DOWNLOAD PDF AC CIRCUIT ANALYSIS PROBLEMS AND SOLUTIONS

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Clementine Gallagher
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1 Chapter 1 : Resistors in Circuits - Practice â The Physics Hypertextbook In AC circuit analysis, if the circuit has sources operating at different frequencies, Superposition theorem can be used to solve the circuit. Please note that AC circuits are linear and that is why Superposition theorem is valid to solve them. It has been my experience that students require much practice with circuit analysis to become proficient. To this end, instructors usually provide their students with lots of practice problems to work through, and provide answers for students to check their work against. While this approach makes students proficient in circuit theory, it fails to fully educate them. They also need real, hands-on practice building circuits and using test equipment. So, I suggest the following alternative approach: Another reason for following this method of practice is to teach students scientific method: Students will also develop real troubleshooting skills as they occasionally make circuit construction errors. Discuss these issues with your students in the same Socratic manner you would normally discuss the worksheet questions, rather than simply telling them what they should and should not do. I never cease to be amazed at how poorly students grasp instructions when presented in a typical lecture instructor monologue format! An excellent way to introduce students to the mathematical analysis of real circuits is to have them first determine component values L and C from measurements of AC voltage and current. The simplest circuit, of course, is a single component connected to a power source! Not only will this teach students how to set up AC circuits properly and safely, but it will also teach them how to measure capacitance and inductance without specialized test equipment. A note on reactive components: Small step-down power transformers work well for inductors at least two inductors in one package! What is the purpose of students taking your course? If your students will be working with real circuits, then they should learn on real circuits whenever possible. If your goal is to educate theoretical physicists, then stick with abstract analysis, by all means! But most of us plan for our students to do something in the real world with the education we give them. In most sciences, realistic experiments are much more difficult and expensive to set up than electrical circuits. Nuclear physics, biology, geology, and chemistry professors would just love to be able to have their students apply advanced mathematics to real experiments posing no safety hazard and costing less than a textbook. Exploit the convenience inherent to your science, and get those students of yours practicing their math on lots of real circuits! Question 2 Which component, the resistor or the capacitor, will drop more voltage in this circuit? Also, calculate the total impedance Ztotal of this circuit, expressing it in both rectangular and polar forms. Page 1

