Bài giảng Radio communications circuits - Chapter 3: Passive components, resonators and impedance matching - Trịnh Xuân Dũng
Dung Trinh, PhD
HCMUT / 2019
Chapter 3
Passive Components, Resonators
and Impedance Matching
Trinh Xuan Dung, PhD
Department of Telecommunications
Faculty of Electrical and Electronics Engineering
Ho Chi Minh city University of Technology
1
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
Contents
1. High frequency characteristics of passive components
2. RLC Networks and Resonators
3. Impedance Matching
4. Exercises
2
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Passive Components
Resistance of a wire above a ground plane:
• Current tends to flow near the external surface of a conductor at high frequencies.
• The current density directed along the axis of a conducting wire is largest at the
surface of the conductor, and falls to small values inside the conductor.
• Most of the current flows within the cylindrical shell within one skin depth from
the surface. The skin depth is denoted by 훿[m]:
1
훿 =
휋푓휇휎
where 휇, 휎 are the permeability and conductivity of the
conductor. 휇0 = 4휋 × 10−7 (퐻/푚)
• The skin depth in copper at 100 kHz, 10 MHz, and 1 GHz
is approximately 0.2mm, 0.02mm, and0.002 mm.
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Passive Components
Resistance of a wire above a ground plane:
• At low frequency:
푙
푙
푅퐷퐶 =
=
휎푆 휋푟2휎
• At higher frequencies, where the skin depth is small compared to the wire radius.
The AC resistance can then be written as:
휋푟2
퐷퐶 휋푑훿
푟
푅퐴퐶 = 푅
= 푅
퐷퐶 2훿
• Obviously, the wire resistance at high frequency is much greater than its DC
resistance, especially at high frequencies.
4
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Passive Components
Example 1: Consider AWG 22 wire. Compute resistance per unit length of the wire
푅퐷퐶 and 푅퐴퐶 at 10MHz and 1GHz.
Solution:
푙
푙
1
푅퐷퐶 =
=
=
= 0.053 Ω/푚
−6
휎푆 휎휋푟2 5.8 × 10 × 휋 × (0.643 2) × 10
7
2
• At 10MHz:
1
1
훿 =
=
≃ 21(휇푚)
휋푓휎휇
휋 × 107 × 5.8 × 107 × 4휋 × 10−7
−3
푟
0.643 2 × 10
푅퐴퐶 = 푅
= 0.053 ×
≃ 0.4 Ω/푚
퐷퐶 2훿
2 × 23 × 10−6
• At 1GHz: 푅퐴퐶 ≃ 4 Ω/푚
AWG (American Wire Gauge) 22 gauge wire is a
typical wire used for connections and component leads
which has diameter 푑 = 25.3푚푖푙푠 or 0.643푚푚.
5
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Passive Components
Inductor of a wire above a ground plane:
• The inductance, per unit length, of a wire with length l, diameter d, and distance h
from a ground plane is (when 푑 ≪ 푙 and ℎ ≪ 푙):
휇0
2ℎ
퐿 =
cosh−1
2휋
푑
휇0 4ℎ
퐿 ≃ ln
• When ℎ 푑 > 1 then the following approximation is useful:
2휋 푑
Note that cosh−1 푥 = ln 푥 + 푥2 − 1
• For values of ℎ/푑 in the range 1 to 100, above equation predicts that L ranges from
2.8nH/cm to 12 nH/cm.
For short wires, the resistance is often small enough to be ignored, however the inductance of the
wire is often significant. In a circuit where impedances are relatively low, the series impedance of
even a short connecting lead may have a significant impact on circuit performance.
This leads to a fundamental rule of RF circuit design:
• At high frequencies it is important to keep the length of interconnecting wires and circuit-board
traces short in order to minimize lead inductance.
• When components are separated by significant distances, interconnections must be treated as
distributed circuit elements, and transmission line models are used to model the conductors that
interconnect components.
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Passive Components
Example 2: A AWG 22 copper wire placed directly on top surface of a PCB with
dielectric thickness of 1.57mm (standard thickness) and a ground plane on the bottom
of the board. Compute the inductance of the wire per unit length, inductive reactance at
100MHz and 1GHz.
