Electricity 4.4.03-01/11 Inductance of solenoids...4.4.03-01/11 Inductance of solenoids Principle: A square wave voltage of low fre- ... BNC-plug/socket 4 mm 07542.26 1 Induction coil,

4.4.03-01/11 Inductance of solenoids

Principle:A square wave voltage of low fre-quency is applied to oscillatory cir-cuits comprising coils and capacitorsto produce free, damped oscillations.The values of inductance are calcu-lated from the natural frequenciesmeasured, the capacitance beingknown.

Tasks:To connect coils of different dimen-sions (length, radius, number ofturns) with a known capacitance Cto form an oscillatory circuit. Fromthe measurements of the natural fre-quencies, to calculate the induc-

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

tances of the coils and determine therelationships between:

1. inductance and number of turns

2. inductance and length

3. inductance and radius.

What you can learn about …

� Lenz’s law� Self-inductance� Solenoids� Transformer� Oscillatory circuit� Resonance� Damped oscillation� Logarithmic decrement� Q factor

Experiment P2440311 with FG-ModuleExperiment P2440301 with oscilloscope

Function generator 13652.93 1Oscilloscope, 30 MHz, 2 channels 11459.95 1Adapter, BNC-plug/socket 4 mm 07542.26 1Induction coil, 300 turns, d = 40 mm 11006.01 1 1Induction coil, 300 turns, d = 32 mm 11006.02 1 1Induction coil, 300 turns, d = 25 mm 11006.03 1 1Induction coil, 200 turns, d = 40 mm 11006.04 1 1Induction coil, 100 turns, d = 40 mm 11006.05 1 1Induction coil, 150 turns, d = 25 mm 11006.06 1 1Induction coil, 75 turns, d = 25 mm 11006.07 1 1Coil, 1200 turns 06515.01 1 1Capacitor /case 1/ 470 nF 39105.20 1 1Connection box 06030.23 1 1Connecting cord, l = 250 mm, red 07360.01 1 1Connecting cord, l = 250 mm, blue 07360.04 1 1Connecting cord, l = 500 mm, red 07361.01 2 2Connecting cord, l = 500 mm, blue 07361.04 2 2Cobra3 Basic Unit 12150.00 1Power supply, 12 V- 12151.99 2RS232 data cable 14602.00 1Cobra3 Universal writer software 14504.61 1Measuring module function generator 12111.00 1PC, Windows® 95 or higher

What you need:

Complete Equipment Set, Manual on CD-ROM includedInductance of solenoids with Cobra3 P24403 01/11

182 PHYWE Systeme GmbH & Co. KG · D-37070 GöttingenLaboratory Experiments Physics

Electricity Electrodynamics

Measurement of the oscillation period with the “Survey Function”.

Set-up of experiment P2440311 with FG-Module

LEP4.4.03

-01Inductance of solenoids

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 24403-01 1

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

PrincipleA 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 calculatedfrom the natural frequencies measured, the capacitance beingknown.

EquipmentInduction coil, 300 turns, d = 40 mm 11006.01 1Induction coil, 300 turns, d = 32 mm 11006.02 1Induction coil, 300 turns, d = 25 mm 11006.03 1Induction coil, 200 turns, d = 40 mm 11006.04 1Induction coil, 100 turns, d = 40 mm 11006.05 1Induction coil, 150 turns, d = 25 mm 11006.06 1Induction coil, 75 turns, d = 25 mm 11006.07 1Coil, 1200 turns 06515.01 1Oscilloscope, 30 MHz, 2 channels 11459.95 1Function generator 13652.93 1Capacitor /case 1/ 470 nF 39105.20 1Adapter, BNC-plug/socket 4 mm 07542.26 1Connection box 06030.23 1Connecting cord, l = 250 mm, red 07360.01 1

Connecting cord, l = 500 mm, red 07361.01 2Connecting cord, l = 250 mm, blue 07360.04 1Connecting cord, l = 500 mm, blue 07361.04 2

TasksTo 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 determinethe relationships between




Set-up and procedureSet up the experiment as shown in Fig. 1 + 2.A square wave voltage of low frequency (f ≈ 500 Hz) is appliedto the excitation coil L. The sudden change in the magneticfield induces a voltage in coil L1 and creates a free dampedoscillation in the L1C oscillatory circuit, the frequency fo ofwhich is measured with the oscilloscope.Coils of different lengths l, diameters 2r and number of turnsN are available (Tab. 1). The diameters and lengths are meas-ured with the vernier caliper and the measuring tape, and thenumbers of turns are given.

