TF EIMLab3 Writeup
- RC Low-pass filter
1-50 kHz filter (20 pnts)
1.)Design a low-pass RC filter with a break point between 1-50 kHz. The break point is the frequency at which the filter starts to attenuate the AC signal. For a Low pass filter, AC signals with a frequency above 1-50 kHz will start to be attenuated (not passed).
- [math]\displaystyle{ \omega_{break} = \frac{1}{RC} }[/math]
- [math]\displaystyle{ \Rightarrow R = \frac{1}{\omega_{break} C } = \frac{1}{25 \times 10^{3} \times 9.45 \times 10^{-9}} = 4,233 \Omega }[/math]
| R | C | [math]\displaystyle{ \omega_B }[/math] | [math]\displaystyle{ \nu_B }[/math] |
| Ohms | Farads | rad/s | Hz |
| [math]\displaystyle{ 1 \times 10^{5} }[/math] | [math]\displaystyle{ 561 \times 10^{-12} }[/math] | 17825 | 2837 |
| [math]\displaystyle{ 96.4 \times 10^{3} }[/math] | [math]\displaystyle{ 561 \times 10^{-12} }[/math] | 18490 | 2943 |
| [math]\displaystyle{ 10.5 }[/math] | [math]\displaystyle{ 1.25 \times 10^{-6} }[/math] | 76190 | 12126 |
| [math]\displaystyle{ 31.3 }[/math] | [math]\displaystyle{ 10.3 \times 10^{-6} }[/math] | 3102 | 494 |
| [math]\displaystyle{ 2.058 \times 10^5 }[/math] | [math]\displaystyle{ 7.73 \times 10^{-10} }[/math] | 6310 | 1004 |
2.)Now construct the circuit using a non-polar capacitor.
3.)use a sinusoidal variable frequency oscillator to provide an input voltage to your filter.
4.)Measure the input [math]\displaystyle{ (V_{in}) }[/math] and output [math]\displaystyle{ (V_{out}) }[/math] voltages for at least 8 different frequencies[math]\displaystyle{ (\nu) }[/math] which span the frequency range from 1 Hz to 1 MHz.
| [math]\displaystyle{ \nu }[/math] | [math]\displaystyle{ V_{in} }[/math] | [math]\displaystyle{ V_{out} }[/math] | [math]\displaystyle{ \frac{V_{out}}{V_{in}} }[/math] |
| Hz | Volts | Volts | |
| 50 | 0.6 | 0.3 | |
| 100 | 0.5 | 0.18 | |
| 250 | 0.5 | 0.075 | |
| 500 | 0.45 | 0.04 | |
| 1000 | 0.4 | 0.017 | |
| 2500 | 0.28 | 0.005 | |
| 5056 | 0.16 | 0.005 | |
- Graph the [math]\displaystyle{ \log \left(\frac{V_{out}}{V_{in}} \right) }[/math] -vs- [math]\displaystyle{ \log (\nu) }[/math]
phase shift (10 pnts)
- measure the phase shift between [math]\displaystyle{ V_{in} }[/math] and [math]\displaystyle{ V_{out} }[/math]
Questions
1.)compare the theoretical and experimentally measured break frequencies. (5 pnts)
[math]\displaystyle{ \omega_{break} = \frac{1}{RC} = \frac{1}{400 \times 10^{3} \times 9.45 \times 10^{-9}} = = 2.6 \times 10^{2} }[/math]
| Theory | Exp | %diff |
2.) Calculate and expression for [math]\displaystyle{ \frac{V_{out}}{ V_{in}} }[/math] as a function of [math]\displaystyle{ \nu }[/math], [math]\displaystyle{ R }[/math], and [math]\displaystyle{ C }[/math]. The Gain is defined as the ratio of [math]\displaystyle{ V_{out} }[/math] to [math]\displaystyle{ V_{in} }[/math].(5 pnts)
[math]\displaystyle{ V - IR -X_CI = =V -(X_R +X_C) I = 0 }[/math]
- [math]\displaystyle{ \Rightarrow }[/math] the capacitor can be added in series with the other resistor
- [math]\displaystyle{ X_{tot} = X_R + X_C }[/math]
It looks like the voltage divider from the resistance section
- [math]\displaystyle{ V_{out}= Re\left [\frac{X_C}{R+X_C} V_{in} \right ]= Re \left [\frac{X_C}{X_R+X_C} V_{in} \right ] }[/math]
To evaluate [math]\displaystyle{ Re[Z] = \sqrt{Z Z^*} }[/math] where[math]\displaystyle{ Z = x+iy }[/math] and [math]\displaystyle{ Z^* = x-iy }[/math]
- [math]\displaystyle{ \frac{V_{out}}{V_{in}} = Re \left [ \frac{\frac{1}{i \omega C}}{R+\frac{1}{i \omega C}} \right ] = \sqrt{\frac{\frac{1}{i \omega C}}{R+\frac{1}{i \omega C}} \frac{\frac{1}{-i \omega C}}{R+\frac{1}{-i \omega C}} } = \frac{1}{\sqrt{1 + \omega^2 R^2 C^2}} }[/math]
Let
- [math]\displaystyle{ \omega_b = \frac{1}{RC} = }[/math] break point (cut off ) frequency
then
- [math]\displaystyle{ \frac{V_{out}}{V_{in}} = \frac{1}{\sqrt{1 + \left ( \frac{\omega}{\omega_b}\right )^2}} }[/math]
3.)Sketch the phasor diagram for [math]\displaystyle{ V_{in} }[/math],[math]\displaystyle{ V_{out} }[/math], [math]\displaystyle{ V_{R} }[/math], and [math]\displaystyle{ V_{C} }[/math]. Put the current [math]\displaystyle{ i }[/math] along the real voltage axis. (30 pnts) 4.)Compare the theoretical and experimental value for the phase shift [math]\displaystyle{ \theta }[/math]. (5 pnts) 5.) what is the phase shift [math]\displaystyle{ \theta }[/math] for a DC input and a very-high frequency input?(5 pnts) 6.) calculate and expression for the phase shift [math]\displaystyle{ \theta }[/math] as a function of [math]\displaystyle{ \nu }[/math], [math]\displaystyle{ R }[/math], [math]\displaystyle{ C }[/math] and graph [math]\displaystyle{ \theta }[/math] -vs [math]\displaystyle{ \nu }[/math]. (20 pnts)