Seminar assignments, Questions - Lab 4 - RC Time Constant and RLC Circuits PDF

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Lab 4 - RC Time Constant and RLC Circuits...

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RC Time Constant and RLC Circuits PHYS – 1004 Winter 2014

Partner: Oscar Zambrano Date Performed: March 13, 2014 Date Submitted: March 20, 2014 TA’s Name: Ken Moats Section: L8A Workstation: 14

Purpose The objective of the first part of this experiment was to study the effect of a resistor and a capacitor in an RC circuit, and the behavior of an RC circuit when the amplitude and the frequency of the function is varied. The goal of the second portion of the lab was to study the oscillations of the RLC circuit and the behavior of the RLC circuit when the frequency was varied.

Theory Part A: RC Circuits Figure 1 is a sample of the setup for part A of the experiment

Figure 1 - Charging an RC Circuit

CHARGING THE CAPACITOR The function generator supplies a voltage which can be labeled V 0 into the RC circuit, then a current I will flow through the circuit and charge the capacitor until its voltage ( V C ) reaches an equivalent voltage of the generator meaning V 0 =V C . In this experiment we were looking to find how V C and I varied with time. Equations 4.1, 4.2, and 4.3 were used in order to find this. −t

(4.1)

I =I 0 e RC

Where I 0 is the initial current at t = 0. This equations shows that the charging current decreases exponentially with time from an initial value of I 0 . The voltage of the capacitor is −t

V C =V 0 −R I 0 e RC Combining these two equations we get the final equation used

(4.2)

−t VC =1−e RC V0

(4.3)

It should be noted that the RC time constant is usually called τ and it is expressed in seconds because of the unit cancellation between the unit ohms ( Ω ) of R and the units of C in farads ( F ).

τ =RC (4.4) In an ideal world with perfect capacitors, it would not allow any current to flow when fully charged. However, the capacitors used in this experiment have a small leakage current. Meaning that when the capacitor is fully charged, there will still be a small current flowing in the circuit, if the capacitor is old it could mean significant errors in calculations.

DISCHARGING THE CAPACITOR Once the capacitor is fully charged and the power supply goes back to 0 volts, the capacitor will discharge through the resistor. Since the voltage of the power supply is 0 V, the voltage across the capacitor can we calculated using −t

VC =e RC V0

(4.5)

COMPLETE CYCLE The current will recharge the capacitor when the function generator goes back to V 0 . The capacitor will never have time to charge or discharge completely if the period of the function generator is shorter than the charging and discharging time.

An example of capacitors in parallel and capacitors in series is below

Figure 2 - Left - RC circuit with two capacitors in series. Right - RC circuit with two capacitors in parallel.

Part B: RLC Circuit An RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. If the capacitor is initially charged an RLC circuit is an example of a damped oscillator. The capacitor discharges through resistor and solenoid, induces current in the solenoid which pushes the current in the same direction and charges the capacitor to opposite polarity. An RLC circuit is shown below

Figure 3 - RLC Circuit

A charged capacitor in a RLC circuit will produce oscillations with angular frequency

ω=

√

1 R2 − 2 LC 4 L (4.6)

A series AC RLC Circuit is an example of driven oscillations. An AC power supply oscillates at a particular frequency, ω , while the natural oscillations of the circuit (without the resistor) are given by

ω=

1 √ LC

(4.7)

Ordinary frequency is given by

ω=2 πf

(4.8)

At the resonant frequency, the impedance is

Z=

√

2 1 L2 ( ω2− ω02 ) + R 2 ω

(4.9)

And the maximum current in the circuit is given by

I=

ωV0

√ L (ω −ω ) + R ω 2

2 2 0

2

2

2

(4.10) Lastly, it is important to note that if the measurements are consistent and within experimental error, the plot should be a straight line with a slope of 1.

t=

|x 1−x 2|

√σ

2 x1

2

+σ x

2

(4.11)

The larger t is, the least likely it is that the two quantities are consistent with each other. If t ≤ 2 , then the measurements are consistent. If t> 2 , then the measurements are inconsistent. For part B the circuit was set up as follows

Figure 4 - Circuit for Damped Oscillations

Apparatus Part A: RC Circuits Materials Used Vernier Differential Voltage Probe (Labeled Charge Sensor in Diagram Function

Range ±6.0 V

Precision ±10.0 V

0.5 Hz to 5

Fine±5%

Generator: BK Precision 4011

MHz in 7 ranges

Figure 5 - Apparatus for Part A Part 1 Note: All Capacitors and Resistors Were Placed on Stand

Figure 6 - Apparatus for two Capacitors in Series Note: All Capacitors and Resistors Were Placed on Stand

