30-Apr-2015: RLC Circuit Response

PURPOSE

The purpose of this experiment was to observe the response of an RLC circuit and analyze the resulting graphs to find missing variables.

PRE-LAB

Figure 1

To begin our lab, we sought to find the damping ratio (ζ) and the natural frequency (ωo) based on the given circuit elements (Figure 1). However, in order to do so, we had to first measure the actual resistance values of the resistors. These measured values are included in the schematics of the circuit in Figure 1. From these values and the other circuit elements, we found the neper frequency (α) and the natural frequency (ωo). Finally, we found the ratio of these two values to find the damping ratio (ζ)

PROCEDURES

Figure 2

After the pre-lab, we constructed the circuit as shown in Figure 2. We had the analog discovery connected to the second resistor (R2) to measure the voltage across it, which we considered to be Vout. Then, we applied a 2 V step input to the circuit at varying frequencies until the circuit was able to reach steady-state in between pulses.

Figure 3

Figure 4: Close up of Figure 3

The resulting graphs are shown above in Figures 3 and 4. From Figure 4, it can be seen that the the overshoot (Mp) was approximately 35 percent. This is because the initial voltage was around 40 mV and the voltage decayed past 0 mV by 14 mV (Mp = 14/40 * 100% = 35%)

28-Apr-2015: Series RLC Circuit Step Response

PURPOSE

The purpose of this lab was to implement what we learned about an RLC circuit in series in a real-life experiment.

PRE-LAB

Before setting up our circuit, we set up a second order differential equation to relate Vout and Vin. We also did some calculations to estimate the damping ratio (ζ), the natural frequency (ωo), and the damped natural frequency (ωd). In order to do so, we had to first find the actual values of the circuit elements. We found the resistances of the resistor and the inductor to be 1.4 Ω and 1.7 Ω, respectively. Furthermore, we found the inductance of the inductor to be 0.999 mH and the capacitance of the capacitor to be 0.437 µF. From these values, we found the necessary variables as shown in Figure 1.

PROCEDURES

Figure 2

To begin our experiment, we constructed the RLC circuit as shown in Figure 2. It consisted of a resistor, an inductor, and a capacitor, all connected in series. Then, we applied a 2 V step input at 1 Hz to the circuit and plotted Vout and Vin on the oscilloscope. The resulting graphs are shown in Figures 3 and 4.

Figure 3: Under-damped circuit (1)

Figure 4: Under-damped circuit (2)

For an under-damped circuit, the response oscillates at the damped natural frequency. In order to find the damped natural frequency, we multiplied the inverse of the period (which we found by analyzing the graphs in Figures 3 and 4) by 2π. We found this value to be 52359.9 rad/s. This experimental value was somewhat close to the theoretical value of 47835.3 rad/s. In fact, the percent difference between the two values was 9.459 percent. Although this number was slightly bigger than what we would have preferred, we believed that the results were acceptable since there were many possible sources of error in this experiment such as the actual capacitance of the capacitor.

Figure 5

For the second part of the lab, we estimated the resistor value that would cause the circuit to be critically damped (Figure 5). As it can be seen from the image, we found this value to be 93.93 Ω. However, since we did not have this resistor value available for use, we chose a 100 Ω resistor instead. The measured value of the resistor (97.8 Ω) is shown in the schematic of the revised circuit in Figure 5.

Figure 6: Critically damped circuit

Figure 6 shows the response of the revised circuit to a 2 V step input oscillating at 1 Hz. 

21-Apr-2015: Inverting differentiator

PURPOSE

The purpose of this lab was to construct an inverting differentiator from an op amp and observe the behavior of its output voltages.

PRE-LAB

Figure 1

Before setting up our circuit, we calculated the theoretical output voltages of our inverting differentiator at three different frequencies. We assumed the input voltage to be a cosine function and found its derivative to predict the output voltage at each frequency. This process is shown in Figure 1.

PROCEDURES

Figure 2

After the pre-lab exercises, we set up the circuit as shown above in Figure 2. The circuit was very similar to an inverting op amp, except that a capacitor was connected between the input voltage and the inverting terminal instead of a resistor. As mentioned before, we applied three different input voltages with three different frequencies to the circuit. We then observed the output voltages on an oscilloscope (Figures 3, 4, and 5).

