Electricity
4.4.03-01/11
Electrodynamics
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 frequency is applied to oscillatory 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.
Set-up of experiment P2440311 with FG-Module
What you need:
Experiment P2440311 with FG-Module
Experiment P2440301 with oscilloscope
Function generator
Oscilloscope, 30 MHz, 2 channels
Adapter, BNC-plug/socket 4 mm
Induction coil, 300 turns, d = 40 mm
Induction coil, 300 turns, d = 32 mm
Induction coil, 300 turns, d = 25 mm
Induction coil, 200 turns, d = 40 mm
Induction coil, 100 turns, d = 40 mm
Induction coil, 150 turns, d = 25 mm
Induction coil, 75 turns, d = 25 mm
Coil, 1200 turns
Capacitor /case 1/ 470 nF
Connection box
Connecting cord, l = 250 mm, red
Connecting cord, l = 250 mm, blue
Connecting cord, l = 500 mm, red
Connecting cord, l = 500 mm, blue
Cobra3 Basic Unit
Power supply, 12 VRS232 data cable
Cobra3 Universal writer software
Measuring module function generator
PC, Windows® 95 or higher
13652.93
11459.95
07542.26
11006.01
11006.02
11006.03
11006.04
11006.05
11006.06
11006.07
06515.01
39105.20
06030.23
07360.01
07360.04
07361.01
07361.04
12150.00
12151.99
14602.00
14504.61
12111.00
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
1
1
1
1
1
1
1
1
1
1
1
1
2
2
1
2
1
1
1
Inductance per turn as a function of the length of the coil at constant radius.
Complete Equipment Set, Manual on CD-ROM included
Inductance of solenoids with Cobra3
P24403 01/11
Tasks:
To connect coils of different dimensions (length, radius, number of
turns) with a known capacitance C
to form an oscillatory circuit. From
the measurements of the natural frequencies, to calculate the induc182 Laboratory Experiments Physics
tances of the coils and determine the
relationships between:
1. inductance and number of turns
2. inductance and length
Measurement of the oscillation period with the “Survey Function”.
3. inductance and radius.
PHYWE Systeme GmbH & Co. KG · D - 37070 Göttingen
LEP
4.4.03
-01
Inductance of solenoids
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 oscillatory 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
Induction coil, 300 turns, d = 40 mm
Induction coil, 300 turns, d = 32 mm
Induction coil, 300 turns, d = 25 mm
Induction coil, 200 turns, d = 40 mm
Induction coil, 100 turns, d = 40 mm
Induction coil, 150 turns, d = 25 mm
Induction coil, 75 turns, d = 25 mm
Coil, 1200 turns
Oscilloscope, 30 MHz, 2 channels
Function generator
Capacitor /case 1/ 470 nF
Adapter, BNC-plug/socket 4 mm
Connection box
Connecting cord, l = 250 mm, red
Connecting cord, l = 500 mm, red
Connecting cord, l = 250 mm, blue
Connecting cord, l = 500 mm, blue
07361.01
07360.04
07361.04
2
1
2
Tasks
To connect coils of different dimensions (length, radius, number of turns) with a known capacitance C to form an oscillatory circuit. From the measurements of the natural frequencies, to calculate the inductances of the coils and determine
the relationships between
1. inductance and number of turns
2. inductance and length
11006.01
11006.02
11006.03
11006.04
11006.05
11006.06
11006.07
06515.01
11459.95
13652.93
39105.20
07542.26
06030.23
07360.01
1
1
1
1
1
1
1
1
1
1
1
1
1
1
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 L1 and creates a free damped
oscillation in the L1C oscillatory circuit, the frequency fo 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 measured with the vernier caliper and the measuring tape, and the
numbers of turns are given.
Fig. 1: Experimental set-up.
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
24403-01
1
LEP
4.4.03
-01
Inductance of solenoids
The following coils provide the relationships between inductance and radius, length and number of turns that we are investigating:
Fig. 2: Set-up for inductance measurement.
1.) 3, 6, 7
2.) 1, 4, 5
3.) 1, 2, 3
L
= f(N)
2
L/N = f( l)
L
= f( r)
As a difference in length also means a difference in the number of turns, the relationship between inductance and number
of turns found in Task 1 must also be used to solve Task 2.
Tab. 1: Table of coil data
Coil No.