Fig. 1: Experimental set-up.

Tab. 1: Table of coil data

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 � L = f(N)

2.) 1, 4, 5 � L/N2 = f( l)

3.) 1, 2, 3 � L = f(r)

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

Fig. 3 shows an measurement example and the oscilloscopesettings:- Input: CH1- Volts/div 10 mV- Time/div <0.1 ms- Trigger source CH1- Trigger mode Norm

The oscilloscope shows the rectangular signal and the dam-ped oscillation behind each peak. Determine the frequency f0of this damped oszillation.

where T is the oscillation period.

f0 �1T

LEP4.4.03


24403-01 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen2

Fig. 2: Set-up for inductance measurement.

Fig. 3: Oscilloscope settings.

Coil No. N Cat. No.

1 300 40 160 11006.01

2 300 32 160 11006.02

3 300 26 160 11006.03

4 200 40 105 11006.04

5 100 40 53 11006.05

6 150 26 160 11006.06

7 75 26 160 11006.07

l

mm2r

mm

NotesThe 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.

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

Theory and evaluationIf a current of strength I flows through a cylindrical coil (sole-noid) of length l, cross sectional area A = p · r2, and numberof turns N, a magnetic field is set up in the coil. When l >> rthe magnetic field is uniform and the field strength H is easyto calculate:

(1)

The magnetic flux through the coil is given by

' = mo · m · H · A (2)

where mo is the magnetic field constant and m the absolutepermeability of the surrounding medium.

When this flux changes, it induces a voltage between the endsof the coil,

(3)

where

(4)

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

Inductivity Equation (4) for the inductance applies only to verylong coils l >> r, with a uniform magnetic field in accordancewith (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 calculatedfrom the natural frequency of an oscillating circuit.

(6)

Ctot. is the sum of the capacitance the known capacitor andthe input capacitance Ci of the oscilloscope.

v0 � 12LCtot.

0 6r

l6 1

L � 2.1 · 10�6 · N2 · r · ar

lb

3>4

L � m0 · m · p · N2 · r2

l

� � L · I �

� � N · m0 · m · A · N

l · I

�

Uind.� � N · £�

H � I · N

l

LEP4.4.03



Fig. 4: Inductances of the coils as a function of the number ofturns, at constant length and constant radius.Double logarithmic plotting.

Fig. 6: Inductance of the coils as a function of the radius, atconstant length and number of turns.Double logarithmic plotting.

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

LEP4.4.03



The internal resistance Ri of the oscilloscope exercises adamping 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 Ctot. = C + Ci and

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

Table 2

The table 3 shows the measured values of the oscillation peri-ods and the corresponding inductance values of the usedcoils calculated according to eq. 7. These Lexp values areplotted in Figs. 4, 5 and 6.

Table 3

Applying the expression

L = A · NB

to the regression line from the measured values in Fig. 4 givesthe exponent

B = 1.95±0.04 ; Btheo = 2 (see Eq. 5)

Now that we know that L ~ N2, we can demonstrate the rela-tionship between inductance and the length of the coil.



C = – 0.82 ± 0.04. ; Ctheo = -0.75



D = 1.86 ± 0.07. ; Dtheo = 1.75

The Equation (5) is thus verified within the limits of error.

L

N2 � A · rD

L

N2 � A · lC

f0 �v0

2p

L �1

4p2 f02 Ctot.

Coil No. N r/m l/m Ltheo/µH

1 300 0.02 0.16 794.65

2 300 0.016 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. Texp./µs Lexp/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

LEP4.4.03

-11Inductance of solenoids with Cobra3


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

PrincipleA 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 calculatedfrom the natural frequencies measured, the capacitance beingknown.

EquipmentCobra3 Basic Unit 12150.00 1Power supply, 12 V 12151.99 2RS 232 data cable 14602.00 1Cobra3 Universal writer software 14504.61 1Cobra3 Function generator module 12111.00 1Induction coil, 300 turns, dia. 40 mm 11006.01 1Induction coil, 300 turns, dia. 32 mm 11006.02 1Induction coil, 300 turns, dia. 25 mm 11006.03 1Induction coil, 200 turns, dia. 40 mm 11006.04 1Induction coil, 100 turns, dia. 40 mm 11006.05 1Induction coil, 150 turns, dia. 25 mm 11006.06 1Induction coil, 75 turns, dia. 25 mm 11006.07 1Coil, 1200 turns 06515.01 1PEK capacitor /case 1/ 470 nF/250 V 39105.20 1Connection box 06030.23 1

