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Coil in the AC circuit with Cobra3
TEP
Related topics
Inductance, Kirchhoff’s laws, Maxwell’s equations, a.c. impedance, phase displacement.
Principle and task
The coil is connected in a circuit with a voltage source of variable frequency. The impedance and phase
displacements are determined as functions of frequency. Parallel and series impedances are measured.
Equipment
1 Cobra3 Basic Unit
2 Power supply, 12 V
1 RS232 data cable
1 Cobra3 Universal Recorder software
1 Cobra3 Function generator module
1 Coil, 300 turns
1 Coil, 600 turns
1 Resistor in plug-in box 47 Ohms
1 Resistor in plug-in box 100 Ohms
1 Resistor in plug-in box 220 Ohms
1 Connection box
2 Connecting cord, 250 mm, red
1 Connecting cord, 250 mm, blue
2 Connecting cord, 500 mm, red
2 Connecting cord, 500 mm, blue
12150-00
12151-99
14602-00
14504-61
12111-00
06513-01
06514-01
39104-62
39104-63
39104-64
06030-23
07360-01
07360-04
07361-01
07361-04
Fig. 1: Experimental set up for the measurement of the coil impedance
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TEP
Coil in the AC circuit with Cobra3
Tasks
1. Determination of the impedance of a coil as a function of frequency.
2. Determination of the inductance of the coil.
3. Determination of the phase displacement between the terminal voltage and total current, as a function
of the frequency in the circuit.
4. Determination of the total inductance of coils connected in parallel and in series.
Set-up and procedure
The experimental set up is as shown in Figs. 1, 2a and 2b.
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.
Fig. 2a: Circuit for measurement of the coil impedance.
Fig. 2b: Circuit for measurement of total current and total voltage.
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Coil in the AC circuit with Cobra3
TEP
Theory and evaluation
If a coil of inductance 𝐿 and a resistor of resistance 𝑅 are connected in a circuit (see Fig. 2), the sum of
the voltage drops on the individual elements is equal to the terminal voltage 𝑈:
𝑈 =𝐼𝑅+𝐿 ∙
𝑑𝐼
𝑑𝑡
,
(1)
where I is the current.
The resistors 𝑅 are selected so that the d. c. resistance of the coil, with a value of 0.2 Ω, can be disregarded. If the alternating voltage 𝑈 has the frequency 𝜔 = 2𝜋𝑓 and the waveform
𝑈 = 𝑈0 cos 𝜔𝑡 ,
Fig. 3: Measuring parameters.
then the solution of (2) is
𝐼 = 𝐼0 𝑐𝑜𝑠 (𝜔𝑡 – 𝜙)
with the phase displacement 𝜙 given by
𝑡𝑎𝑛 𝜙 =
𝜔𝐿
𝑅
(2)
and
𝐼0 =
𝑈0
√𝑅2 + (𝜔𝐿)2
(3)
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Coil in the AC circuit with Cobra3
TEP
It is customary to threat complex impedances as operators 𝑅̂𝑖 :
Coil 𝑅̂𝐿 = 𝑖𝜔𝐿 ,
Ohmic resistance 𝑅̂ = 𝑅.
With parallel connection,
−1
𝑅̂𝑡𝑜𝑡
= ∑ 𝑅̂𝜄−1
𝜄
The real impedance of a circuit is the absolute value of 𝑅̂𝑡𝑜𝑡 and the phase relationship, analogous to (2),
is the ratio of the imaginary part to the real part of 𝑅𝑡𝑜𝑡 .
Task 1
To determine the impedance of a coil as a function of the frequency, the coil is connected in series with
resistors of known value. The frequency is varied until there is the same voltage drop across the coil as
across the resistor (see Fig. 2a). The resistance and impedance values are then equal:
𝑅 = 𝜔𝐿 = 2𝜋𝑓 𝐿
(4)
The measured frequencies for 300 turns and 600 turns coils and for different resistors with the same
voltage drops across the coil as across the resistor are shown in Fig. 4.
Task 2
With the regression line to the measured values of Fig. 4 and the linear statement (see eq. (4))
𝑦 = 𝑎 + 𝑏𝑥
𝑓 = 𝑎 + (1/2𝜋𝐿) 𝑅
We receive for the inductance:
𝐿 = 1/2𝜋𝑏
and with the slopes for 300 turns and 600 turns coils (see Fig. 4):
𝐿(300) = (1.98 ± 0.09) mH
𝐿(600) = (9.1 ± 0.4) mH
Both values are very close to theoretical values of the used inductances 𝐿(300) = 2 mH, 𝐿(600) =
9 mH.
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Coil in the AC circuit with Cobra3
TEP
Fig. 4: Measured frequencies for 300 turns and 600 turns coils and for different resistors when the same
voltage drops across the coil as across the resistor.
Task 3
The phase displacement between the total voltage and the total current can be measured using a circuit
shown in Fig. 2b. Use the "Survey Function" of the Measure Software as it is shown in Fig. 5 for the
measurement of phase displacements.
Plot the phase displacement (see Fig. 6) and the tangent of phase displacement as a function of the Cobra3 function generator frequency (see Fig. 7).
From the regression line to the measured values of Fig. 7 and the linear statement (see eq. (2))
𝑦 = 𝑎 + 𝑏𝑥
𝑡𝑎𝑛(ϕ) = 𝑎 + (2𝜋𝐿/𝑅) 𝑓
We receive for the inductance:
𝐿 = 𝑏𝑅/2𝜋
and with the slopes for 300 turns and 600 turns coils (see Fig. 7):
𝐿(300) = (2.0 ± 0.1) mH
𝐿(600) = (8.6 ± 0.5) mH
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TEP
Coil in the AC circuit with Cobra3
Fig. 5: Measurement of current and voltage amplitudes and of phase displacements with the "Survey
Function".
Both values are very close to theoretical values of the used inductances 𝐿(300) = 2 mH, 𝐿(600) = 9 mH.
Fig. 6: Phase displacement between total current and total voltage for 600 turns coil and 47 ohm resistor
as a function of the frequency.
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Coil in the AC circuit with Cobra3
TEP
Fig. 7: The tangent of phase displacement as a function of frequency for a 600 turns coil.
Task 4
When coils are connected in parallel or in series, care should be taken to ensure that they are sufficiently
far apart, since their magnetic fields influence one another.
As in Task 3, use the "Survey Function" for the measurement of phase displacements and plot the tangent of phase displacement as a function of the frequency (see Fig. 8).
From the slopes of the straight lines for coils connected in parallel in series (see Fig. 8) we receive:
𝐿(300 || 600) = (2.1 ± 0.1) mH
𝐿(300 + 600) = (11.8 ± 0.6) mH
Both values are close to theoretical values of the used inductances:
𝐿(300 || 600) = 1.6 mH
𝐿(300 + 600) = 11 mH.
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TEP
Coil in the AC circuit with Cobra3
Fig. 8: Calculation of the total inductance of coils connected in parallel and in series.
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