Mobile Studio Activity 6 Report

8/9/2019 Mobile Studio Activity 6 Report

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Mobile Studio Activity #6 Adam Steinberger Electric Circuits Section 2

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IntroductionSeries RLC Circuits exhibit transient responses to signals in the form of second order differential

equations. This is due to the dynamic nature of capacitors and inductors, whose voltage and current in a

circuit depend on the previous states of the devices, and the total resistance of the circuit seen by each

device. The response of a second order RLC circuit has a dampening ratio 0, and the shape of thisresponse depends on this value. An effect time constant can be found by approximating the responseof the circuit as a signal exponential curve.

ProcedureThe breadboard configuration for this lab is based off of the two simulated circuit diagrams

found in the Data section on page 3 and page 6. Readings from the Mobile Studio Desktop software

were taken for three different size resistors in the series RLC circuit. For the Operational Amplifier

circuit, the Mobile Studio Desktop software was used to take readings for three different size capacitors.

Readings were also taken for 10k resistors instead of 1k ones.

AnalysisUsing Mesh Analysis of the RLC circuit to find current, it is possible to obtain a second order

differential equation for the response and solve for . Voltage through an inductor is equivalent to . By substituting current across a capacitor with its equivalent

, the inductors voltage becomes

2 2 . Voltage across the capacitor and the resistor are then formulated using Ohms Law and theequation for current across a capacitor. The result is an equation of the form:2

2 + 2

+ 02 = 02, where 2 = / and 02 = 1/. The dampening ratio 0 determines whether the response is overdamped ( > 1), critically damped ( = 1), orunderdamped ( < 1). Methods for solving the differential equation are different for each type ofdampening. For overdamped circuits, replacing the derivate terms with s produces = 11 +22. When the circuit is critically damped, s1 and s2 are equal and the coefficient becomes 2K.Underdamped circuits will produce the complex roots = , which results in the complexequation = 1+ + 2 . This can be reduced to = cos + ,where

= tan1

and

=

2 +

2. For overdamped circuits, the overall time constant will be the

larger euler exponent of the two in the response equation. Critically damped circuits only have one euler

exponent, so this must be the time constant. And in the case of the underdamped response, the limit of

the euler-cosine function will approach the limit of the euler function by itself. Therefore, this euler

exponent must be the time constant.


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ConclusionThe differential approach to Series RLC Circuits also applies to certain types of Op Amp circuits.

The same methods used to solve for these circuits works, and the only difference is that the circuit

analysis that provides the voltage equations used in the differential equations solution is more

complicated. The dampening ratio will still determine the shape of the response, and the time constant

can be found for these circuits as well. The dampening effects are unique to second order circuits, but

exist simply because of the physics of the dynamic devices within the circuit.

Questions

PART A

1. As the resistance of the potentiometer in a series RLC circuit decreases, the circuit becomes less

damped. This means that the effective time constant of the circuit is smaller, so the voltage of

the capacitor takes less time to level off. For circuits with a damping ratio 0 greater than 1,the voltage over the capacitor will be overdamped. If = 1, the natural response of the circuit iscritically damped. And if < 1, the circuit is underdamped. Underdamped responses will takethe shape of a sinusoidal wave dampened by an exponential time constant.

2. The effective time constant of the signal over the capacitor for the series RLC circuit with a 50

resistor is 0.00004.

3. The resistance of the circuit seen by the capacitor is 1587.

PART B

1. When C2 is varied between 0.1F and 100F in the Operational Amplifier circuit while C1 is kept

at 1F, the time constant increases. Also, the amplitude of the sinusoidal dampening of the

circuit decreases with an increase in capacitance. At 0.1F, the response is underdamped. The

1F capacitor has a critically damped response. Also, the 100F capacitor has an overdamped

response.

2. Varying the resistors in the Op Amp circuit (as long as they are varied together) will simplychange the effective time constant of the response wave. By replacing the 1k resistors with

10k, the time constant changes from 10-3

to 10-4

.

3. By replacing only R2 with a 10k resistor, the response becomes overdamped. The effective

time constant also increases.