2 Chapter 2 : Series and Parallel AC Circuits AC Electric Circuits Worksheets Ver E Analysis of Circuits () E Circuit Analysis Problem Sheet 1 - Solutions 1. Circuit (a) is a parallel circuit: there are only two nodes and all four components are connected. Problem Set Overview This set of 34 problems targets your ability to determine circuit quantities such as current, resistance, electric potential difference, power, and electrical energy from verbal descriptions and diagrams of physical situations pertaining to electric circuits. Problems range in difficulty from the very easy and straight-forward to the very difficult and complex. The more difficult problems are color-coded as blue problems. Current When charge flows through the wires of an electric circuit, current is said to exist in the wires. Electric current is a quantifiable notion which is defined as the rate at which charge flows past a point on the circuit. It can be determined by measuring the quantity of charge that flows past a cross-sectional area of a wire on the circuit. The standard metric unit for the quantity current is the ampere, often abbreviated as Amps or A. A current of 1 ampere is equivalent to 1 Coulomb of charge flowing past a point in 1 second. Since the quantity of charge passing a point on a circuit is related to the number of mobile charge carriers electrons which flow past that point, the current can also be related to the number of electrons and the time. To make this connection between the current and the number of electrons, one must know the quantity of charge on a single electron. Like current, resistance is a quantifiable term. The quantity of resistance offered by a section of wire depends upon three variables - the material the wire is made out of, the length of the wire, and the cross-sectional area of the wire. Resistivity values for various conducting materials are typically listed in textbooks and reference books. The main difficulty with the use of the above equation pertains to the units of expression of the various quantities. Thus, the length should be expressed in units of m and the cross-sectional area in m2. Many wires are round and have a circular cross-section. Current is inversely proportional to the overall resistance R of the circuit and directly proportional to the electric potential difference impressed across the circuit. For homes in the United States, this value is close to Volts. Energy is put into a circuit by the battery or the commercial electricity supplier. The elements of the circuit lights, heaters, motors, refrigerators, and even wires convert this electric potential energy into other forms of energy such as light energy, sound energy, thermal energy and mechanical energy. Power refers to the rate at which energy is supplied or converted by the appliance or circuit. It is the rate at which energy is lost or gained at any given location within the circuit. Electricity Costs A commercial power company charges households for the energy supplied on a monthly basis. This unit - a power unit multiplied by a time unit - is a unit of energy. Thus, the task of determining the cost of using a specific appliance for a specified period of time is quite straightforward. The power must first be determined and converted to kilowatts. Equivalent Resistance It is quite common that a circuit consist of more than one resistor. While each resistor has its own individual resistance value, the overall resistance of the circuit is different than the resistance of the individual resistors which make up the circuit. A quantity known as the equivalent resistance indicates the total resistance of the circuit. Conceptually, the equivalent resistance is the resistance that a single resistor would have in order to produce the same overall effect on the resistance as the combination of resistors which are present. So if a circuit has three resistors with an equivalent resistance of 25 ohm, then a single ohm resistor could replace the three individual resistors and have the equivalent effect upon the circuit. The value of the equivalent resistance Req takes into consideration the individual resistance values of the resistors and the way in which those resistors are connected. There are two basic ways in which resistors can be connected in an electrical circuit. They can be connected in series or in parallel. Resistors which are connected in series are connected in consecutive fashion such that all the charge that passes through the first resistor will also pass through the other resistors. In series connection, all of the charge flowing through the circuit passes through all the individual resistors. As such, the equivalent resistance of series-connected resistors is the sum of the individual resistance values of those resistors. Parallel-connected resistors are characterized as having branching locations where charge branches off into the different pathways. The charge which passes through one resistor will not pass through the other resistors. The equivalent resistance of parallel-connected resistors is less than the resistance values of all the individual Page 2

3 resistors in the circuit. While it may not be entirely intuitive, the equation for the equivalent resistance of parallel-connected resistors is given by an equation with several reciprocal terms. It is not unusual that a problem be accompanied by a drawing or a schematic diagram showing the arrangement of batteries and resistors. The drawing and corresponding schematic diagram below represents a series circuit powered by three cells and having three series-connected resistors light bulbs. By imagining a charge leaving the positive terminal of the battery and following its path as it traverses the complete loop, it becomes evident that the charge goes through every resistor in consecutive fashion. As such it meets the criteria of a series circuit. Knowing that the circuit is a series circuit, allows you to relate the overall or equivalent resistance of the circuit to the individual resistance values by the equivalent resistance equation discussed above. Since there is no branching off locations where charge divides into pathways, it can be stated that the current in the battery is equal to the current in resistor 1 is equal to the current in resistor 2 and is equal to the current in resistor This drop in electric potential across each resistor is determined by the current through the resistor and the resistance of the resistor. The electric potential difference across the individual resistors of a circuit is often referred to as a voltage drops. These voltage drops of the series-connected resistors are mathematically related to the electric potential or voltage rating of the cells or battery which power the circuit. If a charge gains 12 volts of electric potential as it passes through the battery of an electric circuit, then it will lose 12 V as it passes through the external circuit. This 12 V drop in electric potential results from a series of individual drops in electric potential as it passes through the individual resistors of the series circuit. A more detailed and exhaustive discussion of series circuits and their analysis can be found at The Physics Classroom Tutorial. Parallel Circuit Analysis The very last problems in this problem set pertain to parallel circuits. Again, it is not unusual that a problem be accompanied by a drawing or a schematic diagram showing the arrangement of batteries and resistors. The drawing and corresponding schematic diagram below represents a parallel circuit powered by three cells and having three parallel-connected resistors light bulbs. By imagining a charge leaving the positive terminal of the battery and following its path as it traverses the complete loop, it becomes evident that the charge reaches a branching location prior to reaching a resistor. At the branching location sometimes referred to as a node, charge follows one of the three possible paths through the resistors. Rather than pass through every resistor, a single charge will pass through a single resistor during a complete loop around the circuit. As such it meets the criteria of a parallel circuit. Knowing that the circuit is a parallel circuit, allows you to relate the overall or equivalent resistance of the circuit to the individual resistance values by the equivalent resistance equation discussed above. As such, the current in the individual pathways will be less than the current outside the pathways. The overall current flow in the circuit and the current in the battery is equal to the sum of the current in the individual pathways. Similar statements can be made of the other branches. Similar to series circuits, any charge leaving the battery must encounter the same drop in voltage as the gain that it encounters when passing through the battery. But unlike series circuits, a charge in a parallel circuit will only pass through one resistor. As such, the voltage drop across that resistor must equal the electric potential difference across the battery. A more detailed and exhaustive discussion of parallel circuits and their analysis can be found at The Physics Classroom Tutorial. Habits of an Effective Problem-Solver An effective problem solver by habit approaches a physics problem in a manner that reflects a collection of disciplined habits. While not every effective problem solver employs the same approach, they all have habits which they share in common. These habits are described briefly here. If needed, they sketch a simple diagram of the physical situation to help visualize it. They equate given values to the symbols used to represent the corresponding quantity e. Where needed, they perform the needed conversion of quantities into the proper unit. The following pages from The Physics Classroom Tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems. Page 3