1
2
3
4
5
6
7
N
2r
mm
l
mm
Cat. No.
300
300
300
200
100
150
75
40
32
26
40
40
26
26
160
160
160
105
53
160
160
11006.01
11006.02
11006.03
11006.04
11006.05
11006.06
11006.07
Fig. 3 shows an measurement example and the oscilloscope
settings:
- 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 damped oscillation behind each peak. Determine the frequency f0
of this damped oszillation.
f0 1
T
where T is the oscillation period.
Fig. 3: Oscilloscope settings.
2
24403-01
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
LEP
4.4.03
-01
Inductance of solenoids
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.
Notes
The distance between L1 and L should be as large as possible so that the effect of the excitation coil on the resonant frequency can be disregarded. There should be no iron components in the immediate vicinity of the coils.
The magnetic flux through the coil is given by
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 (solenoid) of length l, cross sectional area A = p · r2, 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:
HI·
N
l
(1)
' = mo · m · H · A
(2)
where mo 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,
Uind. N · £
N · m0 · m · A ·
N · I
l
L · I
(3)
where
L m0 · m · p ·
N 2 · r2
l
(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 calculated with greater accuracy by an approximation formula
r 3>4
L 2.1 · 10 6 · N 2 · r · a b
l
for 0 6
r
6 1
l
(5)
In the experiment, the inductance of various coils is calculated
from the natural frequency of an oscillating circuit.
v0 Fig. 5: Inductance per turn as a function of the length of coil,
at constant radius.
Double logarithmic plotting.
1
(6)
2LCtot.
Ctot. is the sum of the capacitance the known capacitor and
the input capacitance Ci of the oscilloscope.
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
24403-01
3
LEP
4.4.03
-01
Inductance of solenoids
The internal resistance Ri of the oscilloscope exercises a
damping effect on the oscillatory circuit and causes a negligible shift (approx. 1%) in the resonance frequency.
The inductance is therefore represented by
1
L
4p2 f20 Ctot.
(7)
L = A · NB
to the regression line from the measured values in Fig. 4 gives
the exponent
B = 1.95±0.04 ; Btheo = 2 (see Eq. 5)
v0
where Ctot. = C + Ci and f0 2p
Now that we know that L ~ N2, we can demonstrate the relationship between inductance and the length of the coil.
The table 2 shows the theoretical inductance values of the
used coils calculated according to eq. 5.
Table 2
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.
Applying the expression
The table 3 shows the measured values of the oscillation periods and the corresponding inductance values of the used
coils calculated according to eq. 7. These Lexp values are
plotted in Figs. 4, 5 and 6.
Applying the expression
L
A · lC
N2
to the regression line from the measured values in Fig. 5 gives
the exponent
C = – 0.82 ± 0.04. ; Ctheo = -0.75
Applying the expression
L
A · rD
N2
to the regression line from the measured values in Fig. 6 gives
the exponent
D = 1.86 ± 0.07. ; Dtheo = 1.75
The Equation (5) is thus verified within the limits of error.
Table 3
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.
4
24403-01
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
LEP
4.4.03
-11
Inductance of solenoids with Cobra3
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 oscillatory 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
Power supply, 12 V
RS 232 data cable
Cobra3 Universal writer software
Cobra3 Function generator module
Induction coil, 300 turns, dia. 40 mm
Induction coil, 300 turns, dia. 32 mm
Induction coil, 300 turns, dia. 25 mm
Induction coil, 200 turns, dia. 40 mm
Induction coil, 100 turns, dia. 40 mm
Induction coil, 150 turns, dia. 25 mm
Induction coil, 75 turns, dia. 25 mm
Coil, 1200 turns
PEK capacitor /case 1/ 470 nF/250 V
Connection box
Connecting cord, 250 mm, red
Connecting cord, 250 mm, blue
Connecting cord, 500 mm, red
Connecting cord, 500 mm, blue
PC, Windows® 95 or higher
07360.01
07360.04
07361.01
07361.04
1
1
2
2
Tasks
To connect coils of different dimensions (length, radius, number of turns) with a known capacitance C to form an oscillatory circuit. From the measurements of the natural frequencies, to calculate the inductances of the coils and determine
the relationships between
1. inductance and number of turns
12150.00
12151.99
14602.00
14504.61
12111.00
11006.01
11006.02
11006.03
11006.04
11006.05
11006.06
11006.07
06515.01
39105.20
06030.23
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
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 L1 and creates a free damped
oscillation in the L1C oscillatory circuit, the frequency fo 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 measured with the vernier caliper and the measuring tape, and the
numbers of turns are given.
