The ability to analysis a circuit gives a potential electrical engineer the ability to learn how o problem solve in a theoretical and practical sense which in turn develops industry skills in which will follow them for life, and allow a solid knowledge base for the rest of their career. This report covers the analysis of a DC circuit in order to determine unknown values within a circuit and covers the design process of a DC circuit when specific voltages are required and current and resistors values are unknown.
Introduction This laboratory focuses predominately on DC circuit analysis and design although, as a result of this, many fundamental aspects of electrical engineering ND circuit theory are brought to light. Without the knowledge of circuit theory and fundamental DC circuit analysis and design there is no hope for an electrical engineer to succeed in their job. The knowledge and understanding of these principles in circuit theory are a necessity in order to develop higher order knowledge and skills within an occupational and furthering academic environment.
Considering circuit theory is said to be the corner stone of electrical technology and thus the corner stone of electrical engineering, many fundamentals of circuit analysis and design are required to be elaborated upon or the educational purposes of this report. To begin, “Energy is a scalar quantity associated with the state (or condition) of one or more objects” [ (Walker, 2011) ] which is an important concept to be aware of for later definitions.
The quantity of electricity is said to be electric charge, charge is also said to be ‘conservative’ in that it can be neither created nor destroyed” [ (Smith, 1980) J. Charge is then directly related to current as it is “defined by the electricity transported by one ampere of current in one second” [ (Arizona, 2009) l. Current is thus defined as “the electric charge passing wrought the area A per unit of time” [ (Smith, 1980) ] which as previously stated relates directly to the definition of charge.
From this point, voltage is also but a small step from charge due to its relation, through flow of electric charge. The energy-transfer capability of a flow of electric charge is determined by the electric potential difference or voltage through which the charge moves” C (Smith, 1980) l. Thus from these three consecutive definitions it can already be seen how they are all interrelated and therefore, considering the allow for electric circuits to function, are essential to circuit theory as a base knowledge. It can be considered fairly basic knowledge that a circuit has three basic elements being, current (measured in Ampere’s: A), voltage (measured in Volts: V) and resistance (measured in Ohms: Q).
Resistance is yet to be defined and thus a definition is as follows: In many applications, resistance must be inserted into a circuit. The purpose of this resistance is either to reduce the current or to produce a desired IR voltage drop. The components manufactured with a specific resistance for these uses are called resistors. (Grog, 1977) From these three basic elements of a circuit it was George Ohm who constructed he aptly known Ohm’s Law used in circuit theory. Ohms Law is simply: Voltage = Current x Resistance (V = I R) (V) From this equation the simplistic variations of DC circuit analysis is effectively born.
Ohm’s Law not only allows the basics to be explained in simple terms but also allows for more complex analysis methods such as Node Voltage and Mesh Current analysis to be formed with the assistance of such laws as Kerchiefs Current Law (KCAL) and Kerchiefs Voltage Law (KAVA). KCAL is defined as: The algebraic sum of the currents into any point of the circuit must equal the algebraic sum of all the currents out of that point. Otherwise, charge would accumulate at the point, instead of having a conducting path [ (Grog, 1977) | KAVA is similarly defined as: For each mesh followed continuously in its tracing direction the algebraic summation of all the instantaneous voltage drops is zero (Pike, 1971). Where a mesh is defined as “a loop that does not contain any other loops” [ (Nilsson, 2011)] and a loop is defined as “a path whose last node is in the same as the starting node” [ (Nilsson, 2011) ] and a node is defined as a point where two or more circuit elements join” [ (Nilsson, 2011) ]. KAVA KCAL and Ohm’s Law are all important tools for circuit analysis, especially sing the node voltage method.
A worked example for KCAL and Ohm’s law is included for Figure 1 next; which can also be directly applied to that of KAVA in the same manner in place of KCAL. Figure 1: Simple dual voltage source (VI and V) circuit, with three resistors (RI, RE and RE) and four nodes (IN, NO, NO, NO). From Figure 1, using KCAL it can be said that at node two: Which can then be substituted for Ohm’s law to give: Va-Verb +PVC-Verb= Verb When values are then substituted into equation three, the voltage ‘b’ at node two is equal to 24 volts. The same can be applied to KAVA in nodal voltage analysis, forever, instead of the summation of currents into nodes, the summation of voltages are taken and then Ohm’s Law used to allow for simultaneous or algebraic solving just as completed above. Therefore, from this example it can be seen that KCAL, KAVA and Ohm’s law are very important in the analysis of electrical circuits, especially in the node voltage method and must be a part of the base knowledge for circuit theory of an electrical engineer.
