__Introduction __

During this experiment, the class continued to learn about how voltage varies with frequency. This experiment explored passive filters, which are also known as band pass filters. The following procedures were done with two series RLC circuits and two parallel circuits containing an inductor, a capacitor, and two resistors.

__Procedure__

**Preliminary Calculations:**

- The expression H(jω)=V
_{2}/V_{1}for the circuit shown in Figure 11.3 is H(jω)=1/[1+j((ωL/R)-(1/RCω))]. - |H(jω)|=1/sqrt[1+((1/R)(ωL-(1/ωC))
^{2}]. - The resonant frequency (ω
_{r}) is the frequency at which the response amplitude is at its maximum value. For this circuit is ω_{r}= 6.1kHz. - The bandwidth of this circuit was calculated to be 11.9kHz.

The above steps were repeated for the circuit shown in Figure 11.4.

- H(jω)=1/[1+(R
_{1}/R_{2})+j(ωC-(1/ωL))] - |H(jω)|=1/sqrt[1+((R
_{1}/R_{2}))^{2}+(ωC-(1/ωL))^{2}]. - ω
_{r}=6.1kHz. - The bandwidth was found to be 11.6kHz.

**Main Procedure:**

- The circuit shown in Figure 11.3 was constructed with R=5.1kΩ, L=68mH, C=0.01μF and the sine wave generator was set to generate a 1V peak sinusoid. While varying the frequency from 10Hz to 100kHz, the oscilloscope was used to measure the magnitude of V
_{2}. - The magnitude of V
_{2}versus frequency is plotted below.

Frequency (Hz) | Magnitude (mV) |

10 | 3.36 ± 0.04 |

20 | 6.3 ± 0.04 |

40 | 12.35 ± 0.05 |

80 | 24.2 ± 0.2 |

100 | 30.2 ± 0.2 |

160 | 48 ± 0.2 |

320 | 95.5 ± 0.5 |

640 | 191 ± 1 |

1000 | 296 ± 1 |

1280 | 372 ± 1 |

2560 | 675 ± 1 |

5120 | 960 ± 1 |

10240 | 880 ± 1 |

20480 | 560 ± 1 |

40960 | 276 ± 1 |

81920 | 118 ± 1 |

100000 | 83 ± 0.5 |

- The resonant frequency given by the graph is 5.12kHz. This is different from the theoretical value of 6.1kHz. More points should have been taken around the peak so that the resonant frequency values would have matched.
- The bandwidth of this graph cannot be accurately found because not enough points were taken. The theoretical bandwidth is 11.9kHz, which appears to be a reasonable value for the graph to have.
- Next, PSPICE was used to plot the magnitude of V
_{2}versus frequency.

This PSPICE plot agrees with the experimental plot shown in step “a” above.

- The above steps were then repeated for R=2kΩ.

Frequency (Hz) | magnitude (mV) |

10 | 1.32 ± 0.2 |

20 | 2.68 ± 0.04 |

40 | 5 ± 0.04 |

80 | 9.9 ± 0.1 |

100 | 12.1 ± 0.1 |

160 | 19.4 ± 0.2 |

320 | 37.8 ± 0.2 |

640 | 77.5 ± 0.5 |

1000 | 120.5 ± 0.5 |

1280 | 154 ± 1 |

2560 | 340 ± 1 |

5120 | 825 ± 2 |

10240 | 600 ± 2 |

20480 | 260 ± 1 |

40960 | 112.5 ± 0.5 |

81920 | 47 ± 0.5 |

100000 | 34 ± 0.2 |

- Just like in Step 1, the theoretical resonant frequency equals 6.1kHz. The graph shows that the resonant frequency is equal to 5.12kHz. More points should have been taken around the peak to get a more accurate resonant frequency value.
- The theoretical bandwidth of this circuit is 11.9kHz. This cannot be accurately seen from the graph because not enough data points were taken.
- Next, the PSPICE was used to plot the magnitude of V
_{2}versus frequency. These match the results from step “a” above.

