Transfer
German Jordanian University (GJU) Electrical Circuits Laboratory Section Experiment Kirchhoff's Laws and Maximum Power Transfer Post lab Report Mahmood Hisham Shubbak / / 8 Objectives: To learn KVL and
More informationModule 2. DC Circuit. Version 2 EE IIT, Kharagpur
Module 2 DC Circuit Lesson 5 Nodevoltage analysis of resistive circuit in the context of dc voltages and currents Objectives To provide a powerful but simple circuit analysis tool based on Kirchhoff s
More informationCircuit Theorems Overview Linearity Superposition Source Transformation Thévenin and Norton Equivalents Maximum Power Transfer
Circuit Theorems Overview Linearity Superposition Source Transformation Thévenin and Norton Equivalents Maximum Power Transfer J. McNames Portland State University ECE 221 Circuit Theorems Ver. 1.36 1
More information3.1 Superposition theorem
Many electric circuits are complex, but it is an engineer s goal to reduce their complexity to analyze them easily. In the previous chapters, we have mastered the ability to solve networks containing independent
More informationLecture Notes on DC Network Theory
Federal University, NdufuAlike, Ikwo Department of Electrical/Electronics and Computer Engineering (ECE) Faculty of Engineering and Technology Lecture Notes on DC Network Theory Harmattan Semester by
More informationChapter 10 Sinusoidal Steady State Analysis Chapter Objectives:
Chapter 10 Sinusoidal Steady State Analysis Chapter Objectives: Apply previously learn circuit techniques to sinusoidal steadystate analysis. Learn how to apply nodal and mesh analysis in the frequency
More informationHomework 2. Due Friday (5pm), Feb. 8, 2013
University of California, Berkeley Spring 2013 EE 42/100 Prof. K. Pister Homework 2 Due Friday (5pm), Feb. 8, 2013 Please turn the homework in to the drop box located next to 125 Cory Hall (labeled EE
More informationReview of Circuit Analysis
Review of Circuit Analysis Fundamental elements Wire Resistor Voltage Source Current Source Kirchhoff s Voltage and Current Laws Resistors in Series Voltage Division EE 42 Lecture 2 1 Voltage and Current
More informationOutline. Week 5: Circuits. Course Notes: 3.5. Goals: Use linear algebra to determine voltage drops and branch currents.
Outline Week 5: Circuits Course Notes: 3.5 Goals: Use linear algebra to determine voltage drops and branch currents. Components in Resistor Networks voltage source current source resistor Components in
More informationChapter 2. Engr228 Circuit Analysis. Dr Curtis Nelson
Chapter 2 Engr228 Circuit Analysis Dr Curtis Nelson Chapter 2 Objectives Understand symbols and behavior of the following circuit elements: Independent voltage and current sources; Dependent voltage and
More informationElectric Circuits I. Nodal Analysis. Dr. Firas Obeidat
Electric Circuits I Nodal Analysis Dr. Firas Obeidat 1 Nodal Analysis Without Voltage Source Nodal analysis, which is based on a systematic application of Kirchhoff s current law (KCL). A node is defined
More informationENGG 225. David Ng. Winter January 9, Circuits, Currents, and Voltages... 5
ENGG 225 David Ng Winter 2017 Contents 1 January 9, 2017 5 1.1 Circuits, Currents, and Voltages.................... 5 2 January 11, 2017 6 2.1 Ideal Basic Circuit Elements....................... 6 3 January
More informationKirchhoff's Laws and Circuit Analysis (EC 2)
Kirchhoff's Laws and Circuit Analysis (EC ) Circuit analysis: solving for I and V at each element Linear circuits: involve resistors, capacitors, inductors Initial analysis uses only resistors Power sources,
More informationThevenin Norton Equivalencies  GATE Study Material in PDF
Thevenin Norton Equivalencies  GATE Study Material in PDF In these GATE 2018 Notes, we explain the Thevenin Norton Equivalencies. Thevenin s and Norton s Theorems are two equally valid methods of reducing
More informationChapter 6: SeriesParallel Circuits
Chapter 6: SeriesParallel Circuits Instructor: JeanFrançois MILLITHALER http://faculty.uml.edu/jeanfrancois_millithaler/funelec/spring2017 Slide 1 Identifying seriesparallel relationships Most practical
More informationSeries & Parallel Resistors 3/17/2015 1
Series & Parallel Resistors 3/17/2015 1 Series Resistors & Voltage Division Consider the singleloop circuit as shown in figure. The two resistors are in series, since the same current i flows in both
