Impedance

Impendance

In this lab, we find the impedance of three circuit elements, a resistor, inductor, and capacitor. The circuits are wired according to the diagrams drawn above. The impedance for the three circuit elements can be simply calculated with the use of formulas. The impedance of a resistor is just Z = R, so the impedance is really just the resistance of the resistor. For a capacitor, the impedance can be calculated as Z = 1/(jωC) , where j is the imaginary component, ω is the angular velocity of the input, and C is the capacitance of the capacitor. The impedance of an inductor is Z = jωL, where L is the inductance of the inductor. The phase angles for a resistor, capacitor, and inductor are 0°, -90°, and 90°, respectively.

To find the impedance of these circuit elements experimentally, we can divide the element's voltage phasor by its current phasor so that the impedance is Z = V/I. The voltage across the circuit element of interest can be measured directly with the oscilloscope on WaveForms. However, the current must be measured indirectly but measuring the voltage drop across the first resistor VR, and dividing by 47 Ω (46.9 Ω measured). Since the elements are in series, the current through this resistor must be equal to the current through the other circuit element.

For this experiment, we will apply a 2 V sinusoidal wave and measure the first resistor voltage VR, the calculated current I (which is VR/47), and the voltage across the circuit element V. For each circuit, the voltage input will be varied from 1 kHz, 5 kHz and 10 kHz. Finally a comparison with the theoretical values is done to see how accurate the experiment was.

*  Note: For the graphs, it is important to distinguish the colors of the curves. The voltage across the 47 Ω resistor is the orange curve, the current is the red curve, and the voltage drop across the element of interest is the blue curve. The 47 Ω resistor voltage drop curve is not discussed since we are more interested in the current through it, although it will still be seen in the graphs.

Resistor Circuit

Here we have our first circuit consisting of a 47 Ω (46.9 Ω measured, used throughout the experiment) and a 100 Ω (100.3 Ω measured).

1 kHz

5 kHz

10 kHz

Here we apply a 2 V @ 1 kHz sinusoidal input. The first thing noticed is that the current (red) and 150 Ω resistor voltage drop (blue) are in phase (have 0° phase shift). The measured (denoted by the "m" sub) current, voltage, and impedance phasors match our theoretical values (denoted by the "th" sub) with great accuracy.

RL Circuit

1 kHz

5 kHz

10 kHz

Qualitatively speaking, in the three graphs we see that the current lags the voltage by 90° as expected. Also, the voltage across the inductor increases as the frequency increases.

The theoretical values for impedance, current, and voltage need to be calculated for three frequencies. Doing the algebra with complex numbers becomes very tedious. It is easier to use create FreeMat program to do the calculations. The program above calculates the impedance, current, and inductor voltage in the complex domain and their phasors (second row of each matrix). The program is then modified to calculate the impedance, current, and capacitor current in the next section. Above we see the program (function) Zl is used to calculate these values for a 2 V source, 1e-3 H inductor, 10 kHz frequency, and 46.9 Ω resistor.

Here we have the data organized for the three frequencies. The percent errors for impedance, current and voltage are very high. Reflecting back on the experiment, we never accounted for the resistance of the inductor. The extra resistance in the circuit will be enough for the accuracy to be compromised in the experiment.

RC Circuit

1 kHz

5 kHz

10 kHz

The three graphs show that qualitatively, we see that the current leads the capacitor voltage by 90°, which was expected. Also, as the frequency increases, the capacitor voltage decreases.

Here we see the data for our RC circuit. The % error for impedance, current, and voltage are much better than in the RL circuit. The accuracy of the RC experiment is acceptable.

Very High Frequency

Just to see what happens, we increased the frequency to 1 MHz and found that the capacitor voltage and the current had no phase angle between them. This happens because the capacitor acts like a short at high frequencies. When we look at the formula for capacitor impedance Zc = 1/(jωC), we can see that as the frequency increases, the impedance goes to zero. Similarly, for an inductor, an inductor acts like an open circuit at high frequencies since the impedance is Z = jωL.