LEP4403 01 11Inductance of solenoids

  Electricity Electrodynamics

  4.4.03-01/11 Inductance of solenoids What you can learn about … Lenz’s law Self-inductance Solenoids Transformer Oscillatory circuit Resonance Damped oscillation Logarithmic decrement Q factor Principle: A square wave voltage of low fre- quency is applied to oscillatory cir- cuits comprising coils and capacitors to produce free, damped oscillations. The values of inductance are calcu- lated from the natural frequencies measured, the capacitance being known.

  Set-up of experiment P2440311 with FG-Module What you need: Experiment P2440311 with FG-Module Experiment P2440301 with oscilloscope Function generator 13652.93

  1 Oscilloscope, 30 MHz, 2 channels 11459.95

  1 Adapter, BNC-plug/socket 4 mm 07542.26

  1 11006.01

  1

  

1

Induction coil, 300 turns, d = 40 mm 11006.02

  1

  

1

Induction coil, 300 turns, d = 32 mm 11006.03

  1

  

1

Induction coil, 300 turns, d = 25 mm 11006.04

  1

  

1

Induction coil, 200 turns, d = 40 mm 11006.05

  1

  

1

Induction coil, 100 turns, d = 40 mm 11006.06

  1

  

1

Induction coil, 150 turns, d = 25 mm 11006.07

  1

  

1

Induction coil, 75 turns, d = 25 mm Coil, 1200 turns 06515.01

  1

  

1

Capacitor /case 1/ 470 nF 39105.20

  1

  

1

Connection box 06030.23

  1

  

1

Inductance per turn as a function of the length of the coil at constant radius. 07360.01

  1

  

1

Connecting cord, l = 250 mm, red 07360.04

  1

  

1

Connecting cord, l = 250 mm, blue 07361.01

  2

  

2

Connecting cord, l = 500 mm, red 07361.04

  2

  

2

Connecting cord, l = 500 mm, blue Cobra3 Basic Unit 12150.00

  

1

Power supply, 12 V- 12151.99

  

2

RS232 data cable 14602.00

  

1

Cobra3 Universal writer software 14504.61

  

1

Measuring module function generator 12111.00

  

1

PC, Windows® 95 or higher Complete Equipment Set, Manual on CD-ROM included Inductance of solenoids with Cobra3 P24403 01/11 Tasks:

  To connect coils of different dimen- tances of the coils and determine the sions (length, radius, number of relationships between: turns) with a known capacitance C 1. inductance and number of turns to form an oscillatory circuit. From Measurement of the oscillation period with the “Survey Function”.

  2. inductance and length the measurements of the natural fre- 3. inductance and radius. quencies, to calculate the induc-

  LEP

  1 Function generator 13652.93

  of which is measured with the oscilloscope. Coils of different lengths l, diameters 2 r and number of turns N are available (Tab. 1). The diameters and lengths are meas- ured with the vernier caliper and the measuring tape, and the numbers of turns are given.

  o

  A square wave voltage of low frequency ( f ≈ 500 Hz) is applied to the excitation coil L. The sudden change in the magnetic field induces a voltage in coil L1 and creates a free damped oscillation in the L1C oscillatory circuit, the frequency f

  Set-up and procedure Set up the experiment as shown in Fig. 1 + 2.

  C to form an oscilla- tory circuit. From the measurements of the natural frequen- cies, to calculate the inductances of the coils and determine the relationships between 1. inductance and number of turns 2. inductance and length 3. inductance and radius.

  2 Tasks To connect coils of different dimensions (length, radius, num- ber of turns) with a known capacitance

  1 Connecting cord, l = 500 mm, blue 07361.04

  2 Connecting cord, l = 250 mm, blue 07360.04

  1 Connecting cord, l = 500 mm, red 07361.01

  1 Connecting cord, l = 250 mm, red 07360.01

  1 Connection box 06030.23

  1 Adapter, BNC-plug/socket 4 mm 07542.26

  1 Capacitor /case 1/ 470 nF 39105.20

  1 Oscilloscope, 30 MHz, 2 channels 11459.95

  4.4.03 -01

Inductance of solenoids

  1 Coil, 1200 turns 06515.01

  1 Induction coil, 75 turns, d = 25 mm 11006.07

  1 Induction coil, 150 turns, d = 25 mm 11006.06

  1 Induction coil, 100 turns, d = 40 mm 11006.05

  1 Induction coil, 200 turns, d = 40 mm 11006.04

  1 Induction coil, 300 turns, d = 25 mm 11006.03

  1 Induction coil, 300 turns, d = 32 mm 11006.02

  Induction coil, 300 turns, d = 40 mm 11006.01

  Equipment

  A square wave voltage of low frequency is applied to oscilla- tory circuits comprising coils and capacitors to produce free, damped oscillations. The values of inductance are calculated from the natural frequencies measured, the capacitance being known.