Connecting cord, 250 mm, red 07360.01 1Connecting cord, 250 mm, blue 07360.04 1Connecting cord, 500 mm, red 07361.01 2Connecting cord, 500 mm, blue 07361.04 2PC, Windows® 95 or higher

TasksTo 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 determinethe relationships between




Set-up and procedureSet up the experiment as shown in Fig. 1 + 2.A square wave voltage of low frequency (f ≈ 500 Hz) is appliedto the excitation coil L. The sudden change in the magneticfield induces a voltage in coil L1 and creates a free dampedoscillation in the L1C oscillatory circuit, the frequency fo ofwhich is measured with the Cobra3 interface.Coils of different lengths l, diameters 2r and number of turnsN are available (Tab. 1). The diameters and lengths are meas-ured with the vernier caliper and the measuring tape, and thenumbers of turns are given.

Fig. 1: Experimental set-up.

Tab. 1: Table of coil data

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 � L = f(N)

2.) 1, 4, 5 � L/N2 = f( l)

3.) 1, 2, 3 � L = f(r)

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

NotesThe 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.Connect the Cobra3 Basic Unit to the computer port COM1,COM2 or to USB port (for USB computer port use USB toRS232 Converter 14602.10). Start the measure program andselect Cobra3 Universal Writer Gauge. Begin the measure-ment using the parameters given in Fig. 3.For the measurement of the oscillation period the “SurveyFunction” of the Measure Software is used (see Fig. 4).

LEP4.4.03



Fig. 2: Set-up for inductance measurement. Fig. 3: Measuring parameters.

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

Coil No. N Cat. No.

1 300 40 160 11006.012 300 32 160 11006.023 300 26 160 11006.034 200 40 105 11006.045 100 40 53 11006.056 150 26 160 11006.067 75 26 160 11006.07

l

mm2r

mm

Fig. 4 shows the rectangular signal and the damped oscillati-on behind each peak. Determine the frequency f0 of this dam-ped oszillation,

where T is the oscillation period.

Theory and evaluationIf a current of strength I flows through a cylindrical coil (sole-noid) of length l, cross sectional area A = p · r2, and numberof turns N, a magnetic field is set up in the coil. When l >> rthe magnetic field is uniform and the field strength H is easyto calculate:

(1)

The magnetic flux through the coil is given by

' = mo · m · H · A (2)

where mo is the magnetic field constant and m the absolutepermeability of the surrounding medium.

When this flux changes, it induces a voltage between the endsof the coil,

(3)

where

(4)

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

Inductivity Equation (4) for the inductance applies only to verylong coils l >> r, with a uniform magnetic field in accordancewith (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 calculatedfrom the natural frequency of an oscillating circuit.

(6)

Ctot. is the sum of the capacitance the known capacitor andthe input capacitance Ci of the Cobra3 input.The internal resistance Ri of the Cobra3 input exercises adamping 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 Ctot. = C + Ci and

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

Table 2

The table 3 shows the measured values of the oscillation peri-ods and the corresponding inductance values of the usedcoils calculated according to eq. 7. These Lexp values areplotted in Figs. 5, 6 and 7.

Table 3

f0 �v0

2p

L �1

4p2 f02 Ctot.

v0 � 12LCtot.

0 6r

l6 1

L � 2.1 · 10�6 · N2 · r · ar

lb

3>4

L � m0 · m · p · N2 · r2

l

� � L · I �

� � N · m0 · m · A · N

l · I

�

Uind.� � N · £�

H � I · N

l

f0 �1T

LEP4.4.03



Fig. 5: Inductances of the coils as a function of the number ofturns, at constant length and constant radius.Double logarithmic plotting

Coil No. Texp./µs Lexp/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

Coil No. N r/m l/m Ltheo/µH

1 300 0.02 0.16 794.65

2 300 0.016 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

LEP4.4.03




L = A · NB


B = 1.95±0.04 ; Btheo = 2 (see Eq. 5)

Now that we know that L ~ N2, we can demonstrate the rela-tionship between inductance and the length of the coil.



C = -0.82±0.04 ; Ctheo = -0.75



D = 1.86 ± 0.07. ; Dtheo = 1.75

The Equation (5) is thus verified within the limits of error.

L

N2 � A · rD

L

N2 � A · lC

Fig. 7: Inductance of the coils as a function of the radius, atconstant length and number of turns.Double logarithmic plotting

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

Documents

Electricity 4.4.03-01/11 Inductance of solenoids...4.4.03-01/11 Inductance of solenoids Principle: A square wave voltage of low fre- ... BNC-plug/socket 4 mm 07542.26 1 Induction coil,