Solution:
• We are given ℎ = 1.57푚푚 and 푑 = 0.643푚푚. Because ℎ/푑 > 1, then:
휇0 4ℎ 4휋 × 10−7 4 × 1.57
퐿 ≃
ln ≃
2휋 푑
ln
= 0.46(휇퐻/푚)
2휋
0.643
• At 100MHz:
푍퐿 = 푗휔퐿 = 푗2휋 × 108 × 0.46 × 10−6 = 푗300(Ω/푚)
푍퐿 = 푗휔퐿 = 푗2휋 × 109 × 0.46 × 10−6 = 푗3 (푘Ω/푚)
• At 1GHz:
7
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Resistors
Several types of resistors are used in RF circuits, including wire-wound, carbon
composition, thick film, and thin film units:
• Wire-wound resistors consist of a length of lossy wire that is coiled up to fit into a
small package. This type of resistor is seldom used at RF because they have
relatively large inductance.
• Carbon composition resistors consist of a lossy dielectric material sandwiched
between two conducting electrodes.
• Thick or thin-film resistors consist of a film of conducting material deposited on
an insulating substrate. Film resistors are available in cylindrical packages with
attached connecting leads and also as surface mount devices (SMDs). Thick or
thin film resistors in a surface mount package ( aka, “chip” resistors) are the most
common type for RF applications.
A carbon composition resistor
A wire wound resistor
A Thick film resistor
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Resistors
Over a fairly wide frequency range, resistors can be modeled using the equivalent
circuit shown in Fig.1.
• The inductor represents the inductance associated with the current path through
the resistor.
• The capacitor represents the capacitance between the two electrodes used to
connect the resistor to external circuitry.
The inductance is primarily determined by the length of the current path within the
element. The capacitance is determined by the size and separation of the contact
electrodes, as well as the dielectric permittivity of the material between the electrodes.
Generally, the inductance and capacitance associated with a miniature surface mount
resistor package are on the order of 1 nH and 1 pF, respectively.
Standard equivalent circuit of a resistor
9
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Resistors
Significantly higher inductance would be associated with a part in an axial package
with wire leads.
Even with very short leads, such a package would have a typical inductance value on
the order of 10 nH.
For many purposes, a simpler (standard) model can be used even for resistors with
external connecting leads if the inductance is taken to be the sum of the resistor
inductance and lead inductance and the shunt capacitance is taken to be the sum of
package capacitance and capacitance between the leads.
When the resistance is small (≪ 100Ω), and the frequency is not too high, the series
LR brand of the model has a much lower impedance than the capacitance shunt to it.
The capacitive reactance can be neglected.
Equivalent circuit of a resistor with connecting leads
Equivalent circuit of a small resistor
10
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Resistors
Example 3: Consider a 50Ω resistor with a lead inductance of 퐿 = 10푛퐻, shunt
capacitor 퐶 = 1푝퐹. Draw the magnitude of the impedance when the frequency varies
from 1Hz to 1GHz.
Solution:
• The series inductance acts to increase the impedance at high frequencies.
• The equivalent circuit of a small resistor can be applied to compute the impedance.
11
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Resistors
Example 4: Consider a 10푘Ω resistor with a lead inductance of 퐿 = 10푛퐻, shunt
capacitor 퐶 = 1푝퐹. Draw the magnitude of the impedance when the frequency varies
from 1Hz to 1GHz.
Solution:
• When the resistance is large (≫ 100Ω), and the frequency is not too high the
inductive reactance will be small compared to R and can be neglected.
• The shunt capacitance is dominant and tends to “short out” the resistance, resulting
in a dramatic reduction in the impedance at high frequencies.
Equivalent circuit of a resistor with
high resistance
12
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Resistors
Example 5: Consider a 200Ω resistor with a lead inductance of 퐿 = 10푛퐻, shunt
capacitor 퐶 = 1푝퐹. Draw the magnitude of the impedance when the frequency varies
from 1Hz to 1GHz.
Solution:
• In this case neither of the parasitic reactances is significant compared to 200Ω and
the impedance variation is relatively small up through 1GHz.
• Notice that the impedance of a resistor can be much larger or smaller than the DC
resistance, depending on the resistance value and frequency.
13
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Capacitors
Capacitors are constructed by separating two conducting electrodes by an insulating
medium such as air, or a low-loss dielectric material.
• Loss in the dielectric is modeled as a resistance (푅푝) in parallel with the intrinsic
capacitance.