  Principle

  Law of inductance, Lenz’s law, self-inductance, solenoids, transformer, oscillatory circuit, resonance, damped oscillation, logarithmic decrement, Q factor.

  PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 24403-01 Related topics

  Fig. 1: Experimental set-up.

  LEP

4.4.03 Inductance of solenoids -01

  Fig. 2: Set-up for inductance measurement. The following coils provide the relationships between induct- ance and radius, length and number of turns that we are in- vestigating: 1.) 3, 6, 7 = f(N)

  L

  2

  2.) 1, 4, 5 = L/N f( l)

  3.) 1, 2, 3 = f( r) L

  As a difference in length also means a difference in the num- ber of turns, the relationship between inductance and number of turns found in Task 1 must also be used to solve Task 2.

  Fig. 3 shows an measurement example and the oscilloscope settings:

• Input: CH1 Tab. 1: Table of coil data
• Volts/div 10 mV
• Time/div <0.1 ms

  2 r l Coil No. Cat. No. - Trigger source CH1

  N mm mm

• Trigger mode Norm 1 300

  40 160 11006.01 The oscilloscope shows the rectangular signal and the dam- 2 300

  32 160 11006.02 ped oscillation behind each peak. Determine the frequency f of this damped oszillation.

  3 300 26 160 11006.03 4 200 40 105 11006.04

  1 5 100 40 53 11006.05 f

  T 6 150 26 160 11006.06

  7

  75 26 160 11006.07 where T is the oscillation period.

  Fig. 3: Oscilloscope settings.

  24403-01 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen

  Notes

  I U

  3 >4

  L m · m · p ·

  N

  2

  · r

  2

  l L · I N · m

  · m · A ·

  N l ·

  ind _.

  2

  N · £ H I ·

  N l

  LEP

  4.4.03 -01

Inductance of solenoids

  PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 24403-01

  Fig. 4: Inductances of the coils as a function of the number of turns, at constant length and constant radius.

  Double logarithmic plotting.

  Fig. 6: Inductance of the coils as a function of the radius, at constant length and number of turns.

  Double logarithmic plotting.

  Fig. 5: Inductance per turn as a function of the length of coil, at constant radius.

  · r · a r l b

  · N

  The distance between L1 and L should be as large as possi- ble so that the effect of the excitation coil on the resonant fre- quency can be disregarded. There should be no iron compo- nents in the immediate vicinity of the coils.

  o

  The tolerance of the oscilloscope time-base is given as 4%. If a higher degree of accuracy is required, the time-base can be calibrated for all measuring ranges with the function generator and a frequency counter prior to these experiments.

  Theory and evaluation

  If a current of strength I flows through a cylindrical coil (sole- noid) of length l, cross sectional area A = p · r

  2

  , and number of turns N, a magnetic field is set up in the coil. When l >> r the magnetic field is uniform and the field strength H is easy to calculate:

  (1) The magnetic flux through the coil is given by

  ' = m

  o

  · m · H · A

  (2) where m

  is the magnetic field constant and m the absolute permeability of the surrounding medium. When this flux changes, it induces a voltage between the ends of the coil,

  6

  (3) where (4) is the coefficient of self-induction (inductance) of the coil.

  Inductivity Equation (4) for the inductance applies only to very long coils l >> r, with a uniform magnetic field in accordance with (1).

  In practice, the inductance of coils with l > r can be calcula- ted with greater accuracy by an approximation formula for (5) In the experiment, the inductance of various coils is calculated from the natural frequency of an oscillating circuit.

  (6) C tot. is the sum of the capacitance the known capacitor and the input capacitance

  C

  i of the oscilloscope.

  v

  1

  2LC tot. 0 6 r l 6 1

  L 2.1 · 10

  Double logarithmic plotting.

  LEP

  Applying the expression L = A · N

  2

  The internal resistance R

  i

  of the oscilloscope exercises a damping effect on the oscillatory circuit and causes a negli- gible shift (approx. 1%) in the resonance frequency. The inductance is therefore represented by

  (7) where C tot.

