Non-Ideal Behavior of Components
Todd H. Hubing
Dept. of Electrical and Computer Engineering
Clemson, University
Clemson, SC 29634
USA
email: hubing@clemson.edu
Telephone: 1-864-656-7219
Circuit Schematics
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Resistors
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Resistance
l
R=
ohms
σA
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Quiz Question
The D.C. resistance of a 5-cm trace on a printed circuit board is,
a.) about 100 ohms
b.) about 100 milliohms
c.) less than a milliohm
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DC Resistance of a Printed Circuit Board
Trace
Trace length = 5 cm
Trace width = 0.25 mm
Trace thickness = 0.034 mm
σcu = 5.7 x 107 S/m.
0.05m
R=
= 0.10 ohms
7
−3
−3
( 5.7 × 10 S / m )( 0.25x10 m )( 0.034x10 m )
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Skin Depth
High-frequency electric
fields and currents decay
exponentially with distance
from the surface of a good
conductor.
J ( x) = J S e
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− x πfμσ
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Skin Depth
1
δ=
π f μσ
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meters
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Resistance per unit length of a cable
At 60 Hz
δ60 Hz =
1
π ( 60 ) ( 4π × 10
Rinnerconductorat60Hz =
Routerconductorat60Hz
−7
)( 5.7 × 10 )
7
= 8.6 mm
1
1
=
= 22.3 mΩ / m
2
7
3
−
σ A ( 5.7 × 10 ) π (0.5 × 10 )
1
1
=
=
= 5.6 mΩ / m
7
−3
−3
σ A ( 5.7 × 10 ) 2π ( 5 × 10 )(0.1 × 10 )
total = 22.3 + 5.6 = 28 mΩ / m
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Resistance per unit length of a cable
At 100 MHz
δ 100 MHz =
1
π (10 8 )( 4π × 10 −7 )( 5.7 × 107 )
= 0.0066 mm
Rinnerconductorat100MHz =
1
1
=
= 832 mΩ / m
7
−3
−6
σ A ( 5.7 × 10 ) 2π (0.5 × 10 )( 6.6 × 10 )
Routerconductorat100MHz =
1
1
=
= 85 mΩ / m
7
−3
−6
σ A ( 5.7 × 10 ) 2π ( 4.9 × 10 )( 6.6 × 10 )
total = 832 + 85 = 917 mΩ / m
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Capacitors
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Capacitance
Q
C=
V
E=
Q0
4πε 0 r
Vab = ∫
rb
ra
2
ra < r < rb
volts / m.
Q0
Q0 ⎡ 1 1 ⎤
dr =
− ⎥ volts.
⎢
2
4πε 0 r
4πε 0 ⎣ ra rb ⎦
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farads.
Cab =
Q0
4πε 0
=
Vab ⎡ 1 1 ⎤
⎢r − r ⎥
⎣ a b⎦
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farads.
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Absolute Capacitance
Cabs = lim Cab = 4πε 0 ra
b→∞
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farads.
13
Self and Mutual Capacitance
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Self and Mutual Capacitance
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Inductors
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Inductance
Inductance is a property of current loops!
Ψ = ∫ B ⋅ ds
webers
S
L=
Ψ
I
henries
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Inductance
⎡ ⎛ 8R ⎞
⎤
Lcircle ≈ N 2 Rμ ⎢ln⎜ ⎟ − 2.0⎥ henrys
⎣ ⎝ a ⎠
⎦
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Lsquare loop ≈ N 2
2μ 0 w ⎡ ⎛ w ⎞
⎤
−
ln
0
.
774
⎜
⎟
⎥ henrys
π ⎢⎣ ⎝ a ⎠
⎦
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Quiz Question
The inductance of a 2-cm wide, 10-cm long ground strap is,
a.) about 10 nanohenries
b.) about 100 nanohenries
c.) undefined
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Loop Inductances
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Mutual Inductance
L21 =
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Ψ 21
I1
henries.
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Partial Inductance
L21 =
Ψ 21
L12
henries.
I1
∫ B i ds
1
L21 =
henries.
S2
I1
∫ ( ∇ × A )i ds ∫
=
1
L21 =
S2
L21 =
μ
4π
I1
∫∫
dl1i dl2
R
L11
A1i dl2
I1
henries
henries
A1 = ∫
μ I1
dl webers / m
4π R
μ
Lij =
4π
I
J
∑∑ l
i =0 i = 0
ij
Useful for computer modeling.
Not useful for estimating inductance.
where
lij =
∫
∫
segment i segment j
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dli i dl j
Rij
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Partial Inductance (Branch Inductance)
Lloop = Ltrace + Lvia + Lvia + Lplane
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Resistors
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Do Resistors Have Capacitance and
Inductance?
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Impedance of a 50-Ohm Resistor
0.8 pF
20 nH
48 ohms
Impedance in Ohms
1000
100
10
1
10
100
1000
Frequency in MHz
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Types of Resistors
„ Metal Film
High precision, low cost
„ Composite
Medium precision, good transient immunity
„ Wire Wound
High power, high inductance
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Capacitors
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Do Capacitors Have Resistance and
Inductance?
0.011 μF
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2 nH
15 mohms
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Impedance of a 0.01-μF Capacitor
0.011 μF
15 mohms
2 nH
Impedance in Ohms
100
10
1
0.1
0.01
1
10
100
1000
Frequency in MHz
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What are ESR and ESL?
?!!
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SMT Capacitor Connection Inductance
CB = 3.4 nF
L BULK = 5 nH
L D = 2nH
CBULK = 1μ F
C D = 10nF
Bare Board
100.
10.
Board with
decoupling
1.
0.1
0.1 MHz
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1 MHz
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10 MHz
100 MHz
1 GHz
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Types of Capacitors
„ Ceramic
Low cost, stable, good precision
„ Tantalum
Polarized, good energy density
„ Other Electrolytic
Polarized, good energy density
„ Mica
High-voltage applications
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Inductors
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Do Inductors Have Resistance and
Capacitance?
160 pF
15 mohms
5 μH
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Impedance of a 5-μH Inductor
160 pF
15 mohms
5 μH
Impedance in Ohms
1000
100
10
1
0.1
1
10
Frequency in MHz
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Types of Inductors
„ Ferrite Core
High inductance in small package
„ Air Core
Linear behavior under high-current conditions
„ Common-mode
Impedes common-mode currents while
passing differential-mode currents.
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Ferrites
Fair-Rite 2643000701
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Non-Ideal Behavior of Active Devices
Currents on the lead frame of an RDR memory
module at the third harmonic of the clock frequency.
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Summary
„ All components (when connected to a circuit)
have resistance, capacitance and inductance.
„ The behavior of a component at high
frequencies is usually much different than the
nominal (low-frequency) behavior.
„ The inductance of a low-inductance device is
generally determined by the connection and is
not a property of the device itself.
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