• The inductance associated with the current path through the electrodes and any
connected leads appears in series.
• A resistance in series with the inductor (푅푠) models losses in the electrodes and
leads.
If 푅푝 ≫ 1 휔퐶, the model can be transformed into a series RLC model.
The sum of the ohmic resistance and the transformed dielectric loss resistance is
termed the equivalent series resistance (ESR). The ESR of a capacitor is dominated by
dielectric loss at sufficiently low frequencies:
1
푅푒푠푟 = 푅푠 +
푅푝휔2퐶2
Equivalent RLC model of a capacitor, valid if 푅푝 ≫ 1
Standard equivalent circuit of a capacitor
휔퐶
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Capacitors
The Q of a capacitor at any frequency is the ratio of the reactance and the ESR
푋
푄 =
푅푒푠푟
where the reactance 푋 = 휔퐿 − 1 휔퐶. The dissipation factor, d, is the inverse of Q:
1
푡푎푛훿 = 푑 = 푄 = 푅
푒푠푟
푋
The dissipation factor is also called the loss tangent.
All capacitors will have a series resonant frequency, 푓 = 1
. Above this
푠
2휋 퐿퐶
frequency, the inductive dominates. Hence capacitors are inductive at frequency above
푓 .
푠
Circuit designers make explicit use of the fact that
capacitor impedance is smallest at and near the series
resonant frequency. When a capacitor is used as a DC
bias circuit decoupling element or as an inter-stage DC-
blocking coupling element, it is sometimes possible to
choose the capacitor so that the intended frequency of
operation falls near the series resonant frequency of the
capacitor, where the impedance is smallest.
15
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Capacitors
Example 6: Consider a 0.01휇퐹 capacitor with total lead length of 1푐푚. Given typical
value of lead inductance is 10nH/cm. Compute resonant frequency of the capacitor.
Draw the capacitor’s impedance and reactance for the frequency range from 1Hz to
100MHz.
Solution:
• Series resonant frequency of the capacitor is 15.9MHz.
Reactance for a 0.01휇퐹 capacitor.
Log amplitude of capacitor’s impedance.
16
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Inductors
A reasonably accurate equivalent circuit for an inductor consisting of a coil of wire
wound as a solenoid, around a torus, or in a plane includes
• A series resistance to model the ohmic loss in the wire.
• A shunt capacitance to model distributed capacitance between the turns of the coil
The inductance of the wire results in voltage differences between the different parts of
the coil. This voltage difference sets up an electric field in the air and in any dielectric
material near the coil.
The effect of this stored electric energy can be modeled with a capacitance shunted
across the terminals of the coil. This effective capacitance is called the distributed
capacity of the coil.
The coil resistance can be estimated from the AC resistance of an isolated wire having
the length equal to the total length of wire in the coil.
푛휋푑퐷
푅퐴퐶 = 푟퐷퐶
4훿
• d is the wire diameter, D is the coil
diameter, n is the number of turns in the coil
and 훿 is the skin depth in the wire.
Equivalent circuit of an inductor
17
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Inductors
Example 7: Consider a 10휇퐻 inductor with series resistance of 15Ω and distributed
capacitor of 20pF. Compute the resonant frequency of the inductor. Draw the
inductor’s impedance and reactance for the frequency range from 1Hz to 100MHz.
Solution:
Impedance of the inductor
Reactance for the inductor.
18
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Inductors
In many cases, an inductor is used to provide a DC bias signal to a circuit, but it is
necessary to isolate the bias supply from the circuit at the operating frequency.
In this case, an inductor with parallel resonant frequency near the operating frequency
may be employed to provide a low impedance to DC and large impedance to RF
signals. In such an application the inductor element is referred to as an RF choke.
Example use of RFC
19
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. High frequencies Characteristics of Inductors
When considering a particular inductor for use in a circuit, the designer needs to be
aware of the parallel resonant frequency as well as the “Quality Factor,” or Q, of the
inductor. The Q of an inductor is defined to be the ratio of inductive reactance and
resistance associated with the component
푋
푄 =
푅푠
The higher the Q, the better the inductor approximates an “ideal” component. The Q is
an important parameter if the inductor is to be used in a resonant circuit, filter, or
matching network.
The quality factor for the inductor in example 7
20
Dept. of Telecoms Engineering
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