More informationDC STEADY STATE CIRCUIT ANALYSIS
DC STEADY STATE CIRCUIT ANALYSIS 1. Introduction The basic quantities in electric circuits are current, voltage and resistance. They are related with Ohm s law. For a passive branch the current is: I=
More informationD C Circuit Analysis and Network Theorems:
UNIT1 D C Circuit Analysis and Network Theorems: Circuit Concepts: Concepts of network, Active and passive elements, voltage and current sources, source transformation, unilateral and bilateral elements,
More informationModule 2. DC Circuit. Version 2 EE IIT, Kharagpur
Module 2 DC Circuit esson 8 evenin s and Norton s theorems in the context of dc voltage and current sources acting in a resistive network Objectives To understand the basic philosophy behind the evenin
More informationBasic Electrical Circuits Analysis ECE 221
Basic Electrical Circuits Analysis ECE 221 PhD. Khodr Saaifan http://trsys.faculty.jacobsuniversity.de k.saaifan@jacobsuniversity.de 1 2 Reference: Electric Circuits, 8th Edition James W. Nilsson, and
More informationElectric Circuits II Sinusoidal Steady State Analysis. Dr. Firas Obeidat
Electric Circuits II Sinusoidal Steady State Analysis Dr. Firas Obeidat 1 Table of Contents 1 2 3 4 5 Nodal Analysis Mesh Analysis Superposition Theorem Source Transformation Thevenin and Norton Equivalent
More informationChapter 10 AC Analysis Using Phasors
Chapter 10 AC Analysis Using Phasors 10.1 Introduction We would like to use our linear circuit theorems (Nodal analysis, Mesh analysis, Thevenin and Norton equivalent circuits, Superposition, etc.) to
More informationQUIZ 1 SOLUTION. One way of labeling voltages and currents is shown below.
F 14 1250 QUIZ 1 SOLUTION EX: Find the numerical value of v 2 in the circuit below. Show all work. SOL'N: One method of solution is to use Kirchhoff's and Ohm's laws. The first step in this approach is
More informationPhy301 Circuit Theory
Phy301 Circuit Theory Solved Mid Term MCQS and Subjective with References. Question No: 1 ( Marks: 1 )  Please choose one If we connect 3 capacitors in series, the combined effect of all these capacitors
More informationEE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:0011:15 SEB 1242 Lecture 4 120906 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review Voltage Divider Current Divider NodeVoltage Analysis 3 Network Analysis
More informationCHAPTER FOUR CIRCUIT THEOREMS
4.1 INTRODUCTION CHAPTER FOUR CIRCUIT THEOREMS The growth in areas of application of electric circuits has led to an evolution from simple to complex circuits. To handle the complexity, engineers over
More informationEE 3120 Electric Energy Systems Study Guide for Prerequisite Test Wednesday, Jan 18, pm, Room TBA
EE 3120 Electric Energy Systems Study Guide for Prerequisite Test Wednesday, Jan 18, 2006 67 pm, Room TBA First retrieve your EE2110 final and other course papers and notes! The test will be closed book
More informationDesigning Information Devices and Systems I Spring 2018 Lecture Notes Note 11
EECS 16A Designing Information Devices and Systems I Spring 2018 Lecture Notes Note 11 11.1 Context Our ultimate goal is to design systems that solve people s problems. To do so, it s critical to understand
More informationKirchhoff s laws. Figur 1 An electric network.
Kirchhoff s laws. Kirchhoff s laws are most central to the physical systems theory, in which modeling consists in putting simple building blocks together. The laws are commonly known within electric network
More informationChapter 5 Objectives
Chapter 5 Engr228 Circuit Analysis Dr Curtis Nelson Chapter 5 Objectives State and apply the property of linearity State and apply the property of superposition Investigate source transformations Define
More informationExperiment #6. Thevenin Equivalent Circuits and Power Transfer
Experiment #6 Thevenin Equivalent Circuits and Power Transfer Objective: In this lab you will confirm the equivalence between a complicated resistor circuit and its Thevenin equivalent. You will also learn
More informationmywbut.com Mesh Analysis
Mesh Analysis 1 Objectives Meaning of circuit analysis; distinguish between the terms mesh and loop. To provide more general and powerful circuit analysis tool based on Kirchhoff s voltage law (KVL) only.
