AC Circuits

AC Circuits

Purpose

To build a low pass filter using a capacitor and resistor in series and to investigate how the filter response as a function of the frequency of the input signal.

Objectives

  • To learn how to use of a digital oscilloscope
  • To calculate the impedance of a AC circuit or a resistor and capacitor in series
  • To learn the relation between the voltage across each element in an AC circuit of R and C element in series.
  • To find out the dependence of the voltage across the C element as a function of the frequency of the input signal

Equipment

  • Digital Oscilloscope Tektronix TDS 2001C
  • Function Generator
  • Breadboard, resistors and capacitors
  • 3 BNC cables, 1 banana-banana plug cable

Definitions

A capacitor in an AC circuit has capacitive reactance of:

X_{C}=\frac{1}{\omega C} = \frac{1}{2\pi f C}

An inductor in an AC circuit has inductive reactance of:

X_{L} = \omega L = 2 \pi f L

Activity 1. RC Low Pass Filter

Theory

We built a low-pass filter by connecting a resistor and capacitor in series. The impedance of the circuit is:

Z = \sqrt{R^2+X_C^2}

If the current in the circuit is given by I=I_o \sin \omega t, the maximum voltage across is:

V_o=ZI_o

The maximum voltage across the capacitor is:

V_{oC}=X_C I_o

The ratio between the two then is given as:

\frac{V_{oC}}{V_o} = \frac{X_C}{Z}=\frac{1}{2\pi f C \sqrt{R^2 + 4 \pi^2 f^2 C^2}}

or

 \frac{V_{oC}}{V_o} = \frac{1}{\sqrt{4\pi^2 f^2 C^2 R^2+1}}

When the frequency is zero the ratio is simply 1. When the frequency becomes very large (f \rightarrow \infty), the limit of the ratio goes to zero. The cut-off frequency is that frequency for which the ratio is one half:

\frac{1}{2}=\frac{1}{\sqrt{4\pi^2 f^2 C^2 R^2+1}} \rightarrow f = \frac{1}{2\pi RC}

Experimental Setting

Use a 1.0-kΩ resistor and a 0.1-μF capacitor to connect them in series to the Function Generator and and turn the power on.

Activity 2. Impedance of RLC series circuit