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Data

Voltage across Capacitor for 1k Resistor

= 1 = 103 = 0.1 = 107 = 1 = 103

2 = =103

103 = 106

02 = 1 =1

103107 = 10102/2

0 = 10 = 0

=

0 = 0

= 0 = 00 = 0

=

=

= 22

+ + =

22 +

+ =

22 + 2

+ 02 = 02

22 + 106

+ 1010 = 0

2 + 106 + 1010 = 01 = 9.9 105 1062 = 1.01 104 104 = 11 + 22

= 11000000 + 210000

= 111 + 222

0 = 1 + 2 = 10

= 1061 1042

=

0 = 1071061 1042 = 00.11 0.0012 = 0

1 10.1 0.001 12 =

1

0

12 = 0.011.01

= 0.011000000 + 1.0110000

R1

1k

V1

TD = .0005

TF = 0

PW = .0005

PER = .001

V1 = 0

TR = 0

V2 = 1

L1

1mH

1 2

C1

0.1uF

0

VV-V+


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Voltage across Capacitor for 500 Resistor

= 500

2 = =500

103 = 5 10522 + 2

+ 02 = 02

22 + 5 105

+ 1010 = 0

2 + 5 1 05 + 1010 = 01 = 4.8 105

2 = 2.1 104 = 11 + 22

= 1480000 + 221000

= 111

+ 222

0 = 1 + 2 = 10 = 1074800001 210002 = 0

0.0481 0.00212 = 0

1 10.048 0.0021

12 = 1

0

1

2 = 0.045

1.04

= 0.045480000 + 1.0421000

Voltage across Capacitor for 50 Resistor

= 50

2 = =50

103 = 5 104

2

2 + 2 + 02 = 0222 + 5 104

+ 1010 = 0

2 + 5 1 04 + 1010 = 0 = 2.5 104 9 . 7 1 04

=

= 1+ + 2 = cos +

= tan1

= tan1 9.7 104

2.5 104= 1.319

= 2 + 2 = 2.5 1042 + 9.7 1042 = 105

= 10000025000 cos97000+ 1.319

Effective Time Constant of Capacitor in 50 Series RLC Circuit

= / 10000025000

= 125000

= 4 105


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Resistance of Series RLC Circuit

= 1 = 2 = 2000/ = 1 =

1

2000/107 = 1592

= = 2000/103 = 6.283 = 2 + 2

1 = 12 + 6.283 15922 = 1875500 = 5002 + 6.283 15922 = 166350 = 502 + 6.283 15922 = 1587


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Output Voltage of Op Amp Circuit

1 =

2 +

1 = 11

2 = 2

= 1 1 1

1 = 2 + 1

1

1 = + 22

22 1

= 2 + 1 22

1122 2

2 + 21 + 2 + =

2

2 + 2

1 +

2

1122 +1

1122 =

22 + 20

+ 02 =

20 = 21 + 21122 =1

21 +1

11

02 = 11122

0 = 11122

= 21 + 221122

2 + 20 + 02 = 01 = 0 + 02 1

2 = 0 02 1 = 11 + 22

U1

OPAMP

+

-

OUT

C1

1uF

IC = 0v

C2

{C}

IC = 0v

V1

TD = .0005

TF = 0PW = .0005PER = .001

V1 = 0

TR = 0

V2 = 1

0

R1

1k

R2

1k

PARAMETERS:

C = 1uF

V

i1

i2

iC

vinvout

vout

v1


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Output Voltage for 0.1F Capacitor in Op Amp Circuit