More informationNotes for course EE1.1 Circuit Analysis TOPIC 3 CIRCUIT ANALYSIS USING SUBCIRCUITS
Notes for course EE1.1 Circuit Analysis 200405 TOPIC 3 CIRCUIT ANALYSIS USING SUBCIRCUITS OBJECTIVES 1) To introduce the Source Transformation 2) To consider the concepts of Linearity and Superposition
More informationEE201 Review Exam I. 1. The voltage Vx in the circuit below is: (1) 3V (2) 2V (3) 2V (4) 1V (5) 1V (6) None of above
EE201, Review Probs Test 1 page1 Spring 98 EE201 Review Exam I Multiple Choice (5 points each, no partial credit.) 1. The voltage Vx in the circuit below is: (1) 3V (2) 2V (3) 2V (4) 1V (5) 1V (6)
More informationDiscussion Question 6A
Discussion Question 6 P212, Week 6 Two Methods for Circuit nalysis Method 1: Progressive collapsing of circuit elements In last week s discussion, we learned how to analyse circuits involving batteries
More informationEIT Review. Electrical Circuits DC Circuits. Lecturer: Russ Tatro. Presented by Tau Beta Pi The Engineering Honor Society 10/3/2006 1
EIT Review Electrical Circuits DC Circuits Lecturer: Russ Tatro Presented by Tau Beta Pi The Engineering Honor Society 10/3/2006 1 Session Outline Basic Concepts Basic Laws Methods of Analysis Circuit
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science : Circuits & Electronics Problem Set #1 Solution
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.2: Circuits & Electronics Problem Set # Solution Exercise. The three resistors form a series connection.
More informationIn this lecture, we will consider how to analyse an electrical circuit by applying KVL and KCL. As a result, we can predict the voltages and currents
In this lecture, we will consider how to analyse an electrical circuit by applying KVL and KCL. As a result, we can predict the voltages and currents around an electrical circuit. This is a short lecture,
More informationSinusoidal Steady State Analysis (AC Analysis) Part II
Sinusoidal Steady State Analysis (AC Analysis) Part II Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/
More informationLecture 5: Using electronics to make measurements
Lecture 5: Using electronics to make measurements As physicists, we re not really interested in electronics for its own sake We want to use it to measure something often, something too small to be directly
More informationDC CIRCUIT ANALYSIS. Loop Equations
All of the rules governing DC circuits that have been discussed so far can now be applied to analyze complex DC circuits. To apply these rules effectively, loop equations, node equations, and equivalent
More informationMidterm Exam (closed book/notes) Tuesday, February 23, 2010
University of California, Berkeley Spring 2010 EE 42/100 Prof. A. Niknejad Midterm Exam (closed book/notes) Tuesday, February 23, 2010 Guidelines: Closed book. You may use a calculator. Do not unstaple
More informationCOOKBOOK KVL AND KCL A COMPLETE GUIDE
1250 COOKBOOK KVL AND KCL A COMPLETE GUIDE Example circuit: 1) Label all source and component values with a voltage drop measurement (+, ) and a current flow measurement (arrow): By the passive sign convention,
More informationDesigning Information Devices and Systems I Spring 2018 Homework 7
EECS 6A Designing Information Devices and Systems I Spring 08 Homework 7 This homework is due March, 08, at 3:59. Selfgrades are due March 5, 08, at 3:59. Submission Format Your homework submission should
More informationDEPARTMENT OF COMPUTER ENGINEERING UNIVERSITY OF LAHORE
DEPARTMENT OF COMPUTER ENGINEERING UNIVERSITY OF LAHORE NAME. Section 1 2 3 UNIVERSITY OF LAHORE Department of Computer engineering Linear Circuit Analysis Laboratory Manual 2 Compiled by Engr. Ahmad Bilal
More informationParallel Circuits. Chapter
Chapter 5 Parallel Circuits Topics Covered in Chapter 5 51: The Applied Voltage V A Is the Same Across Parallel Branches 52: Each Branch I Equals V A / R 53: Kirchhoff s Current Law (KCL) 54: Resistance
More informationSolution: Based on the slope of q(t): 20 A for 0 t 1 s dt = 0 for 3 t 4 s. 20 A for 4 t 5 s 0 for t 5 s 20 C. t (s) 20 C. i (A) Fig. P1.
