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magnetic fields? Try putting it near to a speakerplaying load music; does the LED flash with themusic?

7.There is a type of LED called a bi-colour LED(Maplins code: QY83E). These devices are ac-tually two LEDs contained within one pack-age—a red and a green LED. The two LEDs arewired so that one will light (say the red one)when connected one way round and the other(the green) will light when the connections (orthe power) is reversed. If you wire this type ofLED to the generator it is rather nice, as the redwill light when you shake the magnet one wayand the green will light when the magnet fallsback. If the magnets are small they can twist andturn inside the can and so the light changescolour in an interesting way. The changingcolour of the LED shows that this simple gener-ator is in fact an AC generator (see above).

Website informationFor details about the Shake-a-gen:www.creative-science.org.uk/gensimple1.html

For details about the storage device:www.creative-science.org.uk/gensimple2.htmlFor details about LEDs:www.creative-science.org.uk/LEDS.htmlFor details about winding coils:www.creative-science.org.uk/wire.htmlFor details about magnets:www.axminster.co.uk www.maplin.co.uk www.netcomuk.co.uk/~wwl/neodym.html

AcknowledgmentsI would like to thank NESTA for the continued sup-port with my NESTA Fellowship which is givingthe valuable space, time and resources to follow upideas and most importantly to dream up new ones.I would also like to thank Angie Birch from theRough Science Team for giving me a load of those‘damn strong’magnets.

Jonathan HareCreative Science CentreSussex University

Six ways to measure inductanceThe following experiments generate highly con-sistent results with those obtained from theoreti-cal predictions employing formulas derived fromFaraday’s law, the Biot–Savart law and Ampere’slaw. They not only draw on the above theoreticalconcepts in electromagnetism that enable pupils togain a comprehensive understanding of the topic,

Figure 1. The choke of a 65 W fluorescent lamp.

but also provide them with ample opportunitiesto use the CRO for measurements of voltage, timedifference, rate of change of current and decay rate.Undoubtedly, these laboratory exercises, if used asproject investigations, will enhance pupils’ labo-ratory and problem-solving skills.

Five easily available inductors with differentorders of magnitude are used as specimen samples

Figure 2. Geometry of the solenoid.

L E T ’ S I N V E S T I G AT E

Open-ended investigations that really work, with enough detail to start and finish a class.

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Table 1. Sample inductors.

Sample Type Dimensions Calculated value(except sample A)

A Choke of a 65 W, 220 V External dimension of buckle-shaped core 0.6–1.0 H at 50 Hzfluorescent lamp with about 2.5 cm × 4 cm × 12 cm (figure 1) (printed on the case)silicon–iron core

B Close-wound, single-layer, l = 0.28 m, r = 4.18 cm and N = 493 5.26 ± 0.21 mHair-cored solenoid (figure 2)

C Close-wound, single-layer, a = 18.7 mm, b = 23.6 mm, h = 17.4 mm 30.2 ± 0.6 µHrectangular cross-section, and N = 193 (figure 3)wooden-cored toroid

D Circular wire loop l = 1.0 m and d = 0.6 mm (figure 4) 1.67 ± 0.05 µH

E Coaxial cable a = 0.3 mm, b = 2.2 mm and l = 3.06 m 1.22 ± 0.04 µH(figure 5)

The approximate value of sample A is printed on the case. The theoretical values of samples B–E are calculatedusing the following formulas in SI units. The percentage error in each case is obtained by assuming a 1% error inlength measurement and a 1% error in the formula, except for the circular wire loop.

For the solenoid [1]

L (in H) = r2N 2

2.54(9r + 10l)× 10−4 H = 5.26 ± 0.21 mH.

For the toroid [2]

L (in H) = µ0N2h

2πln

b

a= 30.2 ± 0.6 µH.

For the circular loop, we are unable to find a formula to calculate the inductance; the inductance per unit length ofa straight wire [3] is used. The calculated result is a slight overestimation because the flux produced by any smallarc segment tends to cancel, to some extent, the flux produced by the arc subtended by the opposite angle:

L (in H) = µ0l

2πln

(8l

πd− 7

4

)= 1.67 ± 0.05 µH.

For the coaxial cable [4]

L (in H) = µ0l

2πln

(b

a

)= 1.22 ± 0.04 µH.

Figure 3. Geometry of the toroid.

Figure 4. Geometry of the wire loop.

(table 1). With the exception of the last method,which is applicable to inductance down to 1 µH, allthe methods can be used for L > 10 µH. Also, the

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Figure 5. (a) Configuration of a coaxial cable.(b) Coaxial cable with core and outer mesh connectedtogether at one end.

first three methods are applicable only when theresistance of the inductor is negligible comparedwith its reactance within the working frequencyrange, that is when ωL � r .

Method 1. Comparison of voltages acrossR and Z in a LR circuit

1. Measure the resistance r of the inductor (e.g.solenoid) using a multimeter.

2. Connect the solenoid, L, in series with aresistor, R, and a signal generator (SG)with sinusoidal output set at a convenientfrequency f (figure 6). To simplify ourcalculation, ω (2πf ) and R are chosen suchthat R � ωL � r or ωL � R � r . Inour case, the first inequality is employed withR = 24 k� (±5%) and f = 50 kHz becausethe maximum frequency is limited by the SGat 110 kHz.

