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 capacitors to produce free, damped oscillations. The values of inductance are calcu- lated from the natural frequencies measured, the capacitance being known. Tasks: To connect coils of different dimen- sions (length, radius, number of turns) with a known capacitance C to form an oscillatory circuit. From the 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 the relationships 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-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 Induction coil, 300 turns, d = 40 mm 11006.01 1 1 Induction coil, 300 turns, d = 32 mm 11006.02 1 1 Induction coil, 300 turns, d = 25 mm 11006.03 1 1 Induction coil, 200 turns, d = 40 mm 11006.04 1 1 Induction coil, 100 turns, d = 40 mm 11006.05 1 1 Induction coil, 150 turns, d = 25 mm 11006.06 1 1 Induction coil, 75 turns, d = 25 mm 11006.07 1 1 Coil, 1200 turns 06515.01 1 1 Capacitor /case 1/ 470 nF 39105.20 1 1 Connection box 06030.23 1 1 Connecting cord, l = 250 mm, red 07360.01 1 1 Connecting cord, l = 250 mm, blue 07360.04 1 1 Connecting cord, l = 500 mm, red 07361.01 2 2 Connecting cord, l = 500 mm, blue 07361.04 2 2 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 What you need: Complete Equipment Set, Manual on CD-ROM included Inductance of solenoids with Cobra3 P24403 01/11 182 PHYWE Systeme GmbH & Co. KG · D-37070 Göttingen Laboratory Experiments Physics Electricity Electrodynamics Measurement of the oscillation period with the “Survey Function”. Set-up of experiment P2440311 with FG-Module
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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:
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
1. inductance and number of turns
2. inductance and length
3. inductance and radius.
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.
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.
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.
Applying the expression
to the regression line from the measured values in Fig. 5 givesthe exponent
C = – 0.82 ± 0.04. ; Ctheo = -0.75
Applying the expression
to the regression line from the measured values in Fig. 6 givesthe exponent
D = 1.86 ± 0.07. ; Dtheo = 1.75
The Equation (5) is thus verified within the limits of error.
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
1. inductance and number of turns
2. inductance and length
3. inductance and radius.
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).
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.