Problem 1.24 The plot in Fig. P1.24 displays the cumulative charge q(t) that has entered a certain device up to time t. Sketch a plot of the corresponding current i(t). q 20 C 0 1 2 3 4 5 t (s) 20 C Figure
More informationBFF1303: ELECTRICAL / ELECTRONICS ENGINEERING. Alternating Current Circuits : Basic Law
BFF1303: ELECTRICAL / ELECTRONICS ENGINEERING Alternating Current Circuits : Basic Law Ismail Mohd Khairuddin, Zulkifil Md Yusof Faculty of Manufacturing Engineering Universiti Malaysia Pahang Alternating
More informationLecture 5: Using electronics to make measurements
Lecture 5: Using electronics to make measurements As physicists, we re not really interested in electronics for its own sake We want to use it to measure something often, something too small to be directly
More informationAC Circuit Analysis and Measurement Lab Assignment 8
Electric Circuit Lab Assignments elcirc_lab87.fm  1 AC Circuit Analysis and Measurement Lab Assignment 8 Introduction When analyzing an electric circuit that contains reactive components, inductors and
More informationClicker Session Currents, DC Circuits
Clicker Session Currents, DC Circuits Wires A wire of resistance R is stretched uniformly (keeping its volume constant) until it is twice its original length. What happens to the resistance? 1) it decreases
More information6. MESH ANALYSIS 6.1 INTRODUCTION
6. MESH ANALYSIS INTRODUCTION PASSIVE SIGN CONVENTION PLANAR CIRCUITS FORMATION OF MESHES ANALYSIS OF A SIMPLE CIRCUIT DETERMINANT OF A MATRIX CRAMER S RULE GAUSSIAN ELIMINATION METHOD EXAMPLES FOR MESH
More informationR 2, R 3, and R 4 are in parallel, R T = R 1 + (R 2 //R 3 //R 4 ) + R 5. CC Tsai
Chapter 07 SeriesParallel Circuits The SeriesParallel Network Complex circuits May be separated both series and/or parallel elements Combinations which are neither series nor parallel To analyze a circuit
More informationSimple Resistive Circuits
German Jordanian University (GJU) Electrical Circuits Laboratory Section 3 Experiment Simple Resistive Circuits Post lab Report Mahmood Hisham Shubbak 7 / / 8 Objectives: To learn how to use the Unitr@in
More informationElectrical Technology (EE101F)
Electrical Technology (EE101F) Contents Series & Parallel Combinations KVL & KCL Introduction to Loop & Mesh Analysis Frequently Asked Questions NPTEL Link SeriesParallel esistances 1 V 3 2 There are
More informationElectronics Resistive Sensors and Bridge Circuits
Electronics Resistive Sensors and Bridge Circuits Wilfrid Laurier University September 27, 2012 Switches in voltage dividers One of the simplest forms of voltage divider is where one of the elements is
More informationOUTCOME 3  TUTORIAL 2
Unit : Unit code: QCF evel: 4 Credit value: 15 SYABUS Engineering Science /601/1404 OUTCOME 3  TUTORIA Be able to apply DC theory to solve electrical and electronic engineering problems DC electrical
More informationV x 4 V x. 2k = 5
Review Problem: d) Dependent sources R3 V V R Vx  R2 Vx V2 ) Determine the voltage V5 when VV Need to find voltage Vx then multiply by dependent source multiplier () Node analysis 2 V x V x R R 2 V x
More informationCircuits. PHY2054: Chapter 18 1
Circuits PHY2054: Chapter 18 1 What You Already Know Microscopic nature of current Drift speed and current Ohm s law Resistivity Calculating resistance from resistivity Power in electric circuits PHY2054:
More information15EE103L ELECTRIC CIRCUITS LAB RECORD
15EE103L ELECTRIC CIRCUITS LAB RECORD REGISTER NO: NAME OF THE STUDENT: SEMESTER: DEPARTMENT: INDEX SHEET S.No. Date of Experiment Name of the Experiment Date of submission Marks Staff Sign 1 Verification
More informationCURRENT SOURCES EXAMPLE 1 Find the source voltage Vs and the current I1 for the circuit shown below SOURCE CONVERSIONS
CURRENT SOURCES EXAMPLE 1 Find the source voltage Vs and the current I1 for the circuit shown below EXAMPLE 2 Find the source voltage Vs and the current I1 for the circuit shown below SOURCE CONVERSIONS
More informationChapter 2 Direct Current Circuits
Chapter 2 Direct Current Circuits 2.1 Introduction Nowadays, our lives are increasingly dependent upon the availability of devices that make extensive use of electric circuits. The knowledge of the electrical
More informationECE 1311: Electric Circuits. Chapter 2: Basic laws
