RF Circuits
Inductance
• Inductance depends on the physical configuration of the conductor. If
a conductor is formed into a coil, its inductance is increased
• A coil of many turns will have more inductance than one of few turns.
If a coil is placed around an iron core its inductance will increase
• RF coils can be wound on special iron cores or can use an air core by
winding the coil wire on a non-magnetic material (paper)
• At high frequencies a straight piece of wire can have significant
inductance
• The approximate inductance of a single-layer air-core coil may be
calculated as follows
d 2 n2
L (µ H ) =
18 d + 40 l
L = Inductance
d = Diameter
l ≥ o.4 d
l = Length
n = Number of Turns
Example:
A 10 µ H inductor is required. The form on which the coil is wound is
one inch diameter and 1.25 inches long.
d = 1,
l = 1.25,
L = 10
n=
10 [ (18 x 1) + ( 40 x 1.25) ]
1
= 26.1 turns
• A 26 - turn coil would be close enough. Since the coil is 1.25 inches
long the number of turns per inch will be 26.1 / 1.25 = 20.9
• Consulting a wire chart we find that no. 17 enameled wire (or anything
smaller) can be used
• When winding the coil the spacing between turns should be made
uniform
• Most inductance formulas lose accuracy when applied to small coils
because conductor thickness is no longer negligible. The figure below
shows the measured inductance of VHF coils and may be used as the
basis for circuit design
• Machine-wound coils with the diameters and turns per inch given in
tables 4 and 5 below are commonly available
• Forming a wire into a solenoid increases its inductance, and also
introduces distributed capacitance between each turn which is at a
slightly different potential
• At some frequency the effective capacitance will have a reactance
(impedance) equal to that of the inductance and the inductor will show
self-resonance
• Above self-resonance, a coil takes on the reactive properties of a
capacitor
•
At low frequencies the inductance of a straight, round, nonmagnetic wire in free space is given by
2b
L = 0.0002 x b  ln  − 0.75
 a 

L = Inductance in µH
a = Wire radius in mm
b = Length in mm
•
Skin effect reduces the inductance at VHF and above. As the
frequency approaches infinity the 0.75 constant approaches 1.0. As
a practical matter the effect is only a few percent
• A VHF or UHF tank circuit can be fabricated from a wire parallel to a
ground plane with one end grounded. A formula for the inductance is
a follows:
•
Another conductor configuration that is frequently used for
inductors is the flat strip. This arrangement has lower skin effect
loss because it has a higher surface area to volume ratio
  2b 
 w + h  
0
.
5
0
.
2235
+
+
L = 0.00508 x b  ln 

 b  
  w + h
L = Inductance in µ H, b = Length in inches
w = Width in inches, h = Thickness in inches
•
•
•
•
•
For RF circuits the losses in solid iron core coils can be reduced to a
useable level by grinding iron into a powder and mixing it with an
insulating binder
The core is usually shaped in the form of a cylinder (slug) to fit
inside the form on which the inductor is wound
By pushing the slug in and out of the coil the inductance can be
varied over a considerable range
When two coils are araranged with their axis on the same line as
shown below current in coil 1 creates a magnetic field that cuts coil
2 which creates an EMF in coil 2. This EMF is a result of the
mutual inductance between the two coils
Maximum coupling is achieved when one coil is wound over the
other. The coupling is least when two coils are far apart and at right
angles.