3. Use the CRO to measure the peak-to-peakvoltage across Z (VZ) and R (VR).

Figure 6. Circuit for method 1. Comparison ofvoltages VZ and VL (or VR).

Theory and result

From VL = IXL,

VL = VZ

ZωL = VZωL√

R2 + (ωL)2≈ VZ

RωL. (1)

By measurement from the CRO,

VL = 0.42 ± 0.01 V, VZ = 6.00 ± 0.10 V

ω = 2πf = 2π × (50.0 ± 0.05) kHz

and the multimeter

R = 23.97 ± 0.02 k� and r = 12 � � R.

The error contribution comes mainly from voltagemeasurements and adds up to about 4%. Substitutemeasured values of V and R into (1), giving

L = VLR

VZω= 5.34 ± 0.20 mH.

Method 2. Measurement of difference inzero-crossings of VR and VZ in an LRcircuit

1. Measure the resistance r of the inductor (e.g.the toroid) using a multimeter. The resultshows r < 1.0 �.

2. Connect the toroid, L, in series with a resistor,R, and a signal generator with sinusoidaloutput set at frequency f (figure 7(a)). Bythe same token as the solenoid, ω = 2πf andR are chosen such that R � ωL � r . Inour case, we choose R = 200 � (±5%) andf = 100 kHz.

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Figure 7. (a) Circuit for method 2: measurement ofdifference in zero-crossings. (b) Phase relationbetween VL and VR .

3. Use a CRO to display the voltage traces acrossZ (VZ) and R (VR). Adjust the zero levelof both traces so that they coincide with oneanother and the zero line of the screen.

4. Increase the sensitivity of the sweep time andV -gain until the largest possible separation ofthe zero-crossings, �T , can be seen clearly.Check zero levels again if necessary. Record�T .

Theory and result

From the vector diagram shown in figure 7(b),

tan φ = VL

VR

= ωL

R(2)

For small φ,

tan φ ≈ φ = ω�T. (3)

Eliminate ω from (2) and (3),

L = R�T. (4)

Actual measurement shows

R = 200.5 ± 0.05 �

�T = (7.5 ± 0.1) × 0.02 µs = 0.15 ± 0.002 µs.

Therefore, from (4)

L = 30.1 ± 0.5 µH.

The measured φ should be slightly larger thanits true value due to the neglect of r , but thissystematic error will introduce an overestimationof L by no more than 0.1%.

Method 3. Measurement of VZ and slopeof the VR trace in a LR circuit

1. Connect the circuit as shown in figure 8 byinterchanging the positions of the resistor andthe inductor (the toroid) in figure 7(a).

2. Use a CRO to display the voltage traces acrossZ (VZ) and L (VL). Adjust the zero levelof both traces so that they coincide with oneanother and the zero line of the screen.

3. Increase the sensitivity of the V -gain of the VZ

trace and the sweep frequency until the curveappears as an inclined straight line across thewhole screen (figure 9).

4. Adjust the voltage output of the SG to obtainconvenient values for slope measurement.Measure the slope of the VZ trace and theamplitude of the VL trace.

Theory and result

By definition,

Ldi

dt= VL

L

R

dVR

dt= VL.

Figure 8. Circuit for method 3.

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Figure 9. Measurement of slope of VZ trace.

For R � ωL, Z =√

R2 + (ωL)2 ≈ R andVZ ≈ VR . Replacing VR by VZ , and takingmaximum value for both sides of the equation,

L = RVL|max

(dVZ/dt)|max. (5)

Substituting R = 200.5 �, VL|max = (2.9 ±0.1) × 50 mV and

dVR

dt

∣∣∣∣max = 300 ± 3 mV

300 ± 3 ns

= (1.00 ± 0.02) × 106 V s−1

into equation (3),

L = 29.1 ± 1.8 µH.

Method 4. Measurement of decayconstant in a discharging LR loopThis method is suitable for measurement ofinductance larger than 10 µH with little restrictionto the internal resistance r of the inductor, as longas L/r is within the measurable time range of theCRO. The choke of a 65 W fluorescent lamp willbe used to illustrate how this can be done.

1. Connect the circuit as shown in figure 10.A square wave output is desirable but notessential for charging the inductor. Aconvenient value for the external resistor isR0 = 200 (±5%) �.

2. Obtain a V –t trace across the resistor R0

(figure 11).

Figure 10. Circuit to show the decay curve in an LRloop.

Figure 11. The initial V –t trace (the lower portion isdistorted in an actual experiment).

Figure 12. The decaying portion of the V –t trace.

3. Adjust the x-shift, y-shift, V -gain andtimebase of the CRO until a significant portionof the decay curve is displayed on the screen(figure 12). If a sine wave from a lowvoltage power supply is used, only the lowerportion of the decay curve is suitable formeasurement. Record the zero level and thehalf-life of the trace.