02 = 11110.1= 1072/2

0 = 3162 /

= 0.11 + 121110.1 = 0.3162

2 = 0.11 = 3162 0.3162 + 31620.1 1

= 1000 + 3000

2 = 3162 0.3162 31620.1 1= 1000 + 3000

= 11000+3000 + 210003000

=1000 cos3000 +

= tan1 30001000

= 1.25

= 10002 + 30002 = 3162

= 3162

1000 cos

3000

+ 1.25

Output Voltage for 1F Capacitor in Op Amp Circuit

02 = 11111= 1062/2

0 = 1000 /

=

11 + 121111

= 1

2 = 1

1 = 1000 + 10001 1 = 10002 = 1000 10001 1 = 1000

= 21000 0 = 2 = 1

= 12

= 1000

Output Voltage for 100F Capacitor in Op Amp Circuit

02 = 1111100= 1042/2

0 = 100 /

= 1001 + 12111100 = 10

2 = 1001 = 1000 + 100100 1 = 5

2 = 1000 100100 1 = 1995 = 15 + 21995

0 = 1 + 2 = 12 = 100515 199521995

20 = 10051 19952 = 00.00051 0.19952 = 0


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1 10.0005 0.1995

12 = 1

0

12

=

1.0025

0.0025

= 1.00255 0.00251995

Output Voltage for 10k Resistors in Op Amp Circuit

02 = 1101101= 1062/2

0 = 10000 /

= 1

10

+ 10

2101101 = 1

2 = 1

1 = 10000 + 100001 1 = 100002 = 10000 100001 1 = 10000

= 210000

0

= 2

= 1

= 12

= 10000

Output Voltage for 10k Resistors in Op Amp Circuit

02 = 111101= 105

2/

2

0 = 316.2 /

= 11 + 10211101 = 1.739

2 = 3.0251 = 316.2 1.739 + 316.23.025 1

= 100

2 = 316.2 1.739 316.23.025 1= 1000

= 1100 + 21000 0 = 1 + 2 = 1

2 = 11001100 10002100020 = 11001 10002 = 0

0.00011 0.0012 = 0

1 10.0001 0.001

12 = 1

0

12 = 1.11

0.11

= 1.11100 0.111000


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Figure 1 - Mobile Studio Reading: Voltage across 0.1uF Capacitor and 1k Resistor in Series RLC Circuit

Figure 2 - Mobile Studio Reading: Voltage across 0.1uF Capacitor and 500 Resistor in Series RLC Circuit

Figure 3 Mobile Studio Reading: Voltage across 0.1uF Capacitor and 50 Resistor in Series RLC Circuit

Figure 4 - Mobile Studio Reading: Output Voltage for 0.1uF Capacitance in Op Amp Circuit


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Figure 5 - Mobile Studio Reading: Output Voltage for 1uF Capacitance in Op Amp Circuit

Figure 6 - Mobile Studio Reading: Output Voltage for 100uF Capacitance in Op Amp Circuit

Figure 7 - Mobile Studio Reading: Output Voltage for 10k Resistance in Op Amp Circuit

Figure 8 - Mobile Studio Reading: Output Voltage for 1k, 10k Series Resistance in Op Amp Circuit


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Time0.50ms 0.55ms 0.60ms 0.65ms 0.70ms 0.75ms 0.80ms 0.85ms 0.90ms 0.95ms 1.00ms 1.05ms 1.10ms 1.15ms 1.20ms 1.25ms 1.30ms 1.35ms 1.40ms 1.45ms1.50ms

V(R1:1,L1:1) V(C1:2)-1.0V

-0.5V

0V

0.5V

1.0V

1.5V

1k

500

100

1k

500

100

Figure 10 - PSPICE Simulation: Voltage across Resistor for Variable Resistance in Series RLC Circuit

Time0.50ms 0.55ms 0.60ms 0.65ms 0.70ms 0.75ms 0.80ms 0.85ms 0.90ms 0.95ms 1.00ms 1.05ms 1.10ms 1.15ms 1.20ms 1.25ms 1.30ms 1.35ms 1.40ms 1.45ms1.50ms

V(R1:1,L1:1) V(C1:2)-1.0V

-0.5V

0V

0.5V

1.0V

1.5V

1k

1k

100

100

500

500

Figure 9 - PSPICE Simulation: Voltage across Capacitor for Variable Resistance in Series RLC Circuit


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Time0s 0.5ms 1.0ms 1.5ms 2.0ms 2.5ms 3.0ms 3.5ms 4.0ms 4.5ms 5.0ms 5.5ms 6.0ms 6.5ms 7.0ms 7.5ms 8.0ms 8.5ms 9.0ms 9.5ms 10.0ms

V(C1:2)0V

100mV

200mV

300mV

400mV

500mV

600mV

700mV

10k

5k

1k

Figure 12 - PSPICE Simulation: Output Voltage for Variable Resistance in Op Amp Circuit

Time0s 0.5ms 1.0ms 1.5ms 2.0ms 2.5ms 3.0ms 3.5ms 4.0ms 4.5ms 5.0ms 5.5ms 6.0ms 6.5ms 7.0ms 7.5ms 8.0ms 8.5ms 9.0ms 9.5ms 10.0ms

V(U1:OUT)0V

0.2V

0.4V

0.6V

0.8V

1.0V

1.2V

100uF

1uF

0.1uF

Figure 11 - PSPICE Simulation: Output Voltage for Variable Capacitance in Op Amp Circuit


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Figure 14 - MATLAB Plot: Voltage across Capacitor for Variable Resistance in Series RLC Circuit

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R=1k

R=500

Time0s 20ms 40ms 60ms 80ms 100ms 120ms 140ms 160ms 180ms 200ms 220ms 240msV(C1:2)

0V

40mV

80mV

120mV

160mV

200mV

240mVFigure 13- PSPICE Simulation: Output Voltage for 1k, 10k Series Resistance in Op Amp Circuit

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