Figure 7 - Apparatus for two Capacitors in Parallel Note: All Capacitors and Resistors Were Placed on Stand

Part B: RLC Circuit Materials Used Vernier Differential Voltage Probe Function Generator: BK Precision 4011

Range ±6.0 V

Precision ±10.0 V

0.5 Hz to 5 MHz in 7 ranges

Fine±5%

Figure 8 - Circuit for Part B

Observations Part A: RC Circuits Component Green Resistor (Rg) Yellow Resistor (Ry) Green Capacitor (Cg)

Number 160 112 103

Value ± Error

(1200 ±60)Ω (6800 ±680)Ω ( 0.047 ×10−6 ± 0.0047 ×10−6) F

Yellow Capacitor (Cy)

( 0.470 ×10−6 ± 0.047 ×10−6) F

113

Table 1 - Values for Resistors and Capacitors

Configuration Rg - Cg Discharging Rg - Cg Charging Rg – (Cg / Cy Series) Rg – (Cg / Cy Parallel)

RC Measured

RC Calculated (± Error)

( 8.340 ×10 ± 4.536 ×10 ) sec ( 6.773 ×10−5 ± 1.175 × 10−6 ) sec ( 6.185 ×10−5 ± 8.782 × 10−6 ) sec ( 6.173 ×10−4 ± 2.983 ×10−6) sec −5

( 5.64 ×10−5 ± 8.46 × 10−6) sec ( 5.64 ×10−5 ± 8.46 × 10−6) sec ( 5.1272 × 10−5 ±1.2818 ×10−5 ) sec ( 6.204 ×10−4 ± 1.551 × 10−4 ) sec

−6

Table 2 - RC Time Constant Observations

Part B: RLC Circuit

S ECT ION A Equation Plotted:

A=V 0

B=

R 2L

√

C=

1 R2 − 2 LC 4 L

f ( t )= A e− B t sin ( C t + D ) + E

−1.750 ±0.001527

1650 ±1.987 4

−2.558 ×10 ± 1.863 Table 3 - Observations of a Damping Oscillator

SECTION B ω

Frequency (Hz) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

4040 4022 2813 4142 3731 5046 4828 4492 4190 4308 2623 3109 3318 3466 3665 3856

Max V (V)

(rad/s) 25384 25271 17675 26025 2343 31705 30335 28224 26327 27068 16481 19534 20848 21778 23028 24228

0.662 0.659 0.159 0.632 0.432 0.138 0.186 0.325 0.595 0.477 0.141 0.195 0.241 0.286 0.383 0.544

17

3933

24712

0.604

Table 4 - Observations of a Forced Oscillator

Calculations Part A: RC Circuits Calculating Error of Rg Given: Rg = 1200 Ω σ % = 5% Solution:

Required: σ=?  Rg =1200 Ω± 60 Ω

σ =1200 Ω× 5 % σ =1200 Ω× 0.05 σ =60 Ω Calculating Error of Ry  R y =6800 Ω± 680 Ω

Calculating Error of Cy  C y =0.470 ×10−6 F ± 0.047 ×10−6 F

Calculating Error of Cg  C g =0.047 × 10−6 F ± 0.0047 ×10−6 F Calculating RC for Rg and Ry Given:

Rg =1200 Ω± 60 Ω −6 −6 C g=0.047 × 10 F ± 0.0047 ×10 F

Required: RC = ? sec Error:

(

)

−6 60 Ω 0.0047 ×10 F + ×5.64 × 10−5 se RC=R g ×C g 1200 Ω 0.047 ×10−6 F −5 RC= (1200 Ω± 60 Ω) × (0.047 ×10−6 F ± 0.0047 σ =¿ 8.46 ×10 sec −5 RC=5.64 × 10 sec  RC=5.64 × 10−5 sec  ± 8.46× 10−5 sec

Solution:

Calculating RC for Cg  RC= ( 5.1272 × 10−5 ± 1.2818 × 10−5 ) sec Calculating RC for Cy  RC= ( 6.204 ×10−4 ± 1.551 ×10−4 ) sec

σ=

Calculating the Comparison Value for R g - Cg Discharging Required: Given: −5 t =? x 1=8.340 ×10 sec

x 2=¿ 5.64 ×10−5 sec σ 1=4.536 ×10−6 sec σ 2 =¿ 8.46 ×10−6 sec

Equation:

t=

|x 1−x 2|

√σ

2 x1

+σ 2x

2

Solution:

t=

t=

|x 1−x 2|

√σ

2 x1

+σ

The larger t is, the least likely it is that the two quantities are consistent with each other. If t ≤ 2 , then the measurements are consistent. If t> 2 , then the measurements are inconsistent.

