Nodal Analysis
In this lab, we use nodal analysis to determine the voltages across the 22 kΩ and 6.8 kΩ resistors (V_1 and V_2).
We set our reference node at the bottom and use nodal analysis at node B to obtain the first equation (EQ. 1). The assumed current directions are drawn with arrows but not labeled since it is not absolutely necessary. We then obtain three more equations (EQ. 2 - EQ. 4) using nodal analysis. Obtaining the last three equations is simple since the voltage sources are all between the reference node and the remaining nodal voltages to be found. The nodal voltages are just that of the voltage source.
Putting the equations into a matrix, we can solve for the individual nodal voltages. Substitution might have been easier in this case but since we are learning matrices, we chose the matrix method. We then used these nodal voltages to find the voltages across the resistors V_1 and V_2.
We got V_1 = 2.42 V and V_2 = 4.42 V.
Now we wired up the circuit so we can find our experimental values. The three resistors are 6.8 kΩ, 22 kΩ, and 10 kΩ (from left to right). However, the actual measured values for these resistors were 6.66 kΩ, 9.88 kΩ, and 21.8 kΩ (from left to right). The voltage sources were provided by WaveForms with Analog Discovery. The voltages delivered are -3V, -5V, and 5V, from left to right (the red, black, and red leads).
We measured V_1 = 2.41 V and V_2 = 4.38 V. This is pretty close to our original theoretical values. Substituting the values measured for the resistors into the first nodal analysis equation (EQ. 1) and solving for the voltage at node B, we get V_B = -0.5856 V. Now solving again for V_1 and V_2 we get 2.41 V and 4.41 V, respectively.
V_1 has an experimental error of 0.0% and V_2 has an error of 0.7%. We feel very confident about our values.
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