Figure 3: Frequency at 1 kHz

Figure 4: Frequency at 2 kHz

Figure 5: Frequency at 500 Hz

As it can be seen from these graphs, the amplitude of the output voltage increased as the frequency of the input voltage increased. This is what we expected to see based on our pre-lab exercises. As a matter of fact, we compared the theoretical and experimental amplitudes of the output voltages and found their percent differences. These values are shown below in Figure 6.

One last thing to note is that there were not supposed to be any phase differences between the input and output voltages according to the calculations done in the pre-lab. These phase differences can be most likely attributed to the delay between the input of the signal and the response of the circuit.

16-Apr-2015: Passive RC Circuit Natural Response

PURPOSE

The purpose of this experiment was to analyze the natural response of a circuit to estimate its time constant and compare it to the theoretical value.

PRE-LAB

Figure 1

Before constructing our circuit, we did some calculations to find the initial voltage across the capacitor and time constants of the two circuits shown in Figure 1. We found the capacitor voltage using voltage division, while we found the time constants by multiplying the Thevenin resistance (Rth) of the circuit with the capacitance of the capacitor. These were the theoretical values that we would later compare to the experimental values.

PROCEDURES

Figure 2

For the first part of the experiment, we set up the circuit as illustrated in Figure 2. We applied 5 V to the circuit for a few moments to allow it to reach steady-state. Then, we disconnected the voltage source (circled in red) and observed the natural response of the RC circuit, as shown in Figures 3 and 4.

Figure 3: Initial time of response

Figure 4: Time at which the voltage was at 36.79 percent of its initial value

To start our analysis of these graphs, we first calculated how much 36.79 percent of the initial voltage (3.408 V) was, which was 1.254 V. We looked at the graph in Figure 3 to find the time at which the response began (t0 = -61.5 ms). We subtracted this value from the time when the voltage reduced to 36.79 percent of its initial value (tf = -11 ms) to find Δt. This Δt was equal to the time constant, τ, of this circuit. We found this value to be equal to 50.5 ms. We compared this to the theoretical value found in the pre-lab by finding the percent difference, which was 2.956 percent. Since this was less than 5 percent, we concluded that our results were acceptable.

Figure 5: Initial time of response (square wave)

Figure 6: Time at which the voltage reached 36.79 percent of its initial value (square wave)

For the second part of the lab, we applied a square wave with an amplitude of 2.5 V and an offset of 2.5 V to the same circuit as the one used in Part 1 (Figure 2). Since the voltage oscillated between 0 V and 5 V, this input acted as an on and off switch. In other words, the voltage source acted as a short circuit when it was at 0 V. The response of the circuit when the input voltage was at 0 V is shown in Figures 5 and 6. Using the same process as Part 1, we found the time constant to be 15.5 ms. When compared to the value found in the pre-lab (15.24 ms), it is only 1.706 percent bigger. Therefore, we can conclude that we conducted the experiment correctly.

14-Apr-2015: Capacitor Voltage-Current Relations

PURPOSE

The purpose of this experiment was to get us more familiar with the relationship between the voltage and the current across a capacitor.

PROCEDURES

Figure 1

We began this experiment by constructing the set-up shown above in Figure 1. It was a relatively simple circuit as it consisted of a single capacitor, a resistor and a voltage source. We varied the voltage source in terms of shape, frequency, and amplitude and observed how the voltage across the resistor responded to these variations. In addition, we set up a math channel to find the current across the capacitor, which was the same as the current across the resistor since the capacitor and the resistor were in series. Therefore, we concluded that the current across the capacitor was simply the resistor voltage divided by the resistance value of the resistor.

First, we applied a sinusoidal input voltage with an amplitude of 2 V, oscillating at a frequency of 1 kHz. The resulting voltage across the resistor and the current across the capacitor was displayed on an oscilloscope along with the input voltage, as shown in Figure 2 (click to enlarge). The amplitude, frequency, and period of the output voltage were also displayed on the oscilloscope.