2 x2

|8.340 × 10−5 −5.64 ×10−5|

√( 4.536 ×10

−6 2

2

) + ( 8.46 ×10−6 )

t=2.81

The values are not consistent

Calculating the Comparison Value for R g - Cg Charging The values are consistent t=1.33 Calculating the Comparison Value for R g - Cg Discharging The values are consistent t=0.681 Calculating the Comparison Value for R g - Cg Discharging t=¿ 0.0200 The values are consistent

Part B: RLC Circuit Calculating the Comparison Value for A A Measured: −1.750 ±0.001527

t=

t=

A Calculated: 1 V

|x 1−x 2|

√σ

2 x1

+σ 2x

2

|−1.750−1|

√( 0.001527 ) +( 0 ) 2

2

t=1801 Calculating the Comparison Value for B B Measured: 1650 ±1.987

t=67.9 The values are not consistent

B Calculated:

R 2L 100 Ω B= 2 ( 33 mH ) B=1515 B=

Calculating the Comparison Value for C C Measured: −2.558 ×104 ± 1.863

C=

t=¿ 20.1 The values are not consistent

Calculating Angular Frequency Given: L = 33 mH R = Rfgen + RL = 100 Ω

1

(

(33 mH ) ( 0.047 ×10−6 ± 0.0047 ×10−6 )−¿ 10 ¿ 4 (3 C= √ ¿ C=25347 ±10 % C=25347 ±2535

Equation:

ω 0=

Solution:

ω 0=

√

1 R2 − 2 LC 4 L

Required: ω0 = ?

C=( 0.047 ×10−6 ± 0.0047 × 10−6 ) F

ω 0=

C Calculated:

1 √ LC

1 √ LC 1

√ ( 33 mH ) ( 0.047 ×10

−6

± 0.0047 ×10−6 F )

 ω 0=25392± 2540  z

ω 0=25392± 10 % ω 0=25392± 2540

Converting to Ordinary Frequency Given:

Required:

f =? Solution:

1 ( √ LC ) f= 2π

1 ( 25392 ± 2540 ) f= Hz 2π f =4041 Hz ± 2540 Hz

Calculation the Comparison Value for ω0 t=¿ 0.000787

Equation:

1 ( √ LC ) f= 2π

 f =4041 Hz ±2540 Hz

Results The purpose of the first part of the lab was to study RC circuits. The RC time constant was calculated for each of the different circuit set ups: charging and discharging and also capacitors in series and parallel. All of the values were consistent as t was less than 2, except for the discharging of the capacitor, that value was not consistent with a t value of 2.81. The second portion of the experiment was to study RLC circuits. We studied damped oscillation and forced oscillation. For the damped oscillation we generated a curve which had the equation

−Bt f ( t )= A e sin ( C t + D )+ E , the values of A, B and C in the

equation were compared to a calculated value of A, B, and C, however each was found to be inconsistent. For the forced oscillation an angular frequency was calculated as 25392± 2540 Hz and that was then converted to the ordinary frequency of 4041 Hz ± 2540 Hz, which was used as the frequency on the function generator. The t value for the angular frequency calculation was very small, 0.000787, and therefore very consistent.

Discussion In this experiment the two tools that were used were a voltage sensor and a function generator, each of which can be used in practical situations. Function generators can be used to measure the ESR of a capacitor in circuit, and also to troubleshoot. In the first part of the lab the values were for the most part consistent with the measured values, other than the discharging, however it was very close to 2 which gives confidence in the results. A capacitor stores energy, a capacitor is essentially two conductors, usually conduction plates separated by an insulator with connection wires connected to the two conducting plates. Circuits with capacitors exhibit frequency-dependent behavior so that circuits that amplify certain frequencies can be built. For the first part I thought that the time constant for the capacitors in parallel and the capacitors in series would be different, I thought the one in parallel would be quicker after looking at the RC calculated, however the capacitor in series had a slightly faster time constant. To slow down the charging and discharging cycle, increase the time constant. To do this add in resistors in front of the capacitors to slow down charge and add in resistors to the circuit behind the capacitors to slow down discharge. A smartphone can have as many as 500 capacitors inside it. The second part of this experiment was to study RLC circuits. The values when compared with their calculated values were not very consistent. I think this can be explained because the values for A and C in

the equation of the damped oscillation were negatives, and the calculated values were positives which caused for the large t value. However if the negatives are not considered the values are much more consistent. I think the calculations for section A of part B of the experiments are not correct. I think that the values should actually be very consistent with each other as the equation that was fit, looked as though it was an accurate fit. The second section of part B was much better in consideration of the calculations. The values found for the frequency and angular frequency checked with that of the values found on the graph. The t value calculated was much smaller than two, meaning the values are very consistent....