Calculating Ztheory for Different Frequencies

Calculating Ztheory for Different Frequencies

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Introduction:

An RLC circuit consists of a resistor, inductor, and capacitor in either series or parallel configuration. In a sequence RLC circuit, the modern-day is the identical via every detail due to the fact in collection the cutting-edge is steady. Unlike in series circuit where the whole voltage is the addition of the voltage through every detail, in an RLC circuit a simple summation of the voltages does no longer same the total voltage. The general voltage in an RLC circuit is a vector sum, this takes place due to the fact the voltage across the resistor is in-section with the current, the voltage throughout the inductor is leading the contemporary by using 90°, and the voltage throughout the capacitor is lagging the modern-day by 90°. The section attitude of each of the voltages is unique this means that that a sum of the voltages isn’t always viable. To locate the full voltage, a phasor diagram is used.

A phasor diagram is a scaled representation of AC portions that have an importance and path on a coordinate device. The contemporary in the phasor diagram is a vector along the tremendous x-axis. The voltage of the resistor is alongside the x-axis inside the identical course as the present day. The voltage of the inductor is along the advantageous y-axis and the capacitor is along the terrible y-axis. Since the voltages for the inductor and the capacitor are in the contrary path, a simple subtraction of these to portions might bring about the y aspect of the total voltage. The x factor of the whole voltage is the vector quantity of the resistor. Using the Pythagorean Theorem, the full voltage of the circuit can be acquired. The perspective between the whole voltage vector and the x-axis is known as the phase angle.

The transition of the oval shape determined in an oscilloscope to a line is the visible effect of being “in-segment” at resonance. The oval shape appears while out-of-segment and the line seems whilst in-phase. This is used to set a positive frequency the resonance frequency. In the experiment, a screw turned into adjusted to make the transition among the oval and line at a frequency, which became specified the resonance frequency.

Below is the RCL circuit used inside the test with L as the inductor, C as the capacitor, and R because the resistor. The equations visible beneath have been used to calculate the theoretical impedance, Eq. 1, the segment angles, Eq. 2, and the inductance on the resonance frequency, Eq. Three. The variables visible represent certain quantities, Z is impedance, RL is the inductor resistance, XL and Xc are the inductive and capacitance reactance. The section perspective is Ф and fres is the resonance frequency.

Figure 1. AC Circuit Diagram

(1)

(2)

(3)

Objective:

The motive of this lab is to examine and calculate the impedance in an RLC circuit with the aid of gazing the relationship between the voltage and present day and segment perspective. In the primary segment of the experiment, the RLC circuit changed into measured at 2400 Hz, its resonance frequency, to locate the in-section. Using the values acquired for the resonance frequency, the Zres may be received. The voltage values throughout the resistor have been acquired and used to discover the impedance. Using the impedance formulation, the theoretical impedance can be obtained. The segment perspective values for sure precise frequencies have been received to pick out if main, lagging, or to be in-section at those frequencies. Using the records obtained within the facts desk, the trade in impedance throughout every frequency may be observed.

Procedures

  1. Setting up the Oscilloscope

o Turn on the oscilloscope

o Make connections from the AC Circuit container to the oscilloscope as shown in Figure 1.

  • Connect a wire from point A to Ch1 on the oscilloscope
  • Connect a cord from factor B to CH2 on the oscilloscope
  • Connect a twine from factor D to floor at the oscilloscope
  1. Setting up the Function Generator

o Make connections from the AC Circuit to the Model 4011A function generator

  • Connect a twine from factor F to the tremendous terminal at the feature generator
  • Connect a cord from point E to the terrible terminal on the function generator

o Select the sine wave function, flip the DC Offset and the -20db transfer to their off position.

O Adjust the output degree to its midrange position.

O Select the 5k variety and modify the frequency to 2400Hz ± five Hz the use of the coarse and first-class knobs.

  1. Adjusting the oscilloscope

¾ NOTE: A sample like an oval ought to be visible

o Adjust the Channel 1 and Channel 2 Volts/Div knobs placed on the oscilloscope

  • If the pattern resembles on oval
  • Adjust the screw on the circuit box till it becomes a line
  • If the pattern resembles a line
  • Verify that the circuit is connected correctly, via adjusting the screw barely to look that an oval starts off evolved to form then regulate the screw again to get the line

o Disconnect the wires from factor A, B, and D from the oscilloscope and AC circuit container.

O Turn off the oscilloscope

  1. Measuring voltage across the resistor

¾ NOTE: Do no longer modify the screw setting at the inductor.

O Connect a black cord to the COMS socket and a crimson wire to the VΩ socket on the multi-meter

o Adjust the multi-meter to 2V ~ AC placing.

O Connect the ends of the black and crimson twine from the multi-meter to factor A and D respectively.

O Using the output degree knob placed at the characteristic generator, set the voltage throughout the AC circuit container to 1V ~ AC.

O At 2400 Hz measure the AC voltage throughout resistor R, by means of shifting the black and crimson wire from factor A and D to point B and D. Record the cost within the facts table as VR for f = 2400 Hz.

O To discover IR, use Ohm’s law with R = 100Ω.

¾ NOTE: This IR is the modern thru the complete circuit.

O To find Zmeas, use the system provided inside the information table and use VZ = 1V.

Five. Completing the Data Table

o Find the VR for all the frequencies inside the data desk.

O Repeat step four for every. Make certain that the voltage across the circuit is 1V with the aid of measuring the voltage across the AC circuit field for each time you convert the frequency.