  1 4p

  L

  4.4.03 -01

Inductance of solenoids

  to the regression line from the measured values in Fig. 4 gives the exponent B = 1.95±0.04 ; B

  B

2 C tot.

  values are plotted in Figs. 4, 5 and 6. Table 3

  ./µs L

  0.16 537.75 3 300 0.013 0.16 373.91 4 200

  0.02 0.105 484.38 5 100 0.02 0.053 202.22 6 150 0.013

  0.16

  93.48

  7 75 0.013

  0.16

  23.37 Coil No.

  T

  exp

  exp

  theo

  /µH 1 119.94 776.09 2 97.42 512.01

  3 78.24 330.25

  4 94.77 448.53

  5 62.88 213.31

  6

  39.27

  83.20

  7

  20.19

  21.99

  /µH 1 300 0.02 0.16 794.65 2 300 0.016

  N r/m l/m L

  exp

  = -0.75 Applying the expression to the regression line from the measured values in Fig. 6 gives the exponent

  L

  Table 2 The table 3 shows the measured values of the oscillation peri- ods and the corresponding inductance values of the used coils calculated according to eq. 7. These

  theo

  = 2 (see Eq. 5) Now that we know that

  L ~ N

  2

  , we can demonstrate the rela- tionship between inductance and the length of the coil. Applying the expression to the regression line from the measured values in Fig. 5 gives the exponent

  C = – 0.82 ± 0.04. ; C

  theo

  D = 1.86 ± 0.07. ; D

  Coil No.

  theo

  = 1.75 The Equation (5) is thus verified within the limits of error. L N

  2 A · r D

  L N

  2 A · l C

  and The table 2 shows the theoretical inductance values of the used coils calculated according to eq. 5.

  i

  = C + C

  24403-01 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen

  f

  f v 2p

  LEP

  ®

  1 Connection box 06030.23

  1 Connecting cord, 250 mm, red 07360.01

  1 Connecting cord, 250 mm, blue 07360.04

  1 Connecting cord, 500 mm, red 07361.01

  2 Connecting cord, 500 mm, blue 07361.04

  2 PC, Windows

  95 or higher

  1 Coil, 1200 turns 06515.01

  Tasks

  To connect coils of different dimensions (length, radius, num- ber of turns) with a known capacitance C to form an oscilla- tory circuit. From the measurements of the natural frequen- cies, to calculate the inductances of the coils and determine the relationships between 1. inductance and number of turns 2. inductance and length 3. inductance and radius.

  Set-up and procedure Set up the experiment as shown in Fig. 1 + 2.

  A square wave voltage of low frequency ( f ≈ 500 Hz) is applied to the excitation coil L _{. The sudden change in the magnetic} field induces a voltage in coil L

  1 and creates a free damped oscillation in the L1C oscillatory circuit, the frequency f

  o

  of which is measured with the Cobra3 interface. Coils of different lengths l, diameters 2 r and number of turns N are available (Tab. 1). The diameters and lengths are meas- ured with the vernier caliper and the measuring tape, and the numbers of turns are given.

  1 PEK capacitor /case 1/ 470 nF/250 V 39105.20

  1 Induction coil, 75 turns, dia. 25 mm 11006.07

  4.4.03 -11

Inductance of solenoids with Cobra3

  1 Power supply, 12 V 12151.99

  PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 24403-11 Related topics

  Law of inductance, Lenz’s law, self-inductance, solenoids, transformer, oscillatory circuit, resonance, damped oscillation, logarithmic decrement, Q factor.

  Principle

  A square wave voltage of low frequency is applied to oscilla- tory circuits comprising coils and capacitors to produce free, damped oscillations. The values of inductance are calculated from the natural frequencies measured, the capacitance being known.

  Equipment

  Cobra3 Basic Unit 12150.00

  2 RS 232 data cable 14602.00

  1 Induction coil, 150 turns, dia. 25 mm 11006.06

  1 Cobra3 Universal writer software 14504.61

  1 Cobra3 Function generator module 12111.00

  1 Induction coil, 300 turns, dia. 40 mm 11006.01

  1 Induction coil, 300 turns, dia. 32 mm 11006.02

  1 Induction coil, 300 turns, dia. 25 mm 11006.03

  1 Induction coil, 200 turns, dia. 40 mm 11006.04

  1 Induction coil, 100 turns, dia. 40 mm 11006.05

  Fig. 1: Experimental set-up.

  LEP

4.4.03 Inductance of solenoids with Cobra3 -11 Fig. 2: Set-up for inductance measurement.

  Fig. 3: Measuring parameters.

  Tab. 1: Table of coil data

  2 r l Coil No. Cat. No.

  N mm mm As a difference in length also means a difference in the number 1 300

  40 160 11006.01 of turns, the relationship between inductance and number of 2 300 32 160 11006.02 turns found in Task 1 must also be used to solve Task 2. 3 300 26 160 11006.03 4 200

  40 105 11006.04

  Notes

  5 100

  40 53 11006.05 The distance between L1 and L should be as large as possi- 6 150

  26 160 11006.06 ble so that the effect of the excitation coil on the resonant fre-

  7

  75 26 160 11006.07 quency can be disregarded. There should be no iron compo- nents in the immediate vicinity of the coils.