The Lab Exercise Figure 2 will be referenced significantly throughout this section of the report as it was analyses in terms of node voltages and then used as a design tool in terms f resistance and current when node voltages were known. Figure 2: Circuit constructed for analysis of ‘Va’, ‘V’ and PVC’ then secondly for design purposes concerning resistor and current values when voltages were known. For the initial analysis of Figure 2, the node voltage method was selected and conducted via the use of KCAL in conjunction with Ohm’s Law. This was conducted on nodes one, two and three, with node four acting as the reference node. Equations for the currents leaving each node was formed as per KCAL and then Ohm’s Law substituted to allow the currents to be put in terms of resistance and Olathe as per Figure Xi’s worked example and below.
IN Va- Vass + Vary+ Va- Vicar+Va- Verb=O Knob- Vary+ Verb+ V- Vicar=O NV- Verb+ Vicar+ PVC- Vary=O Equations four, five and six were then re-arranged to be in terms of Va, V and PVC in order to be solved simultaneously via Gaussian elimination, in terms of resistance (Table 1 exhibits the resistor values used to solve the equations below). Resistors/Voltage Source Value I RI kill ROR ROR kill ROR ask I ROR ask I ROR kill vs. Table 1 : Values of resistors and voltage source for Figure 2 circuit. As a result of the mathematics behind simultaneously solving for the node latest the following values were calculated (as per Table 2). Node I Voltage Calculated I Node I-Va 1 10/v I Ended-V 15/v I Node-PVC 15/3 v I Table 2: Solved voltages for each node of the circuit in Figure 2.
The process of obtaining these voltages for each node concluded the analysis of the laboratory as the unknowns of the circuit were solved in terms of known resistance values. Which in turn lead to the ability to solve fully understand the circuit despite the fact on first inspection only resistance values and a source value were known. On completion of the analysis of Figure 2, a design task was undertaken in order to determine resistor values as a result of knowing node voltages. Table 3 exhibits the known node voltages for this aspect of the laboratory below. Node I Voltage Given I Voltage Source | 5 V I Node I-Va 13 V I Node-V 12 v I Node-PVC I Table 3: Given values for node voltages in order to allow for individual resistor values to be calculated as part of a design process.
In order to solve for the resistance values, equations four, five and six were again utilized, however, in this instance the known values for node voltages were substituted into the equations in order to obtain solely resistance oriented equations (equations seven, eight and nine). IN -ERR+ ERR+ AIR+ERR=O NO ERR+ AIR=ERR NO AIR=ERR+ERR From this point each node current was required to be rank order in terms of highest to lowest current so that resistor values could be correctly calculated in order to ensure the currents flowed as the design in Figure 2 specified. Currents was rank ordered through circuit analysis of known voltage drops over resistors and ratios of current to voltage when the current was forced to split into parallel parts of the circuit. For example, equation 11 below states that current four must e greater than current six as the voltage drop over resistor four is two volts where as the voltage drop over resistor six is only one volt, thus the statement holds true as a result of Ohm’s Law. IN IL=ii+ii+ii where, ii> NO ii=ii+ii where, ii> ii (A) (10) NO ii=ii+ii (12) Equations 10,11 and 12 are the equations representing where the current splits into parallel parts of the circuit, and must do in a manner befitting of the voltage drops which are the constraints over the specific resistors.
However, as this design problem has more unknowns than equations to be solved as the analysis was, degrees of freedom have been found. Degrees of freedom simply meaning that the problem has infinite solutions and in order for it to be solved, at some point in the solution, arbitrary values for certain elements of the circuit will be required to be chosen. In this solution for Figure 2, there are four degrees of freedom and thus four arbitrary values of current will be required to be chosen in order to solve the resistor values for the given voltages.