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 12150-00 1 Power supply, 12 V 12151-99 1 RS232 data cable 14602-00 1 Cobra3 Universal Recorder software 14504-61 1 Cobra3 Function generator module 12111-00 1 Induction coil, 300 turns, dia. 40 mm 11006-01 1 Induction coil, 300 turns, dia. 32 mm 11006-02 1 Induction coil, 300 turns, dia. 25 mm 11006-03 1 Induction coil, 200 turns, dia. 40 mm 11006-04 1 Induction coil, 100 turns, dia. 40 mm 11006-05 1 Induction coil, 150 turns, dia. 25 mm 11006-06 1 Induction coil, 75 turns, dia. 25 mm 11006-07 1 Coil, 1200 turns 06515-01 1 PEK capacitor /case 1/ 470 nf/250 V 39105-20 1 Connection box 06030-23 1 Connecting cord, 250 mm, red 07360-01 1 Connecting cord, 250 mm, blue 07360-04 2 Connecting cord, 500 mm, red 07361-01 2 Connecting cord, 500 mm, blue 07361-04 PC, Windows 95 or higher Fig. 1: Experimental set-up www.phywe.com P2440311 PHYWE Systeme GmbH & Co. KG All rights reserved 1
TEP 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 f 0 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. 2 PHYWE Systeme GmbH & Co. KG All rights reserved P2440311
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). www.phywe.com P2440311 PHYWE Systeme GmbH & Co. KG All rights reserved 3
TEP 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 f 0 of this damped oszillation, f 0 = 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 = μ 0. μ. H. A (2) where μ 0 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, U ind. = N. ϕ 4 PHYWE Systeme GmbH & Co. KG All rights reserved P2440311
Inductance of solenoids with Cobra3 TEP = N. μ 0. μ. A. N l. I = L. I (3) where L = μ 0. μ. π. N2. r 2 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 L = 2.1 10 6 N 2 r r l 3/4 for 0 < r r < 1 (5) In the experiment, the inductance of various coils is calculated from the natural frequency of an oscillating circuit. ω 0 = 1 (6) LC tot. C tot. is the sum of the capacitance the known capacitor and the input capacitance C i of the Cobra3 input. The internal resistance R i 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 4π 2 f 0 2 C tot. (7) where C tot. = C + C 1 and f 0 = ω 0 2π The table 2 shows the theoretical inductance values of the used coils calculated according to eq. 5. www.phywe.com P2440311 PHYWE Systeme GmbH & Co. KG All rights reserved 5
TEP 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 L exp 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 6 PHYWE Systeme GmbH & Co. KG All rights reserved P2440311
Inductance of solenoids with Cobra3 TEP Fig. 6: Inductance per turn as a function of the length of coil, at constant radius. Double logarithmic plotting Applying the expression L = A N B to the regression line from the measured values in Fig. 5 gives the exponent B = 1.95 ± 0.04 ; B theo = 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 www.phywe.com P2440311 PHYWE Systeme GmbH & Co. KG All rights reserved 7
TEP Inductance of solenoids with Cobra3 to the regression line from the measured values in Fig. 6 gives the exponent C = 0.82 ± 0.04 ; C theo = 0.75 Applying the expression L N 2 = A rd to the regression line from the measured values in Fig. 7 gives the exponent D = 1.86 ± 0.07 ; D theo = 1.75 The Equation (5) is thus verified within the limits of error. 8 PHYWE Systeme GmbH & Co. KG All rights reserved P2440311