ECE 1311: Electric Circuits Chapter 2: Basic laws Basic Law Overview Ideal sources series and parallel Ohm s law Definitions open circuits, short circuits, conductance, nodes, branches, loops Kirchhoff's
More informationCome & Join Us at VUSTUDENTS.net
Come & Join Us at VUSTUDENTS.net For Assignment Solution, GDB, Online Quizzes, Helping Study material, Past Solved Papers, Solved MCQs, Current Papers, EBooks & more. Go to http://www.vustudents.net and
More informationNetwork Topology2 & Dual and Duality Choice of independent branch currents and voltages: The solution of a network involves solving of all branch currents and voltages. We know that the branch current
More informationMAE140  Linear Circuits  Fall 14 Midterm, November 6
MAE140  Linear Circuits  Fall 14 Midterm, November 6 Instructions (i) This exam is open book. You may use whatever written materials you choose, including your class notes and textbook. You may use a
More informationKirchhoff Laws against NodeVoltage nalysis and Millman's Theorem Marcela Niculae and C. M. Niculae 2 on arbu theoretical high school, ucharest 2 University of ucharest, Faculty of physics, tomistilor
More informationNotes on Electricity (Circuits)
A circuit is defined to be a collection of energygivers (batteries) and energytakers (resistors, light bulbs, radios, etc.) that form a closed path (or complete path) through which electrical current
More informationLab E3: The Wheatstone Bridge
E3.1 Lab E3: The Wheatstone Bridge Introduction The Wheatstone bridge is a circuit used to compare an unknown resistance with a known resistance. The bridge is commonly used in control circuits. For instance,
More informationAS and A Level Physics Cambridge University Press Tackling the examination. Tackling the examination
Tackling the examination You have done all your revision and now you are in the examination room. This is your chance to show off your knowledge. Keep calm, take a few deep breaths, and try to remember
More informationNotes for course EE1.1 Circuit Analysis TOPIC 10 2PORT CIRCUITS
Objectives: Introduction Notes for course EE1.1 Circuit Analysis 45 Reexamination of 1port subcircuits Admittance parameters for port circuits TOPIC 1 PORT CIRCUITS Gain and port impedance from port
More informationSinusoidal Steady State Analysis (AC Analysis) Part I
Sinusoidal Steady State Analysis (AC Analysis) Part I Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/
More informationBasic Electricity. Unit 2 Basic Instrumentation
Basic Electricity Unit 2 Basic Instrumentation Outlines Terms related to basic electricitydefinitions of EMF, Current, Potential Difference, Power, Energy and Efficiency Definition: Resistance, resistivity
More informationECE2262 Electric Circuits. Chapter 5: Circuit Theorems
ECE2262 Electric Circuits Chapter 5: Circuit Theorems 1 Equivalence Linearity Superposition Thevenin s and Norton s Theorems Maximum Power Transfer Analysis of Circuits Using Circuit Theorems 2 5. 1 Equivalence
More informationDesigning Information Devices and Systems I Spring 2015 Note 11
EECS 16A Designing Information Devices and Systems I Spring 2015 Note 11 Lecture notes by Edward Wang (02/26/2015). Resistors Review Ohm s law: V = IR Water pipe circuit analogy: Figure 1: Water analogy
More informationA tricky nodevoltage situation
A tricky nodevoltage situation The nodemethod will always work you can always generate enough equations to determine all of the node voltages. The method we have outlined well in almost all cases, but
More informationDesigning Information Devices and Systems I Fall 2018 Lecture Notes Note Introduction: Opamps in Negative Feedback
EECS 16A Designing Information Devices and Systems I Fall 2018 Lecture Notes Note 18 18.1 Introduction: Opamps in Negative Feedback In the last note, we saw that can use an opamp as a comparator. However,
More informationCapacitance. A different kind of capacitor: Work must be done to charge a capacitor. Capacitors in circuits. Capacitor connected to a battery
Capacitance The ratio C = Q/V is a conductor s self capacitance Units of capacitance: Coulomb/Volt = Farad A capacitor is made of two conductors with equal but opposite charge Capacitance depends on shape
More informationCHAPTER 4. Circuit Theorems
