Day 24 5/12/16: Impedance and Admittance, KCL and KVL in Frequency Domains and Equivalent Impedance

We first learned more about impedance, which is the equivalent of resistance for alternating current. We learned that impedance has both a real and imaginary part, with the real portion being the resistance of the resistor and imaginary portion being the reactance of the the capacitor and inductor. The reactance of an inductor is jwl and that of a capacitor is 1/jwC. We also learned that at very high frequencies, an inductor acts like an open circuit and a capacitor acts like a short circuit. We also learned about admittance, which is the inverse of impedance. It contains a real portion, which is the conductance, and an imaginary part, which is called susceptance. The sum of the conductance and susceptance is the inverse of the sum of resistance and reactance. We then did two problems involving finding current and voltage using impedances.

After these two problems, we did a lab titled "Impedance", which is explained in more detail in the "LAB" section below. We then reviewed Kirchhoff's laws in circuits with alternating current, which are used exactly the same as in dc current. KVL and KCL were found to hold for phasors as well.

We then learned about impedance combinations. It was found that impedances in series add, and the equivalent impedance in parallel is 1 over the sum of the inverse of the individual impedances. We then did a problem involving finding the equivalent impedance.

Lastly, the concept of phase shifters was gone over slightly. A problem was then done using phase shifters.


LECTURE:

Above is shown the magnitude of the impedance and the angle formed by the phasor as functions of resistance and reactance. This is found using polar coordinates.

The objective of the above problem is to find the current in the circuit and to find the voltage across the capacitor. First, the impedance of the circuit was found, and the current was obtained by dividing the voltage by the impedance of the circuit. The voltage of the capacitor was then obtained by multiplying the current by the impedance of the capacitor alone.

In the above problem, the objective was to find the voltage across the capacitor. This was done by multiplying the current by the impedance of the capacitor.

The objective is to find the equivalent impedance of this circuit. The impedance of the capacitor and resistor were added, and then the equivalent impednance was found by inversing the sum of the inverse of the impedance of the inductor and the previous impedance combination found.

In this problem, the voltage across the inductor was the objective to determine. First, the equivalent impedance for the inductor and capacitor was determined. Then, a voltage divider was used to find the voltage across this equivalent impedance. Because the capacitor and inductor are in parallel, it is know that the voltage across both is equal.

In this derivation, the phase shift and gain for a specific case when the resistor is equal to the reactance of a capacitor in an RC circuit was found. The phase shift was determined to be -45 degrees and the gain 0.5 of the input voltage. 

LAB:

Purpose:

The purpose of this experiment is to determine the impedance of a resistor, inductor and capacitor experimentally. Another purpose is to determine the gain of each component experimentally and the phase shifts between the current.

Prelab:

In the prelab, the theoretical current through the circuit, the voltage across the circuit elements under analysis (resistor, capacitor, inductor), the gain as a result of each element and the phase shift were determined. Above is the calculations for the resistor and the inductor. The calculations for the capacitor are shown below:

All of this data was then summarized in the table below:

As a correction, the phase shift for the inductor should be 90 degrees instead of 82.4, and the phase shift for the capacitor should be -90 degrees instead of -7.9.

Apparatus:

The equipment of this experiment consists of a breadboard, an analog discovery, resistors, a capacitor, an inductor, wires and a laptop with waveforms software.

Procedure:

First, all of the above schematics consisting of  RL, RC and resistor circuits shown above were built on one breadboard. Then, using the circuit with resistors only, a sinusoidal input with an amplitude of 2 V and an offset of 0 V was applied. This voltage input was applied at frequencies of 1 kHz, 5 kHz, and 10 kHz. The oscilloscope channel 1 was then used to measure the current through the circuit by measuring the voltage across the 47 ohm "internal" resistor then dividing that value by its resistance. The voltage across the resistor under analysis was measured using channel 2 of the oscilloscope. An image of the oscilloscope for the resistor circuit is shown below:

Using the oscilloscope window, the current through the circuit, the gain and the phase shift were determined, which is shown in the data section. The same procedure was done three times for this circuit, with each time at a different frequency.

The same procedure seen above was performed for the inductor at all three different frequencies as well. The oscilloscope for the inductor is shown below:

Again, the same procedure was done for the capacitor verbatim. The oscilloscope window for the capacitor is shown below:

Data:

Above is the data obtained from the oscilloscope window for the resistor circuit. The same data was measured at all of the three different applied frequencies of 1 kHz, 5 kHz and 10 kHz. This data includes the current function, the voltage function across the resistor under analysis, the experimental gain of the circuit and the phase shift between the voltage of the circuit element and current.

Below is the same data at the same frequencies for the inductor and capacitor, respectively..

Inductor

Capacitor

Analysis/Conclusion:

Analyzing the data of the resistor circuit, the current measured through the circuit for each trial (13.39 mA, 13.65 mA, and 13.4 mA for the 1 kHz, 5 kHz and 10 kHz) is very similar to the theoretical (13.6 mA), which shows that the analysis and set up were correct. The slight deviations are due to uncertainties in the values of the circuit elements. Because the currents are similar, the voltages are also similar. The experimental gains obtained (0.695 for 1 kHz, 5 kHz and 0.68 for the 10 kHz) are also very similar to the theoretical gain of 0.68, which is expected. Again, the deviations are due to uncertainties in the element values and error in measuring values in the oscilloscope. Lastly, the phase shift for both values is 0, which is expected for a resistor.

Analyzing the data for the inductor, the experimental current for 1 kHz (40.5 mA) is similar to the theoretical (42.2 mA), and the difference is due to human uncertainty when obtaining values from the oscilloscope and uncertainty in the values of the circuit elements. It was found that the current and voltage varies depending on the frequency, which is expected because the voltage and current are related and time-dependent by v = Ldi/dt. Because the voltage differs depending on the frequency, the gain also differs. However, for the 1 kHz, the obtained gain (0.127) is close to the theoretical gain of 0.133.  Lastly, the experimental phase shift (88.2 degrees) is also close to the theoretical (90 degrees).

Analyzing the data for the capacitor, the voltage, current and gain also differ depending on the frequency, because they are time-dependent and related by i = Cdv/dt. However, analyzing at 1 kHz frequency, the experimental current (4.7 mA) is deviating from the theoretical (5.8 mA). It is unknown exactly why, but it is thought that it could be due to a large uncertainty in the capacitance, since the experimental gain and theoretical gain turned out to be exactly the same (0.99). In addition, the experimental phase shift is also exactly equal to the theoretical (-90 degrees).

Above is the experimental and theoretical impedances for each circuit element under the different applied frequencies, along with the percent differences between these frequencies. The percent differences are large for some of the capacitor values and for the inductor values because of uncertainty in the element values. Because the elements we were working with are small, any small change results in a big difference in impedance.