  1. Finding Capacitance, Inductance, and Zres

o The capacitance may be found on the front aspect of the AC circuit container.

O The Zres the impedance at resonance is identical to the Zmeas on the frequency 2400 Hz.

O The Inductance is calculated the use of the method furnished: fres2 = 1/(4π2LC).

Data Table 1

Vin =VZ= 1V R = 100Ω

Frequency,

f (Hz)

 VR (rms)

Volts

 I (rms)

Amps

 Zmeas

Z=VZ/I (ohms)

 Ztheory

(ohms)
100 0.019 0.00019 5263.16 4541.41
200 0.038 0.00039 2631.58
300 0.057 0.00058 1754.39
500 0.097 0.00098 1030.93
800 0.16 0.00166 625.0 503.33
1200 0.27 0.00279 370.37
1600 0.42 0.00443 238.10
2000 0.64 0.0065 156.25 152.31
2200 0.72 0.00732 140.85
2400 0.74 0.00758 135.14 134.14
2600 0.71 0.0073 140.85
2800 0.66 0.00669 151.52
3500 0.43 0.00459 232.56 198.2
4000 0.34 0.00364 294.12
4500 0.29 0.00302 344.83
5000 0.25 0.00259 400 330.63

Calculations:

Zres=135.14 Hz, therefore RL=135.14– 100 = 35.14Hz

The capacitor in the circuit is C = 0.35µF or 3.5X10-7F with XL = XC when at 2400 Hz

First, 2πfL = [1/(2πfC)] therefore, L = 1/(2πf)2C

This means that L = 1/(2π(2400))2(3.5X10-7) = 1.25X10-2 H

Using this information, we can calculate Ztheory for different frequencies

For 100 Hz:

XL = 2πfL = 2π(100 Hz)(1.25X10-2H) = 7.89 Ω

XC = 1/(2πfC) = 1/2π(100Hz)(3.5X10-7F) = 4547.28 Ω

Ztheory = √((R+RL)2 + (XL-XC)2) = √((100+35.14)2+(4547.28-7.89)2)=4541.41Ω

For 800 Hz:

XL = 2πfL = 2π(800 Hz)(1.25X10-2H) = 62.83 Ω

XC = 1/(2πfC) = 1/2π(800Hz)(3.5X10-7F) = 568.41 Ω

Ztheory = √((R+RL)2 + (XL-XC)2) = √((100+35.14)2+(568.41-62.83)2)=503.33 Ω

For 2000 Hz:

XL = 2πfL = 2π(2000 Hz)(1.25X10-2H) = 157.1 Ω

XC = 1/(2πfC) = 1/2π(2000Hz)(3.5X10-7F) = 227.36 Ω

Ztheory = √((R+RL)2 + (XL-XC)2) = √((100+35.14)2+(227.36-157.1)2)=152.31 Ω

For 3500 Hz:

XL = 2πfL = 2π(3500 Hz)(1.25X10-2H) = 274.89Ω

XC = 1/(2πfC) = 1/2π(3500Hz)(3.5X10-7F) = 129.92 Ω

Ztheory = √((R+RL)2 + (XL-XC)2) = √((100+35.14)2+(129.92-274.89)2)=198.2 Ω

For 5000 Hz:

XL = 2πfL = 2π(5000 Hz)(1.25X10-2H) = 392.7 Ω

XC = 1/(2πfC) = 1/2π(5000Hz)(3.5X10-7F) = 90.95 Ω

Ztheory = √((R+RL)2 + (XL-XC)2) = √((100+35.14)2+(392.7-90.95)2)=330.63 Ω

The following graph displays the relationship between Impedance (Z) vs frequency (Hz) using Zmeas and Ztheory.

Analysis of Data:

From the graph we can conclude:

  1. At what frequency does Zmeas have its minimum value?
  • At 2400 Hz
  1. At resonance describe the phase relationship of the voltage and current.
  • They are both in phase with each other because at resonance XL will equal XC.
  1. Also, at resonance describe the relationship between Inductive and Capacitive reactance.
  • Since voltage and current are in phase with each other, XL equals XC.
  1. At resonance which components of the AC circuit contributes most to the total impedance of the circuit.
  • The AC circuit contributes most to the total impedance of the circuit when resonance is at 135.14 Ω
  1. If the circuit used in this experiment were to be adjusted for a resonant frequency of 3500 Hz what value would the Inductor need to be set to?
  • L=1/(2πf)2C so L=1/(2π(3500)2(3.5X10-7)) = 3.71X10-3 H

Part B:

At frequencies of 800 Hz, 2400 Hz, and 3500 Hz, determine the phase angle, φ, does the current lead, lag, or is in phase with the voltage?

Frequency Phase angle Lead, Lag, In phase
100 Hz Φ1=-88.29 Lead
800 Hz Φ2= -75.03° Lead
2000 Hz Φ3=-27.47 Lead
2400 Hz Φ4= 0° In Phase
3500 Hz Φ5= 47.00° Lag
5000 Hz Φ6=65.87° Lag

Summary

By calculating the impedance of the AC circuit with an RLC collection at a couple of frequencies with a regular voltage, we had been able to discover the resistance and inductance for the corresponding frequency. Based in this fact, we observed that the frequency of the circuit became in resonance at 2400 Hz and at this frequency, the impedance turned into 135.15Ω. We also determined that the voltage and currents are in segment from the information displayed in our graph as well as the software of the equation Φ= tan-1(RL/ZC). This lab in the long run taught us that a circuit’s impedance is the lowest while a circuit is set to its resonance frequency and could increase whilst that frequency is changed.

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