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- Part 1 was then repeated for the circuit shown in Figure 11.4 with R
_{1}=5.1kΩ, R_{2}=20kΩ, L=68mH and C=0.01μF.

Frequency (Hz) | magnitude (mV) |

10 | 16.1 ± 0.05 |

20 | 16.2 ± 0.05 |

40 | 16.5 ± 0.05 |

80 | 17.5 ± 0.1 |

100 | 18.2 ± 0.1 |

160 | 20.9 ± 0.1 |

320 | 30.5 ± 0.2 |

640 | 54.5 ± 0.2 |

1000 | 85 ± 0.5 |

1280 | 108 ± 0.5 |

2560 | 236 ± 1 |

5120 | 635 ± 2 |

10240 | 428 ± 1 |

20480 | 174 ± 1 |

40960 | 83 ± 0.5 |

81920 | 40.8 ± 0.2 |

100000 | 33.8 ± 0.2 |

- The resonant frequency is ω
_{r}=6.1kHz. The plot shows the resonant frequency to be 5.12kHz. As can be seen from the plot, more points should have been taken near the peak to give a more accurate resonant frequency. - The theoretical bandwidth is 11.6kHz. This cannot be accurately confirmed from the graph because more data points should have been taken. However, 11.6kHz does appear to be a reasonable value to be obtained from the plot.
- PSPICE was used to plot the magnitude of V
_{2}versus frequency. These results match the results from step “a” above.

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- Part 3 was reapeted with R
_{1}=10kΩ and R_{2}=10kΩ.

Frequency (Hz) | Magnitude (mV) |

10 | 8.5 ± 0.05 |

20 | 8.6 ± 0.05 |

40 | 8.6 ± 0.05 |

80 | 9.1 ± 0.1 |

100 | 9.4 ± 0.1 |

160 | 10.8 ± 0.1 |

320 | 15.8 ± 0.2 |

640 | 27.9 ± 0.2 |

1000 | 43.2 ± 0.2 |

1280 | 55 ± 0.2 |

2560 | 121.5 ± 0.5 |

5120 | 372 ± 1 |

10240 | 228 ± 1 |

20480 | 89.5 ± 0.5 |

40960 | 43.5 ± 0.5 |

81920 | 20.8 ± 0.2 |

100000 | 16.9 ± 0.1 |

- The resonant frequency is ω
_{r}=6.1kHz. This matches the graph but, it is difficult to see because more points should have been plotted near the peak. The graph gives a resonant frequency of 5.12kHz. - The theoretical bandwidth of this circuit is 11.6kHz, which cannot be accurately confirmed from this plot because too few data points were taken.
- PSPICE was used to plot the magnitude of V
_{2}versus frequency. These results match the results from step “a” above.

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__Discussion __

While doing this experiment, several calculations needed to be done which involved concepts learned in the lecture. The students needed to be able to find H(jω), |H(jω)|, resonant frequency (ω_{r}) and bandwidth. Not knowing how to apply these mathematical models would have drastically slowed down the process of completing the experiment.

All four circuits had the same resonant frequency because they had the same capacitor and inductor values. The theoretical and experimental values were different by about 16%. This is because not enough data points were taken as the values approached the peak. There was a similar problem when finding bandwidth from the graphs. Not enough points were plotted to accurately find the bandwidth. Graphs can be a helpful way to make sure the experimental values of resonant frequency and bandwidth match the theoretical values. Therefore, in future experiments like this, several more data points must be taken near the cutoff and peak frequencies.

__Conclusion__

In this lab, the students observed how the output voltage varies with frequency in band pass filters. The students gained experience taking measurements with the oscilloscope, as well as, using several mathematical models to make calculations. It was observed just how important it is to take an adequate number of data points so that the theoretical values can be verified with the experimental values.