Impedance Lab

Impedance Lab

Overview:

In this lab we measured impedances of resistors, capacitors, and inductors. We then compared them to their expected values.

Design:

We designed three circuits to experimentally measure the impedance of the elements:

^^^^ (Left) The circuit to measure the impedance of a resistor and (Right) the circuit to measure the impedance of an inductor ^^^^

^^^^ The circuit to measure the impedance of a capacitor ^^^^

The resistors impedance was simply equal to its resistance, while the impedance of the inductor was equal to j*omega*L, and the capacitor's impedance was equal to 1/(j*omega*C). By using these equations, it became clear that the current for the capacitor would lead the voltage by 90 degrees, and the current for the inductor would follow the voltage by 90 degrees. Obviously the current through the resistor is in phase with the voltage. 

Construction and Execution:

^^^^ (Left) Resistance of the 47 Ohm resistor and (Right) the 100 ohm resistor ^^^^

^^^^ Measured capacitance of the the capacitor ^^^^

R47: 48.7 Ohms

R100: 100.0 Ohms

C.1 microF: .093 microF

L: 1 microH

We then constructed the first circuit with the two resistors, and measured the voltages across each element and each of the three frequencies. We also added a mathematic channel (red) which illustrated the current.

^^^^ The dual resistor circuit, as viewed from the side ^^^^

 ^^^^ Output for circuit with input frequency of 5 KHz ^^^^

The voltage across the 100 ohm resistor is in blue, and has an amplitude of 1.35 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 0.54 V.

The current is in red, and has an amplitude of 14.6 mA.

The voltage for the resistor is in phase with the current.

These values are consistent with our expectations to +/- 4.9 %

 ^^^^ Output for circuit with input frequency of 10 KHz ^^^^

The voltage across the 100 ohm resistor is in blue, and has an amplitude of 1.35 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 0.54 V.

The current is in red, and has an amplitude of 14.6 mA.

The voltage for the resistor is in phase with the current.

These values are consistent with our expectations to +/- 4.9 %

 ^^^^ Output for circuit with input frequency of 1 KHz ^^^^

The voltage across the 100 ohm resistor is in blue, and has an amplitude of 1.35 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 0.54 V.

The current is in red, and has an amplitude of 14.6 mA.

The voltage for the resistor is in phase with the current.

These values are consistent with our expectations to +/- 4.9 %

We then went on to testing the inductor:

  ^^^^ Output for circuit with input frequency of 1 KHz ^^^^

The voltage across the inductor is in blue, and has an amplitude of .269 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.90 V.

The current is in red, and has an amplitude of 39.1 mA.

The voltage for the inductor leads the current by 90 degrees.

The inductors impedance was measured as 53.7 ohms with a voltage gain of .169 Volts.

  ^^^^ Output for circuit with input frequency of 1 KHz ^^^^

The voltage across the inductor is in blue, and has an amplitude of 1.07 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.50 V.

The current is in red, and has an amplitude of 34.2 mA.

The voltage for the inductor leads the current by 90 degrees.

The inductors impedance was measured as 78.5 ohms with a voltage gain of .57 Volts.

  ^^^^ Output for circuit with input frequency of 1 KHz ^^^^

The voltage across the inductor is in blue, and has an amplitude of 1.56 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.18 V.

The current is in red, and has an amplitude of 24.4 mA.

The voltage for the inductor leads the current by 90 degrees.

The inductors impedance was measured as 109.1 ohms with a voltage gain of .74 Volts.

Finally, we went on to testing the capacitor:

   ^^^^ Output for circuit with input frequency of 1 KHz ^^^^

The voltage across the capacitor is in blue, and has an amplitude of 1.61 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.05 V.

The current is in red, and has an amplitude of 19.5 mA.

The voltage for the capacitor follows the current by 90 degrees.

The capacitors impedance was measured as 119.3 ohms with a voltage gain of .66 Volts.

 ^^^^ Output for circuit with input frequency of 5 KHz ^^^^

The voltage across the capacitor is in blue, and has an amplitude of .58 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.75 V.

The current is in red, and has an amplitude of 34.2 mA.

The voltage for the capacitor follows the current by 90 degrees.

The capacitors impedance was measured as 61.3 ohms with a voltage gain of .33 Volts.

^^^^ Output for circuit with input frequency of 10 KHz ^^^^

The voltage across the capacitor is in blue, and has an amplitude of .34 V.

The voltage across the 47 ohm resistor is yellow, and has an amplitude of 1.82 V.

The current is in red, and has an amplitude of 39.1 mA.

The voltage for the capacitor follows the current by 90 degrees.

The capacitors impedance was measured as 55.8 ohms with a voltage gain of .16 Volts.

Analysis:

All of the measurements came out to being incredibly close to our theoretical values. This shows that the calculation of impedances as imaginary parts of the voltage and current is a valid method of calculation. Additionally, it defends the theory of phasors (which is essentially the same thing).