Name
Partner
ECE331 Electromagnetics Laboratory
Parasitic Inductance
Introduction
Common sense dictates that active components have finite bandwidth, meaning that above some
frequency the output will begin to attenuate. It is important to realize, however, that passive components also have finite bandwidth. While a transistor circuit’s bandwidth may be determined by
undesired capacitance, a passive circuit’s bandwidth may be determined by undesired inductance.
The purpose of this lab is to investigate this effect by measuring the parasitic inductance of typical
ceramic capacitors.
Parasitic inductances are typically too small to measure with an RLC meter. To get around
this, the inductance will be deduced from the resonant frequency of a series RLC circuit. The test
setup is shown in Figure 1.
B
Coaxial
Cable
50Ω
Coaxial
Cable
Coaxial
Cable
C
Signal
Source
Oscilloscope
A
50Ω
50Ω
R
Figure 1: Test circuit
It is apparent upon inspection of the circuit in Figure 1 that the parasitic inductance “seen”
by the capacitor is not only the parasitic lead inductance, but also the series inductance of the
coaxial cable connecting the signal source to the filter input. To separate these two values, several
capacitor lead-lengths may be measured to produce a linear function L(d) where L is the series
combination of the two inductances and d is the length of the capacitor leads. Since the internal
inductance of the capacitor is nearly zero, the y-intercept of this function will equal the inductance
of the coaxial cable.
Theory
The inductance per unit length of coaxial cable (neglecting the skin effect) is
µ◦ 1 1 b
′
Lcoax =
+ ln
π 8 2 a
(1)
where b is the radius of the outer conductor and a is the radius of the inner conductor.
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The inductance per unit length of the capacitor leads is approximately
µ◦ 1
x
L′cap ≈
+ ln
π 4
r
(2)
where x is the separation between the leads and the ground plane and r is the radius of the lead
wire. Note that this equation was derived for a pair of cylindrical wires, and applies to ground-plane
circuits only in as much as the return current flows directly underneath the leads.
Equipment
Test filters, 2 meter coaxial cable, pair of 1 meter coaxial cables, signal source, oscilloscope, multimeter, BNC T-connector, ruler.
Procedure
1. Obtain the measurements a, b, r and x by asking the instructor and record them in Table 1.
2. Use equations 1 and 2 to calculate the inductance per unit length for the coaxial cable and the
capacitor leads. Put L′coax in units of nH per meter and L′cap in units of nH per centimeter.
Record these two values in Table 1.
a
b
L′coax
r
x
L′cap
Table 1: Predictions
vIN (t)
vOUT (t)
L = Lcoax + Lcap
C
Req
Figure 2: Filter circuit
3. Find the transfer function of the filter circuit shown in Figure 2, neglecting the parasitic
inductance. Write the equation below.
HIdeal (s) =
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4. Derive the frequency response from equation 3 using the substitution s = jω. Write the
expression below.
HIdeal (jω) =
(4)
5. From either equation 3 or 4, calculate the corner frequency, assuming Req = 10Ω and C = 1nF.
Record the value below.
fc =
(5)
6. Sketch the magnitude of the frequency response in the space below. What type of filter does
this circuit realize?
7. Find the transfer function of the circuit shown in Figure 2, this time including the parasitic
inductance. Write the equation below.
HReal (s) =
(6)
8. Derive the frequency response from equation 6. Write the expression below.
HReal (jω) =
(7)
9. Calculate the center frequency of the filter by setting the equivalent impedance of the capacitor
and parasitic inductor to zero. Assume that Req = 10Ω, C = 1nF, and L = 250nH.
f0 =
(8)
10. Calculate the corner frequencies of the filter by setting |HReal (jω)| =
√
by verifying that ω0 = ωc1 ωc2 . Record the values below.
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√1 .
2
Check your result
fc1 =
(9)
fc2 =
(10)
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11. Sketch the magnitude of the frequency response in the space below. What type of filter does
this circuit realize?
12. Compare the behavior of the filter with and without the parasitic inductance. Would an
engineer who neglected the parasitic inductance in his or her analysis be surprised by the
behavior of the physical circuit?
13. Pick one of the available filters and measure R and C using a multimeter or RLC meter.
Calculate the equivalent resistance Req of the filter resistor (R) in parallel with the oscilloscope
input resistance (50Ω). Record the values in Table 2.
14. Measure the distance d between the BNC jacks on the filter and record the value in Table 2.
15. Calculate what the corner frequency of the filter would be without the parasitic inductance.
Put the value in Table 2.
16. Test the filter by building the circuit shown in Figure 1. Choose coaxial cables such that the
length of each signal path is approximately equal. This is important to cancel out the delays
introduced by the spatial variation of the voltage along the cables. Be certain the oscilloscope
inputs are set to 50Ω.
17. Carefully find the center frequency of the filter by comparing the input and output waveforms
on the scope. Look at both magnitude and phase. Note that the input voltage does not
remain constant due to the 50Ω output impedance of the function generator. Calculate the
inductance from the center frequency. Record both values in Table 2.
18. Complete steps 13 through 17 two more times, each with a different filter.
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Filter
C
R
Req
d
fc (Ideal)
f0
L
Table 2: Measurements
19. Plot f0 and L versus d on graph paper.
20. Perform a linear regression on the L(d) data. If you have access to a TI-89 Graphing Calculator, you may use the following instructions, replacing d1,d2,d3,L1,L2,L3 with your measurements from Table 2.
{d1, d2, d3} -> d
{L1, L2, L3} -> l
LinReg d,l
regeq(dp)
regeq(d)
The instruction regeq(dp) will return the equation itself, while the instruction regeq(d) will
return the inductances calculated using the equation, to demonstrate how well the regression
worked.
MathCad can also perform the linear regression for you.
Write the resulting equation below.
L(d) ≈
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Questions
1. Give a brief analysis of the data in the space below. Do the measurements match your
predictions?
2. Suppose you are designing a passive filter. Develop a test to determine if you need to include
the parasitic inductance in your circuit model.
3. Printed circuit board designers strive to place power supply decoupling capacitors as close to
the IC packages as possible. Why would they do this?
Attach your sketches, plots, and any hand or MathCad calculations to this handout and submit it
to the instructor.
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