Next, we applied another sinusoidal voltage to the circuit. We kept the amplitude constant and just increased the frequency to 2 kHz. The resulting output graphs are illustrated in Figure 3 (click to enlarge). As it can be seen from the image, the amplitude of the resistor voltage actually decreased as a result of the higher input frequency.

Finally, we applied a triangular voltage with a frequency of 100 Hz and an amplitude of 4 V. The graph of the output voltage was close to rectangular in shape. The curved parts of the graph can be attributed to the delay of the response of the resistor voltage to the abrupt changes of the input voltage.

9-Apr-2015: Temperature Measurement System Design

PURPOSE

The purpose of this experiment was to design a circuit that would result in an output voltage of at least 2 V.

PROCEDURES

Figure 1

To begin, we measured the resistance values of the different circuit elements of the Wheatstone bridge (Figure 1). For example, the resistance of the thermistor was 12.1 kΩ at room temperature and decreased to 10.8 kΩ at ~37°C. 

Figure 2

Figure 3

Then, we set up the circuit as shown in Figure 2. We applied 5 V (Vs) to the circuit and adjusted the resistance of the potentiometer until the output of the circuit was 0 V i.e. balanced the bridge (Figure 3). We also measured the output voltage when the thermistor was warmed up to approximately 37°C, which was -244 mV. 

Figure 4

Next, we designed a difference amplifier as displayed in Figure 4. The design had a theoretical gain of the 15.29, which we believed was sufficient enough to meet the design requirements (Vout =         |(Gain)(Vab)| = |(15.29)(-0.244)| = |-3.73 V| > 2V).

Figure 5

Afterwards, we set up the circuit as shown in Figure 5 and applied two voltages, Va and Vb. We kept Va constant at 300 mV and varied Vb from 100 mV to 500 mV in 50 mV increments. We then plotted the resulting output voltage (Vout) with respect to the potential difference between the two voltage sources (Vab). This graph is illustrated below in Figure 6.

Figure 6: Vout vs Vab

The resulting graph shows that the gain of the circuit is -15.34. If we ignore the negative sign, which was most likely a result of incorrect wiring, we can see that this value was very close to the theoretical value of 15.29. In fact, the percent difference between the values was only 0.327 percent. Therefore, we concluded that our circuit was functioning properly.

Figure 7

Finally, we combined the two circuits together to end up with the circuit in Figure 7. We applied 5 V to the circuit and measured an output voltage of 0 V at room temperature (~25°C). Then, we warmed the thermistor up with our hands and observed an output voltage of -3.56 V. According to the theoretical gain of the difference amplifier, the output voltage was supposed to be -3.73 V. Therefore, the percent difference between the theoretical and experimental values was 4.558 percent. Since the percent difference was less than 5 percent, we concluded that our design was a success.

31-Mar-2015: Summing Amplifier/Difference Amplifier

PURPOSE

The purpose of this experiment was to utilize the theoretical knowledge we gained on summing amplifiers and put it into practice.

PRE-LAB

Figure 1

Figure 2

To begin, we were instructed to design a circuit that performed an addition of two signals. We drew out the schematics of this circuit in Figure 1. As it can be seen from the image, we chose two 3.6 kΩ resistors for the input resistance and a 1.8 kΩ resistor for the output resistance. We did this because we wanted the output voltage to be about half of the sum of the two input voltages (refer to Figure 2).

PROCEDURES

Figure 3

Figure 4

Before implementing our design, we measured the actual resistances of the three resistors. We redrew the schematics of the circuit with these measured values (Figure 3). Then, we set up our circuit as shown in Figure 4 (click to enlarge). The input resistors are encircled in red, while the output resistor is marked with green. The voltage sources are labeled with the black circles, and the positive and negative op amp supplies are within the orange circles. The output voltage that resulted from this set-up was measured with the multimeter as shown with the purple circles.

After constructing the circuit, we applied voltages across the two terminals. We varied Va from -4 V to +5 V, as shown in Figure 6, and kept Vb constant at 1 V. As it can be seen in Figure 5, when we set Va at -4.0 V, the output voltage was 1.49 V. This value was very close to the theoretical voltage that we found by using the equation displayed in Figure 2 (they were equal to the hundredth decimal place; the rest of the digits are not shown in the data table above).