4 Chapter 3 : Electric Circuit Analysis/Circuit Analysis Quiz 1 - Wikiversity Notes: It has been my experience that students require much practice with circuit analysis to become proficient. To this end, instructors usually provide their students with lots of practice problems to work through, and provide answers for students to check their work against. Resistor, coil and capacitor are connected in series. Quantities that we want to determine: We evaluate the current amplitude. For this calculation, we know all the quantities from assignment. Electric current flowing through all the components connected in series is of the same size; however the voltage on the components is out of phase with the current. To obtain the phase difference shift between voltage and current we use a phasor diagram. How to draw a phasor diagram A phasor is an "arrow" that we use to plot the current and voltage values on individual components of the circuit into a phasor diagram. Its magnitude reflects the amplitude of the voltage or current, and its direction indicates the phase angle. Drawing a phasor diagram for a series circuit: We plot the values of voltage and current on individual components in the AC circuit into the phasor diagram. The current is of the same size on all the components, the phasor of current Im is therefore the same for all the components and is usually drawn in the positive direction of the x-axis. The phasor of voltage is on the resistor UR parallel to the current phasor, because the phase difference between the voltage and current is zero â in this case voltage and current are in phase. In the figure the phasor is illustrated by green. In the figure, this phasor is represented by yellow. Therefore we draw the phasor pointing downward â that is in negative direction of the y-axis. This phasor is represented by pink. The amplitude of the overall voltage is obtained by a "vector sum" of phasors of the voltage on individual components. First, we subtract the voltage on the capacitor UC from the voltage on the coil UL in the picture drawn in purple. Then we add this vector and the vector of the voltage on the resistor UR. The phasor of the voltage amplitude of the entire circuit is represented by light blue. In the following figures the phasor diagrams are not illustrated by the same colours. Derivation of the formula for total impedance Z from the phasor diagram To get the total impedance Z from the phasor diagram, instead of the voltage on individual components in the phasor diagram we plot the inductance XL, the capacitance XC and the resistance R. To calculate the impedance Z we use the rectangular triangle we can see in the phasor diagram. The impedance Z is evaluated by using Pythagorean theorem. The size of the impedance Z is however not affected By substituting the relations of inductance and capacitance we obtain: In the first case the numerator says that we consider the case when the voltage leads the current similarly as on the coil. In the second case, on the other hand, the current leads the voltage. We choose a suitable relationship either from the phasor diagram, where we can see the phase difference between the voltage and current, or we choose one of the relations and interpret the result through the sign of the resulting value. If you choose, for example, the second formula for expressing the phase difference and the resulting value has a plus sign, then the current leads the voltage. However, if the resulting value of the phase difference is negative, then the voltage leads the current. Numerical substitution The current amplitude: The voltage on individual components of the circuit is: Answer In the series RLC circuit the amplitude of the current is approximately: The phase difference between the voltage and the current is about: Page 4