CHAPTER 4 Circuit Theorems The growth in areas of application of electrical circuits has led to an evolution from simple to complex circuits. To handle such complexity, engineers over the years have developed
More informationResistor. l A. Factors affecting the resistance are 1. Crosssectional area, A 2. Length, l 3. Resistivity, ρ
Chapter 2 Basic Laws. Ohm s Law 2. Branches, loops and nodes definition 3. Kirchhoff s Law 4. Series resistors circuit and voltage division. 5. Equivalent parallel circuit and current division. 6. WyeDelta
More informationConcepTest PowerPoints
ConcepTest PowerPoints Chapter 19 Physics: Principles with Applications, 6 th edition Giancoli 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for
More informationFrom this analogy you can deduce some rules that you should keep in mind during all your electronics work:
Resistors, Volt and Current Posted on April 4, 2008, by Ibrahim KAMAL, in General electronics, tagged In this article we will study the most basic component in electronics, the resistor and its interaction
More informationChapter 4: Techniques of Circuit Analysis
Chapter 4: Techniques of Circuit Analysis This chapter gies us many useful tools for soling and simplifying circuits. We saw a few simple tools in the last chapter (reduction of circuits ia series and
More informationModule 1, Add on math lesson Simultaneous Equations. Teacher. 45 minutes
Module 1, Add on math lesson Simultaneous Equations 45 minutes eacher Purpose of this lesson his lesson is designed to be incorporated into Module 1, core lesson 4, in which students learn about potential
More informationNORTHERN ILLINOIS UNIVERSITY PHYSICS DEPARTMENT. Physics 211 E&M and Quantum Physics Spring Lab #4: Electronic Circuits I
NORTHERN ILLINOIS UNIVERSITY PHYSICS DEPARTMENT Physics 211 E&M and Quantum Physics Spring 2018 Lab #4: Electronic Circuits I Lab Writeup Due: Mon/Wed/Thu/Fri, Feb. 12/14/15/16, 2018 Background The concepts
More informationECE2262 Electric Circuits
ECE2262 Electric Circuits Equivalence Chapter 5: Circuit Theorems Linearity Superposition Thevenin s and Norton s Theorems Maximum Power Transfer Analysis of Circuits Using Circuit Theorems 1 5. 1 Equivalence
More information... after Norton conversion...
Norton 's Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single current source and parallel resistance connected to a load.
More informationModule 2. DC Circuit. Version 2 EE IIT, Kharagpur
Module DC Circuit Lesson 4 Loop Analysis of resistive circuit in the context of dc voltages and currents Objectives Meaning of circuit analysis; distinguish between the terms mesh and loop. To provide
More informationChapter 2 Resistive Circuits
1. Sole circuits (i.e., find currents and oltages of interest) by combining resistances in series and parallel. 2. Apply the oltagediision and currentdiision principles. 3. Sole circuits by the nodeoltage
More informationStudy Notes on Network Theorems for GATE 2017
Study Notes on Network Theorems for GATE 2017 Network Theorems is a highly important and scoring topic in GATE. This topic carries a substantial weight age in GATE. Although the Theorems might appear to
More informationExperiment 2: Analysis and Measurement of Resistive Circuit Parameters
Experiment 2: Analysis and Measurement of Resistive Circuit Parameters Report Due Inclass on Wed., Mar. 28, 2018 Prelab must be completed prior to lab. 1.0 PURPOSE To (i) verify Kirchhoff's laws experimentally;
More informationChapter 5: Circuit Theorems
Chapter 5: Circuit Theorems This chapter provides a new powerful technique of solving complicated circuits that are more conceptual in nature than node/mesh analysis. Conceptually, the method is fairly
More informationTwoPort Networks Admittance Parameters CHAPTER16 THE LEARNING GOALS FOR THIS CHAPTER ARE THAT STUDENTS SHOULD BE ABLE TO:
CHAPTER16 TwoPort Networks THE LEARNING GOALS FOR THIS CHAPTER ARE THAT STUDENTS SHOULD BE ABLE TO: Calculate the admittance, impedance, hybrid, and transmission parameter for twoport networks. Convert
More informationHomework 3 Solution. Due Friday (5pm), Feb. 14, 2013
University of California, Berkeley Spring 2013 EE 42/100 Prof. K. Pister Homework 3 Solution Due Friday (5pm), Feb. 14, 2013 Please turn the homework in to the drop box located next to 125 Cory Hall (labeled
More information