4. Disconnect the circuit and measure the totalresistance of the discharge loop.

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Theory and result

The inductor is charged by the SG and dischargedthrough R0 periodically in the above circuit. Inthe reverse-bias half-cycle, the connection to thesource is cut off. The collapsing magnetic field inthe coil generates an induced emf and produces atransient current in the upper LR loop.

By Kirchhoff’s law,

Ldi

dt+ iR = 0

where R is the total resistance in the dischargeloop. Solving this equation,

i = i0 e− RL

t

orV = iR0 = V0 e− R

Lt .

By direct measurement

R = R0 + r = 213 ± 0.2 �.

When V = 12V0, T1/2 = 2.0 ± 0.1 ms,

12 = e− R

LT1/2 or ln 2 = R

LT1/2.

Therefore

L = R

ln 2× T1/2 = 615 ± 31 mH.

Method 5. Measurement of resonancefrequency ω = 1/

√LC with a driver

source1. Connect the circuit as shown in figure 13(a).2. Change the supply frequency and observe

variations in the VR and V2 traces.3. Adjust for maximum amplitude of the VR

trace and record the supply frequency.4. Measure the value of C using standard

methods, if necessary.

Theory and result

The current in a RLC circuit is

I = VZ

Z= VZ√

(R + r)2 + (ωL − 1/ωC)2

where r is the resistance of the inductor.

Figure 13. (a) Circuit for method 5: measurement ofresonance frequency. (b) Phase relation of VR , VL andVC in a RLC series circuit.

When the frequency of the source is equal tothe resonance frequency, i.e.

ω = 1√LC

(6)

a maximum current will flow in the circuit, givingrise to

VR = IR = VZ

ZR = VZR√

R2 + r2.

If C and R are chosen such that 1/ωC � R

at the resonance frequency, VC is approximatelyequal to V2. From the phasor diagram of voltagevectors (figure 13(b)), VR and V2 (VC) should bealmost 90◦ out of phase. This condition providesa double check to ensure that the variation in VR isnot caused by variations in the output voltage dueto changes in loading of the SG.

Substitute measured results, C = 1.00 ±0.02 µF and ω = 2πf = 2π ×(2.20±0.01) kHz,into equation (6):

L = 1

(2π × 2.20 × 103)2 × 1.00 × 10−6

= 5.23 ± 0.15 mH.

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Figure 14. Circuit to generate electrical resonance in an isolated LC loop.

Method 6. Measurement of resonancefrequency ω = 1/

√LC by natural

oscillation

When the target inductance is below 10 µH, thestray inductance in connecting wires and wire-wound resistors may introduce a sizable errorin inductance measurement. This error can beminimized by soldering the inductor, say a singlewire loop (sample D) or a few metres of coaxialcable (sample E), directly to the leads of a knowncapacitor. The capacitor is then charged witha periodic pulse and the frequency of naturaloscillation of the isolated LC loop is measuredduring the discharging half-cycle [5].

1. Measure the capacitance of C2, if necessary.

2. Connect the circuit as shown in figure 14.

3. Adjust the frequency and voltage of the squarewave generator and the resistance of therheostat until the trace of an underdampedelectrical LC oscillation is observed on theCRO screen.

4. Record the frequency of oscillation.

Theory and result

The circuit can be divided into three parts. Part A, adifferential circuit made up of C1 and R1, convertsthe square wave from the SG into sharp pulses.Part C is the LC loop. By adjusting the outputvoltage of the SG and/or R1, C2 is charged tosome convenient value below 0.6 V in the chargingcycle. It is isolated by the diode in Part B from therest of the circuit and oscillates with its naturalfrequency in the discharging half-cycle.

By measurement, C2 = 0.31 ± 0.01 µF. Forthe 1 m wire loop fwl = (2.40 ± 0.05) × 105 Hz,

Lwl = 1

ω2C

= 1

(2π × 2.40 × 105)2 × 0.31 × 10−6

= 1.42 ± 0.11 µH

For the 3 m coaxial cable fcc = (2.60 ± 0.05) ×105 Hz,

Lwl = 1

ω2C

= 1

(2π × 2.60 × 105)2 × 0.31 × 10−6

= 1.21 ± 0.10 µH.

References[1] Jordan E C 1985 Reference Data for Engineers:

Radio, Electronics, Computer andCommunications 7th edn (Indianapolis:Howard W Sams) ch 6-2

[2] Resnick R and Halliday D 1966 Physics (NewYork: Wiley) p 902

[3] Harnwell G P 1949 Principles of Electricity andElectromagnetism (New York: McGraw-Hill)p 330

[4] Johnk C T A 1988 Engineering ElectromagneticFields and Waves (New York: Wiley) p 303

[5] Mak S Y 1994 The RLC circuit and thedetermination of inductance Phys. Educ. 29pp 94–7

Se-yuen MakDepartment of Curriculum and Instruction,Faculty of Education, The Chinese University ofHong Kong, Shatin, NT, Hong KongE-mail: symak@cuhk.edu.hk