5 Chapter 4 : circuit analysis The circuit is connected to an AC voltage source with amplitude 25 V and frequency 50 Hz. Determine the amplitude of electric current in the circuit and a phase difference between the voltage and the current. It has been my experience that students require much practice with circuit analysis to become proficient. To this end, instructors usually provide their students with lots of practice problems to work through, and provide answers for students to check their work against. While this approach makes students proficient in circuit theory, it fails to fully educate them. They also need real, hands-on practice building circuits and using test equipment. So, I suggest the following alternative approach: Another reason for following this method of practice is to teach students scientific method: Students will also develop real troubleshooting skills as they occasionally make circuit construction errors. Discuss these issues with your students in the same Socratic manner you would normally discuss the worksheet questions, rather than simply telling them what they should and should not do. I never cease to be amazed at how poorly students grasp instructions when presented in a typical lecture instructor monologue format! An excellent way to introduce students to the mathematical analysis of real circuits is to have them first determine component values L and C from measurements of AC voltage and current. The simplest circuit, of course, is a single component connected to a power source! Not only will this teach students how to set up AC circuits properly and safely, but it will also teach them how to measure capacitance and inductance without specialized test equipment. A note on reactive components: Small step-down power transformers work well for inductors at least two inductors in one package! What is the purpose of students taking your course? If your students will be working with real circuits, then they should learn on real circuits whenever possible. If your goal is to educate theoretical physicists, then stick with abstract analysis, by all means! But most of us plan for our students to do something in the real world with the education we give them. In most sciences, realistic experiments are much more difficult and expensive to set up than electrical circuits. Nuclear physics, biology, geology, and chemistry professors would just love to be able to have their students apply advanced mathematics to real experiments posing no safety hazard and costing less than a textbook. Exploit the convenience inherent to your science, and get those students of yours practicing their math on lots of real circuits! Suppose, though, that the output signal is stuck at o lagging the source voltage, no matter where the potentiometer is set. Hide answer A broken connection between the right-hand terminal of the potentiometer and the bridge could cause this to happen: It is essential, of course, that students understand the operational principle of this circuit before they may even attempt to diagnose possible faults. You may find it necessary to discuss this circuit in detail with your students before they are ready to troubleshoot it. The only bad part about this is that doing complex-number arithmetic by hand can be very tedious. Some calculators, though, are able to add, subtract, multiply, divide, and invert complex quantities as easy as they do scalar quantities, making this method of AC circuit analysis relatively easy. This question is really a series of practice problems in complex number arithmetic, the purpose being to give you lots of practice using the complex number facilities of your calculator or to give you a lot of practice doing trigonometry calculations, if your calculator does not have the ability to manipulate complex numbers! Chapter 5 : Solved Problems - A Source of Free Solved Problems General Solution for RLC Circuit Power in AC Circuits Sample Problem. Chapter 6 : AC Network Analysis Network Analysis Techniques Worksheets Alternating-Current Circuits AC Sources In Chapter 10 we learned that changing magnetic flux can induce an emf according to Faraday's law of induction. Chapter 7 : Electrical Circuits Archives - Solved Problems Page 5

6 Solutions--Ch. 13 (AC & DC Circuits) Solution: According to Ohm's Law, the voltage across a resistor is equal to the current through the resistor times the resistance of the resistor, or V. Chapter 8 : AC analysis intro 1 (video) Khan Academy The solutions for end of the segment self-assessment problems are explained in just as much detail as the case studies and sample problem in the pertaining segments. Chapter 9 : Example Problems on DC Circuits Both AC and DC circuits can be solved and simplified by using these simple laws which is known as Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL). Also note that KCL is derived from the charge continuity equation in electromagnetism while KVL is derived from Maxwell - Faraday equation for static magnetic field (the. Page 6

Superposition theorem

Superposition theorem This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,