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).

$\omega_{break} = \frac{1}{RC}$

$\Rightarrow R = \frac{1}{\omega_{break} C } = \frac{1}{25 \times 10^{3} \times 9.45 \times 10^{-9}} = 4,233 \Omega$

R	C	$\omega_B$	$\nu_B$
Ohms	Farads	rad/s	Hz
$1 \times 10^{5}$	$561 \times 10^{-12}$	17825	2837
$96.4 \times 10^{3}$	$561 \times 10^{-12}$	18490	2943
$10.5$	$1.25 \times 10^{-6}$	76190	12126
$31.3$	$10.3 \times 10^{-6}$	3102	494
$2.058 \times 10^5$	$7.73 \times 10^{-10}$	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 $(V_{in})$ and output $(V_{out})$ voltages for at least 8 different frequencies$(\nu)$ which span the frequency range from 1 Hz to 1 MHz.

$\nu$	$V_{in}$	$V_{out}$	$\frac{V_{out}}{V_{in}}$
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

1. Graph the $\log \left(\frac{V_{out}}{V_{in}} \right)$ -vs- $\log (\nu)$

phase shift (10 pnts)

1. measure the phase shift between $V_{in}$ and $V_{out}$

Questions

1.)compare the theoretical and experimentally measured break frequencies. (5 pnts)

$\omega_{break} = \frac{1}{RC} = \frac{1}{400 \times 10^{3} \times 9.45 \times 10^{-9}} = = 2.6 \times 10^{2}$

Theory	Exp	%diff

2.) Calculate and expression for $\frac{V_{out}}{ V_{in}}$ as a function of $\nu$, $R$, and $C$. The Gain is defined as the ratio of $V_{out}$ to $V_{in}$.(5 pnts)

$V - IR -X_CI = =V -(X_R +X_C) I = 0$

$\Rightarrow$ the capacitor can be added in series with the other resistor

$X_{tot} = X_R + X_C$

It looks like the voltage divider from the resistance section

$V_{out}= Re\left [\frac{X_C}{R+X_C} V_{in} \right ]= Re \left [\frac{X_C}{X_R+X_C} V_{in} \right ]$

To evaluate $Re[Z] = \sqrt{Z Z^*}$ where$Z = x+iy$ and $Z^* = x-iy$

$\frac{V_{out}}{V_{in}} = Re \left [ \frac{\frac{1}{i \omega C}}{R+\frac{1}{i \omega C}} \right ] = \sqrt{\frac{\frac{1}{i \omega C}}{R+\frac{1}{i \omega C}} \frac{\frac{1}{-i \omega C}}{R+\frac{1}{-i \omega C}} } = \frac{1}{\sqrt{1 + \omega^2 R^2 C^2}}$

Let

$\omega_b = \frac{1}{RC} =$ break point (cut off ) frequency

then

$\frac{V_{out}}{V_{in}} = \frac{1}{\sqrt{1 + \left ( \frac{\omega}{\omega_b}\right )^2}}$

3.)Sketch the phasor diagram for $V_{in}$,$V_{out}$, $V_{R}$, and $V_{C}$. Put the current $i$ along the real voltage axis. (30 pnts) 4.)Compare the theoretical and experimental value for the phase shift $\theta$. (5 pnts) 5.) what is the phase shift $\theta$ for a DC input and a very-high frequency input?(5 pnts) 6.) calculate and expression for the phase shift $\theta$ as a function of $\nu$, $R$, $C$ and graph $\theta$ -vs $\nu$. (20 pnts)

Forest_Electronic_Instrumentation_and_Measurement