Inductance of solenoids with Cobra3
TEP
Related topics
Law of inductance, Lenz’s law, self-inductance, solenoids, transformer, oscillatory circuit, resonance,
damped oscillation, logarithmic decrement, 𝑄 factor.
Principle and task
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
1 Cobra3 Basic Unit
1 Power supply, 12 V
1 RS232 data cable
1 Cobra3 Universal Recorder software
1 Cobra3 Function generator module
1 Induction coil, 300 turns, dia. 40 mm
1 Induction coil, 300 turns, dia. 32 mm
1 Induction coil, 300 turns, dia. 25 mm
1 Induction coil, 200 turns, dia. 40 mm
1 Induction coil, 100 turns, dia. 40 mm
1 Induction coil, 150 turns, dia. 25 mm
1 Induction coil, 75 turns, dia. 25 mm
1 Coil, 1200 turns
1 PEK capacitor /case 1/ 470 nF/250 V
1 Connection box
1 Connecting cord, 250 mm, red
1 Connecting cord, 250 mm, blue
2 Connecting cord, 500 mm, red
2 Connecting cord, 500 mm, blue
PC, Windows® 95 or higher
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
07360-01
07360-04
07361-01
07361-04
Fig. 1: Experimental set-up
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Inductance of solenoids with Cobra3
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
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 𝑓0 of which is measured with the Cobra3 interface.
Coils of different lengths 𝑙, diameters 2𝑟 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. 2: Set-up for inductance measurement.
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Inductance of solenoids with Cobra3
TEP
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/𝑁 2 = 𝑓(𝑙)
3.) 1, 2, 3 → 𝐿
= 𝑓 (𝑟)
Fig. 3: Measuring parameters.
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).
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Inductance of solenoids with Cobra3
Fig. 4: Measurement of the oscillation period with the “Survey Function”.
Fig. 4 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.
Theory and evaluation
If a current of strength I flows through a cylindrical coil (solenoid) of length l, cross sectional area 𝐴 =
𝜋 ・ 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:
𝐻=𝐼.
𝑁
𝑙
(1)
The magnetic flux through the coil is given by
𝜙 = 𝜇0 . 𝜇 . 𝐻 . 𝐴
(2)
where 𝜇0 is the magnetic field constant and 𝜇 the absolute permeability of the surrounding medium.
When this flux changes, it induces a voltage between the ends of the coil,
𝑈𝑖𝑛𝑑. = −𝑁 . ϕ̇
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Inductance of solenoids with Cobra3
= −𝑁 . 𝜇0 . 𝜇 . 𝐴 .
𝑁
𝑙
TEP
. 𝐼̇
= − 𝐿 . 𝐼̇
(3)
where
𝐿 = 𝜇0 . 𝜇 . 𝜋 .
𝑁2. 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 ∙ ( )
𝑙
for 0 <
r
r
<1
(5)
In the experiment, the inductance of various coils is calculated from the natural frequency of an oscillating circuit.
𝜔0 =
1
(6)
√𝐿𝐶𝑡𝑜𝑡.
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.
The inductance is therefore represented by
𝐿=
1
4𝜋2 𝑓02 𝐶𝑡𝑜𝑡.
(7)
where 𝐶𝑡𝑜𝑡. = 𝐶 + 𝐶1 and 𝑓0 =
𝜔0
2𝜋
The table 2 shows the theoretical inductance values of the used coils calculated according to eq. 5.
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Inductance of solenoids with Cobra3
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.
Fig. 5: Inductances of the coils as a function of the number of turns, at constant length and constant radius.
Double logarithmic plotting
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Inductance of solenoids with Cobra3
Fig. 6: Inductance per turn as a function of the length of coil, at constant radius.
Double logarithmic plotting
Applying the expression
𝐿 = 𝐴 ∙ 𝑁B
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 ~ 𝑁 2 , we can demonstrate the relationship between inductance and the length
of the coil.
Applying the expression
𝐿
= 𝐴 ∙ 𝑙𝑐
𝑁2
Fig. 7: Inductance of the coils as a function of the radius, at constant length and number of turns.
Double logarithmic plotting
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Inductance of solenoids with Cobra3
to the regression line from the measured values in Fig. 6 gives the exponent
𝐶 = −0.82 ± 0.04 ; 𝐶𝑡ℎ𝑒𝑜 = −0.75
Applying the expression
𝐿
𝑁2
= 𝐴 ∙ 𝑟D
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.
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