Inductor_Design_Lab_notes_19207

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19207 - EE272 Electromagnetism

Inductor Design Laboratory

1. Instructions

• Please read ALL of this document before starting the experimental work.

• You are provided with an exam booklet to act as the ‘Logbook’ for these experiments.Please write your name and registration number on the front page before you begin.

• The exercise will be marked based on the Results Form which is provided as aseparate document. Please ensure that your name and registration number are writtenon the Results Form as well. All results must be supported by evidence containedin the ‘Logbook.’

• You must hand in the ‘Logbook’ and Results Form together for marking. These should

be submitted to the EEE Resource Centre before 4:30 pm on Monday 28th

March 2011.Ideally, if you manage to complete all working during the laboratory class, you mayhand in directly to staff at the end.

2. Introduction

The aim of this laboratory is to use design equations arising from electromagnetic theory todesign and construct a toroidal inductor. The inductance value will be determined indirectly

by measuring the resonant frequency of an LC circuit.

Students will be assessed individually on this exercise and it will count for 10% of the classmark for 19207.

3. Theory

3.1 Toroidal Inductor

An example of a toroidal inductor is shown in Fig. 1. The equation for inductance L of thistype of coil is given by

e

er

l AN

L2

0 µ µ = (1)

where:µ 0 = 4 π × 10 -7 H m -1 (permeability of free space) µ r = relative permeability of the core materialN = number of turns of wire wound onto the core Ae = effective cross-sectional area of the core l e = effective path length around the circular centre line of the core.



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N turns l e

Fig. 1 A toroidal inductor consists of a ‘doughnut’ shaped magnetic corewound with insulated copper wire.

3.2 LC resonance

When an inductor and capacitor are connected either in series or in parallel, their responsewill exhibit a resonance at the specific frequency where their reactive impedances havethe same magnitude. This is because (for ideal L and C ) their impedances will be purelyimaginary and, while the inductive impedance will have a positive sign, the capacitiveimpedance will be negative. The resonant frequency is given by the well-known equation:

LC f r

π 2

1= (2)

In the experiments you will measure resonance using one of the four test circuits shown inFig. 2. With the knowledge that the combined impedance of two components havingimpedances Z 1 and Z 2 respectively in series is given by:

21 Z Z Z S += for series combination (3)

21

21

Z Z Z Z

Z P +×= for parallel combination (4)

Decide for each of the circuits (a) – (d) in Fig. 2 whether you would expect to see amaximum or a minimum in the output voltage V O when the circuit is resonant.



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V I V O

R = 1 k L

C

(a)

V I V O

R = 1 k

L C

(b)

V I V OR = 1 k

L C

(c)

V I V OR = 1 k

L

C

(d)Fig. 2 Various connections of an LCR circuit. Input voltage is denoted by V I and output

voltage by V O. The voltage transfer function (or ‘gain’) of the circuit ata given frequency f would be given by V O/V I .

3.2 ‘Q’ factor of a resonant circuit

The quality factor ‘ Q’ of a tuned circuit is given by the equation:

LU

r

f f f Q −= (5)

where LU f f − represents the bandwidth of the circuit in Hz, which is the differencebetween the upper half-power frequency and the lower half-power frequency. Q is the ratioof resonant frequency to bandwidth, so it is a measure of the ‘sharpness’ of the resonance.A circuit with a high quality factor Q will have a narrow response – that is, it will be moreselective in its range of frequencies, as illustrated in Fig. 3.

0

0.2

0.4

0.6

0.8

1.0

frequency

f L f U

f r

1/√2

gain

(a)

0

0.2

0.4

0.6

0.8

1.0

frequency

f L f U

f r

1/√2

gain

(b)

Fig. 3 (a) A high Q resonance, and (b) a resonance with lower Q.



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6. Having gained a feel for what to expect at step 5, carry out measurements at arange of frequency points that will allow you to characterise the resonant responseby plotting the ratio V O/V I against frequency. Use the log-axis graph paper provided

in the Results Form – the response will look better with frequency on a logarithmicscale.

7. Use the plot from step 6 to determine the resonant frequency and use this value toestimate the actual value of the inductor you have constructed.

8. The tolerance on the value of the capacitor provided is ± 1%. Determine the bestaccuracy (as ± X µH) that could be claimed for the inductance value youdetermined at step 7 (neglecting all other sources of inaccuracy).

9. Calculate the Q factor for the resonant circuit using equation (5).

10. Now connect up the LCR circuit shown in Fig. 2(a), repeat the frequency responsemeasurement and plot your results, following a similar procedure to steps 5 and 6.

11. Estimate the resonant frequency in this second configuration and use it to derive asecond value for your inductance L.

12. If the resonance had to be set very accurately to 1.000 MHz, using a fixedcapacitance and varying the number of turns on the coil is not practical. Estimatethe change in resonant frequency that would be caused by taking off a single turnfrom the inductor you have constructed.

At this point, make sure that you have completed all the working required to complete theResults Form. If so, please proceed as follows:

• Add a second, smaller winding to the core to make it into a transformer. The secondwinding should have either 1/3 or 1/4 the number of turns of the existing winding.Verify experimentally that the voltage step-up / step-down capabilities of thetransformer in the MHz frequency range are in accordance with the turns ratio.

• Now that you have experienced the pleasures of hand-winding a toroid, imagine whatit would be like to attempt 500 turns! Can you think how these coils could be woundby a machine? See www.youtube.com/watch?v=98q4Ic6UL7c for an example.

FINALLY: PLEASE REMOVE THE WIRE FROM YOUR CORE(S) BEFORE LEAVING SOTHEY ARE READY FOR THE NEXT GROUP – THANKS.



v3, 10 March 2011

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