Frequency Response of R, L and C
Author Name: Leandrou Vasilis
OBJECTIVES
This report aims to:
- Confirm that the resistance of a resistor is free of frequency for frequencies in the audio range.
- Notice that the resistance of an inductor becomes larger linearly with the rise in frequency.
- Prove that the resistance of the capacitor become less non-linearly with the lessing frequency.
BACKGROUND THEORY
In this experiment will be used a DMM, an Oscilloscope, a function generator, resistors (100Ω, 1ΚΩ), a 10mH inductor and capacitors (0.1μf, 0,47μf)
- The DMM read resistance, voltage and current with a digital display.
- The oscilloscope is an instrument that will display the variation of a voltage with time on a flat screen monitor.
- A function generator typically expands on the skills of the audio oscillator by supplying a square wave and triangular waveform with an increased frequency range.
- Capacitor is an element constructed simply of two surfaces separated by the air gap. The capacitor displays its true characteristics only when a change in the voltage or current is made in the network.
EQUIPMENT
· Digital Multimeter (Brand: Good Will Instruments Co. Ltd, Model: GDM-8135, Serial Number: CF-922334)
· Oscilloscope (Brand: HAMEG, Model: HM 203-6, Serial Number: 46/87 Z33418)
· Function generator (Model: TG 550)
· 10mH Inductor
· capacitors (0.1μf, 0,47μf)
· resistors (100Ω, 1ΚΩ)
EXPERIMENTAL METHOD AND PROCEDURE
Part 1
The function generator was connected in series with the DMM and the 1ΚΩ resistor. The oscilloscope was connected on the resistor. The total current was measured and then the resistance of the resistor was calculated by using R=VRrms / Irms (table 1). To find out that the resistance is frequency independent keep the voltage of the resistor constant and changed the frequency while monitoring the current.
Part 2
The function generator was connected in series with the 100Ω resistor and the 10mH inductor. The oscilloscope was connected on the inductor. The voltage across the resistor was measured, the current and XL was calculated using Ip-p = VR / R, XL = VL / I, XL = 2πfL (table 3). Replacing the resistor by an inductor, verify that the voltage across the inductor will be kept constant while the frequency of the voltage monitor is varying. The reactance of an inductor increases linearly with increases in frequency.
Part 3
The function generator was connected in series with the 100Ω resistor and the 0.1μF capacitor. The oscilloscope was connected on the capacitor. The voltage across the resistor was measured and the Xc was calculated (table 3).
OBSERVATIONS
Frequency |
VR |
VRrms |
Irms |
R=VRrms / Irms |
50Hz |
4V |
1.414V |
1.36mA |
1.039KΩ |
100Hz |
4V |
1.414V |
1.36mA |
1.039KΩ |
200Hz |
4V |
1.414V |
1.36mA |
1.039KΩ |
500Hz |
4V |
1.414V |
1.36mA |
1.039KΩ |
1000Hz |
4V |
1.414V |
1.36mA |
1.039KΩ |
Table 1 frequency response of the resistor
Frequency |
VL |
VR |
Ip-p = VR / R |
XL = VL / I |
XL = 2πfL |
1kHz |
4V |
5.6V |
56.34mA |
71Ω |
62,8Ω |
3kHz |
4V |
2.1V |
21.12mA |
189Ω |
188.4Ω |
5kHz |
4V |
1.25V |
12.57mA |
318Ω |
319Ω |
7kHz |
4V |
1V |
10mA |
400Ω |
439Ω |
10kHz |
4V |
0.8V |
8.04mA |
497Ω |
628Ω |
Table 2 frequency response of inductor
Table 3 frequency response of the capacitor
Frequency |
VC |
VR |
I |
Xc = Vc / I |
Xc = 1 / (2πfC) |
100Ηz |
4V |
30mV |
300μΑ |
13.33ΚΩ |
15.9ΚΩ |
200Hz |
4V |
52mV |
520μΑ |
7.7ΚΩ |
7.9ΚΩ |
300Hz |
4V |
80mV |
800μΑ |
5ΚΩ |
5.3ΚΩ |
400Hz |
4V |
100mV |
1mA |
4ΚΩ |
3.97ΚΩ |
500Hz |
4V |
130mV |
1.3mA |
3.07ΚΩ |
3.18ΚΩ |
800Hz |
4V |
200mV |
2mA |
2ΚΩ |
1.98ΚΩ |
1000Hz |
4V |
270mV |
2.7mA |
1.48ΚΩ |
1.59ΚΩ |
2000Hz |
4V |
520mV |
5.2mA |
769.2Ω |
795.7Ω |
Data discussion
The resistance of a carbon resistor is not changed by frequency except for extremely high frequencies.
The reactance of an inductor is linearly dependent on the frequency applied.
If we double the frequency we double the reactance.
When the frequencies are very low, the reactance is very small.When the frequencies increase the reactance increases to a very large value.
For dc conditions is used the short-circuit representation.
For very high frequencies XL can be used an open-circuit approximation.
On the table 1 seeing that the current has the same value when the frequency changes. I=1.36mA when the frequency is 50Hz or 1000Kz.
On the table 2 seeing that as the frequency increases the voltage across the resistor decrease and XL increase. When the frequency is 1 KHz VR=5.6V and XL=71Ω and when the frequency is 7 KHz VR=1V and XL=400Ω
On the table 3 seeing that as the frequency increase the voltage across the resistor increase the current increase but Xc is decrease.
To find out that the resistance is frequency independent keep the voltage of the resistor constant and changed the frequency while monitoring the current.
Replacing the resistor by an inductor, verify that the voltage across the inductor will be kept constant while the frequency of the voltage monitor is varying. The reactance of an inductor increases linearly with increases in frequency.
Error Analysis
There is a difference between XL = VL / I and XL = 2πfL when the frequency is changed. The difference is very small. When f=3 KHz XL = VL / I= 189Ω and 188.4Ω
The same happens with Xc=Vc / I and Xc = 1 / (2πfC). When f =300Hz Xc=Vc / I=5ΚΩ Xc = 1 / (2πfC) = 5.3ΚΩ
RECOMMENDATIONS
The resistance of a carbon resistor is not changed by frequency except for extremely high frequencies.
The reactance of an inductor is linearly dependent on the frequency applied.
If we double the frequency we double the reactance.
When the frequencies are very low, the reactance is very small.
When the frequencies increase the reactance increases to a very large value.
For dc conditions is used the short-circuit representation.
For very high frequencies XL can be used an open-circuit approximation.