Fig. 1: Experimental set-up.
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
24403-11
1
LEP
4.4.03
-11
Inductance of solenoids with Cobra3
Fig. 2: Set-up for inductance measurement.
Fig. 3: Measuring parameters.
Tab. 1: Table of coil data
Coil No.
N
2r
mm
l
mm
Cat. No.
1
2
3
4
5
6
7
300
300
300
200
100
150
75
40
32
26
40
40
26
26
160
160
160
105
53
160
160
11006.01
11006.02
11006.03
11006.04
11006.05
11006.06
11006.07
The following coils provide the relationships between inductance and radius, length and number of turns that we are investigating:
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 number
of turns, the relationship between inductance and number of
turns found in Task 1 must also be used to solve Task 2.
Notes
The distance between L1 and L should be as large as possible so that the effect of the excitation coil on the resonant frequency can be disregarded. There should be no iron components 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 to
RS232 Converter 14602.10). Start the measure program and
select Cobra3 Universal Writer Gauge. Begin the measurement using the parameters given in Fig. 3.
For the measurement of the oscillation period the “Survey
Function” of the Measure Software is used (see Fig. 4).
Fig. 4: Measurement of the oscillation period with the “Survey Function”.
2
24403-11
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
LEP
4.4.03
-11
Inductance of solenoids with Cobra3
Fig. 4 shows the rectangular signal and the damped oscillation behind each peak. Determine the frequency f0 of this damped oszillation,
f0 1
T
The inductance is therefore represented by
L
1
4p f20 Ctot.
(7)
2
where Ctot. = C + Ci and f0 where T is the oscillation period.
Theory and evaluation
If a current of strength I flows through a cylindrical coil (solenoid) of length l, cross sectional area A = p · r2, 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:
N
HI·
l
(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 absolute
permeability of the surrounding medium.
When this flux changes, it induces a voltage between the ends
of the coil,
Uind. N · £
N · m0 · m · A ·
N · I
l
L · I
where
L m0 · m · p ·
N ·r
l
2
(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 calculated with greater accuracy by an approximation formula
r 3>4
L 2.1 · 10 6 · N 2 · r · a b
l
for 0 6
r
6 1
l
The table 2 shows the theoretical inductance values of the
used coils calculated according to eq. 5.
Table 2
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
The table 3 shows the measured values of the oscillation periods and the corresponding inductance values of the used
coils calculated according to eq. 7. These Lexp values are
plotted in Figs. 5, 6 and 7.
Table 3
(3)
2
v0
2p
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.
Fig. 5: Inductances of the coils as a function of the number of
turns, at constant length and constant radius.
Double logarithmic plotting
(5)
In the experiment, the inductance of various coils is calculated
from the natural frequency of an oscillating circuit.
v0 1
2LCtot.
(6)
Ctot. is the sum of the capacitance the known capacitor and
the input capacitance Ci of the Cobra3 input.
The internal resistance Ri of the Cobra3 input exercises a
damping effect on the oscillatory circuit and causes a negligible shift (approx. 1%) in the resonance frequency.
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen
24403-11
3
LEP
4.4.03
-11
Inductance of solenoids with Cobra3
Fig. 6: Inductance per turn as a function of the length of coil,
at constant radius.
Double logarithmic plotting
Fig. 7: Inductance of the coils as a function of the radius, at
constant length and number of turns.
Double logarithmic plotting
Applying the expression
to the regression line from the measured values in Fig. 6 gives
the exponent
L = A · NB
C = -0.82±0.04 ; Ctheo = -0.75
to the regression line from the measured values in Fig. 5 gives
the exponent
B = 1.95±0.04 ; Btheo = 2 (see Eq. 5)
Now that we know that L ~ N2, we can demonstrate the relationship between inductance and the length of the coil.
Applying the expression
L
A · lC
N2
4
24403-11
Applying the expression
L
A · rD
N2
to the regression line from the measured values in Fig. 7 gives
the exponent
D = 1.86 ± 0.07. ; Dtheo = 1.75
The Equation (5) is thus verified within the limits of error.
PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen