Lab Experiment #4: Fundamentals of Electric Circuits
Goals
The main objective of this laboratory practice is to gain understanding on the basic laws that describe electric circuits and their application.
Background and Theory
Ohm’s Law. According to Alexander and Sadiku (2013), “Ohm’s Law states that the voltage v across a resistor is directly proportional to the current i flowing through the resistor” (p. 31). The constant of proportionality is the element’s resistance, R, which depends on several factors that will not be analyzed in this report. The equation for Ohm’s Law is then: v=iR (where v: voltage, i: current and R: resistance).
The units for these variables are Volts (V), Amperes (A) and Ohms (Ω). This equation is fundamental in the study of electric circuits, as all subsequent laws are derived from it.
Kirchhoff’s Current Law (KCL). This law states that the algebraic sum of currents entering a node (or a closed boundary) is zero. KCL’s general equation is: n=1Nin=0
In this equation, N represents the number of branches connected to the node and in is each of the currents entering or leaving the node (point of connection between branches).
Kirchhoff’s Voltage Law (KVL). KVL is similar to KCL. According to this law, the algebraic sum of all voltages around a closed path (or loop) is zero. The mathematical representation of this law is as follows (where M is number of voltages in the loop and vm is each individual voltage): m=1Mvm=0
Parallel Connection. A parallel connection is one in which the elements are connected to the same nodes, thus having equal voltage. The equivalent resistance of resistors connected in parallel can be calculated through the following equation: 1Req=1R1+1R2++1RN
Series Connection. In a series connection, elements are connected sequentially, thus the current flowing through them is the same. The equivalent resistance of resistors connected in series can be calculated as follows: Req=R1+ R2++RN= n=1NRn
Thevenin’s Theorem. Alexander and Sadiku (2013) explain that this theorem “states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTH in series with a resistor RTH, where VTH is the open-circuit voltage at the terminals and RTH is the input or equivalent resistance at the terminals when the independent sources are turned off”.
Experiment part 1 - Resistors. The first procedure required selecting two resistors with values greater than 10 kΩ and three resistors with values ranging from 300 Ω to 10 kΩ and corroborate their actual values through measurement. Five resistors which complied with the requirements were selected based on their color code, and a FLUKE multimeter was subsequently used to corroborate their values.
Experiment part 2 – Mesh and Nodal Analysis. Using the selected resistors, the circuits shown in the manual were constructed in the lab, setting the voltage and current sources to the indicated values. Several current and voltage measurements were then performed. To measure current, the multimeter must be set to the “ammeter” function, and be connected in series within the circuit. To measure voltage, the multimeter must be set to the “voltmeter” function and be connected in parallel to the element or group of elements being measured. Successively, the measured values were compared to theoretical values calculated using the previously explained basic laws of electric circuits.
Experiment part 3 – Thevenin’s Theorem. To corroborate Thevenin’s Theorem, this experiment required to construct a circuit composed of a voltage source and four resistors, three of which would be considered part of the system and one would be considered the “load”. The current flowing through the load resistor, RL was measured. Subsequently, upon removal of the load resistor, measurements for open-circuit voltage at the load’s terminals and equivalent resistance (which also required the disconnection of the power source) were obtained. To continue the experiment, the three fixed resistors were replaced by a single resistor of a value equal to the obtained equivalent resistance in the previous step, which was connected in series with the previously used voltage source. The load resistor was also connected in series, and the current flowing through it was measured again. The result was compared to the previous measurement, and to the calculated value.
Results and Analysis
Experiment part 1 – Resistors. The selected resistors are shown in Table 1.
Experiment part 2 – Mesh and Nodal Analysis
Mesh Analysis
*Calculated using the resistors’ measured values
The mesh analysis for the resulting circuit, using the resistors’ measured values for calculation purposes, is as follows:
Mesh 1: -6 V+1.77kΩ+9.04 kΩi1-9.04 kΩi2+10 V=0
10.81 kΩi1-9.04 kΩi2=-4 V
Mesh 2: -10 V-9.04 kΩi1+9.04 kΩ+2.39 kΩi2-2.39 kΩi3=0
-9.04 kΩi1+11.43 kΩi2-2.39 kΩi3=0
Mesh 3:-2.39kΩi2+2.39 kΩ+23.97 kΩi3=0
-2.39kΩi2+26.36 kΩi3=0
The solutions for the system of equations are:
i1=0.00115 A;i2=0.00182 A;i3=0.00017 A
Also, i2-i1=0.00032 A; i3-i2= -0.00165 A
*Calculated using the resistors’ measured values
Current passing through R1: 10V-7.79V1.77 kΩ=1.248587 mA
Current passing through R2: 7.79V9.04 kΩ=0.861726 mA
Current passing through R3: 1.46V2.39 kΩ=0.610879 mA
Current passing through R4: 6V-1.46V23.97 kΩ=0.189403 mA
Current passing through R5: 7.79V-1.46V15.02 kΩ=0.42143808 mA
Experiment part 3 – Thevenin’s Theorem
Initial Circuit
Equivalent Thevenin Circuit
Final Measurement
Calculations:
Initial circuit: Current through RL cannot be calculated as the value of RL has not been provided.
Equivalent Resistance RTH:
RTH=R1*R3R1+R3+R2= 1.77 kΩ*2.39 kΩ1.77 kΩ+2.39 kΩ+9.04 kΩ=10.05 kΩ
According to the provided value of voltage (0.567 V), the current flowing through RL is:
IL= VTHRL+RTH =0.567 VRL+10.05 kΩ
References
Alexander, C., & Sadiku, M. (2013). Fundamentals of Electric Circuits (5th ed.). New York: McGraw Hill.