  Connect the Cobra3 Basic Unit to the computer port COM1, The following coils provide the relationships between induct-

  COM2 or to USB port (for USB computer port use USB to ance and radius, length and number of turns that we are in- RS232 Converter 14602.10). Start the measure program and vestigating: select Cobra3 Universal Writer Gauge. Begin the measure-

  1.) 3, 6, 7 = L f(N) ment using the parameters given in Fig. 3.

  2

  2.) 1, 4, 5 = f( l) L/N

  For the measurement of the oscillation period the “Survey 3.) 1, 2, 3 = f( r)

  L Function” of the Measure Software is used (see Fig. 4).

  Fig. 4: Measurement of the oscillation period with the “Survey Function”.

  24403-11 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen

  LEP

Inductance of solenoids with Cobra3

  4.4.03 -11

  Fig. 4 shows the rectangular signal and the damped oscillati- The inductance is therefore represented by on behind each peak. Determine the frequency of this dam- f

  1 ped oszillation, (7)

  L

  2

  2

  4p f C

  tot.

  1 f v

  T where = and C C + C f

  tot. i

  2p where T is the oscillation period.

  The table 2 shows the theoretical inductance values of the

  Theory and evaluation used coils calculated according to eq. 5.

  If a current of strength I flows through a cylindrical coil (sole-

  2 Table 2

  noid) of length , and number l, cross sectional area A = p · r of turns

  N, a magnetic field is set up in the coil. When l >> r the magnetic field is uniform and the field strength H is easy Coil No. /µH

  N r/m l/m L

  theo

  to calculate: 1 300

  0.02 0.16 794.65 N

  2 300 0.016 0.16 537.75 (1)

  H I · l 3 300 0.013 0.16 373.91

  The magnetic flux through the coil is given by 4 200 0.02 0.105 484.38 5 100 0.02 0.053 202.22

  · m · (2) H · A

  ' = m

  o

  6 150 0.013

  0.16

  93.48 where m is the magnetic field constant and m the absolute

  o

  7 75 0.013

  0.16

  23.37 permeability of the surrounding medium. When this flux changes, it induces a voltage between the ends The table 3 shows the measured values of the oscillation peri- of the coil, ods and the corresponding inductance values of the used coils calculated according to eq. 7. These values are

  L

  exp plotted in Figs. 5, 6 and 7.

  U N · £ ind .

  N · m · ·

  A ·

  I N · m Table 3 l

  (3) Coil No. /µs /µH L · I

  T L

  exp. exp

  1 119.94 776.09 where

  2

  2

  2 97.42 512.01 ·

  N r · m · p · (4)

  L m l 3 78.24 330.25

  4 94.77 448.53 is the coefficient of self-induction (inductance) of the coil.

  5 62.88 213.31 Inductivity Equation (4) for the inductance applies only to very

  6

  39.27

  83.20 long coils l >> r, with a uniform magnetic field in accordance

  7

  20.19

  21.99 with (1). In practice, the inductance of coils with l > r can be calcula-

  Fig. 5: Inductances of the coils as a function of the number of ted with greater accuracy by an approximation formula turns, at constant length and constant radius.

  3 >4

  Double logarithmic plotting r

  6

  2

  · · N r ·

  L 2.1 · 10 a b l r for (5)

  0 6 6 1 l In the experiment, the inductance of various coils is calculated from the natural frequency of an oscillating circuit.

  1 v (6)

  2LC tot. is the sum of the capacitance the known capacitor and

  C tot. the input capacitance of the Cobra3 input.

  C

  i

  The internal resistance of the Cobra3 input exercises a R

  i

  damping effect on the oscillatory circuit and causes a negli- gible shift (approx. 1%) in the resonance frequency.

  PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 24403-11

  LEP

  theo

  Double logarithmic plotting Fig. 6: Inductance per turn as a function of the length of coil, at constant radius.

  Fig. 7: Inductance of the coils as a function of the radius, at constant length and number of turns.

  2 A · l C

  L N

  2 A · r D

  = 1.75 The Equation (5) is thus verified within the limits of error. L N

  theo

  D = 1.86 ± 0.07. ; D

  = -0.75 Applying the expression to the regression line from the measured values in Fig. 7 gives the exponent

  C = -0.82±0.04 ; C

  4.4.03 -11

Inductance of solenoids with Cobra3

  , we can demonstrate the rela- tionship between inductance and the length of the coil. Applying the expression to the regression line from the measured values in Fig. 6 gives the exponent

  2

  L ~ N

  = 2 (see Eq. 5) Now that we know that

  theo

  to the regression line from the measured values in Fig. 5 gives the exponent B = 1.95±0.04 ; B

  B

  Applying the expression L = A · N

  24403-11 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen

  Double logarithmic plotting