Resonance
•
The frequency at which a series circuit is resonant is that at which
XL = X C
f =
•
•
•
•
1
2π
LC
A number of plots of current versus frequency for different values
of R can be seen below
The shape of the curve is determined by the ratio of reactance to
resistance
A “sharp” circuit will respond a great deal more to the resonant
frequency than other frequencies. A “broad” circuit will respond
almost equally to a band of frequencies centered around the
resonant frequency. Both types of circuits are useful
Most diagrams of resonant circuits show only inductance and
capacitance; no resistance is indicated. However, resistance is
present in the wire of the coil and due to dielectric losses in the
capacitor at higher frequencies
•
The value of reactance of either the inductor or capacitor at the
resonant frequency divided by the series resistance in the circuit is
called the Q (Quality factor) of the circuit
Q = Quality factor
Q=
X
R
X = Reactance of coil or capacitor
R = Series resistance
•
•
•
The unloaded Q of a circuit is determined by the inherent resistance
associated with the components
When a voltage of the resonant frequency is inserted in series in a
resonant circuit the voltage that appears across either the inductor or
capacitor is higher than the applied voltage. The large current
through the reactance of the inductor and capacitor causes large
voltage drops. The voltage across either element is QE where E is
the applied voltage
The -3 dB bandwidth (bandwidth at 0.707 relative response) is
given by
Bandwidth −3dB =
fo
Q
Filters
• RF filters are commonly built using standard topologies (Butterworth,
Chebyshev, elliptical, etc.) and passive IC networks
• These filters can be designed using standard tables and computer
simulations
• The designs are easily scaled for different frequencies and impedance
levels
•
The relationship between load resistance (RA), load reactance (XA),
line impedance (Zo) (assuming negligible line resistance) and
reflection coefficient is as follows
ρ=
•
•
•
( RA − Z o ) 2 + X A 2
( Z o − RA ) 2 + X A 2
If RA = Zo and XA = 0 → ρ = 0
This represents the matched condition where all the energy is
transferred to the load
If RA = 0 → ρ = 1
This means all the power is reflected back to the source
If reflections exist, a voltage standing wave pattern will result. The
raito of the maximum voltage on the line to the minimum voltage
(provided the line is longer than a quarter wavelength) is defined as
the voltage standing wave ratio (VSWR)
•
•
Since ISWR = VSWR it is common to simply use the term SWR
The SWR is related to the reflection coefficient as follows
SWR =
•
•
1+ ρ
1− ρ
ρ=
SWR − 1
SWR + 1
Power loss in a transmission line varies logrithmically with the
length. It is customary to express line loss in dB per unit length
Addition loss occurs when the transmission line is not matched
properly. The loss as a function of VSRW is shown below
Classes of Amplifier Service
• Class A amplifier is chosen to permit power supply current to flow
over the entire 360o of the input signal cycle. Class A amplifier is
linear, however the power supply efficiency is poor (< 50 % typically
25 %)
• Class B amplifier is chosen to permit power supply current to flow
over most of the 360o of the input signal cycle. Class B amplifier has
some non-linearity, however the power supply efficiency is good (typ.
60 %) and no supply current flows when no input signal is applied
• Class C amplifier is chosen to permit power supply current to flow
only in narrow pulses corresponding to the peaks of the input signal.
Class C amplifiers are extremely non-linear . High Q tank circuits are
required to suppress unwanted frequency components. Principle asset
is high efficiency (typ. 85 %). Can also be used as a frequency
multiplier
• Even higher frequency can be achieved with specialized Class D and
Class E amplifiers
Frequency Scaling
• To scale he frequency and component values to the 10 - 100 or 100 1000 MHz range multiply all tabulated frequencies by 10 or 100
respectively, and divide all C and L values by the same number. The
gain and SWR data remains the same
• To scale to 1 - 10 KHz, 10 - 100 KHz, or 0.1 - 1.0 MHz range, divide
frequencies by 1000, 100, or 10 and multiply component values by the
same number
Impedance Scaling
• If the desired new impedance level differs from 50 Ω by a factor of
0.1, 10 or 100, the 50 Ω designs are scaled by shifting the decimal
points of the component values. For example, if the impedance level
is increased by ten (to 500 Ω) the decimal point of the capacitors is
shifted to the left one place and the decimal point of the inductors is
shifted to the right one place
• A simple algorithm can be used for scaling by a factor of 1.2, 1.5 or
1.86
• The standing wave ratio (SWR) indicates the level of signal reflection.
For RF applications, SWR values less than 1.2 are recommended to
minimize undesired reflections
Coupled Resonators
• Coupled resonators are frequently encountered in RF circuits.
Applications include simple filters, oscillator tuned circuits and
antennas
• This circuit can be applied when it is desired to match a low-value load
resistance (such as found in a mobile whip antenna) to a more practical
value
• As the frequency of operation is increased, discrete components must
become physically smaller, eventually a point is reached where other
forms of networks must be used. Also, the Q of the devices
themselves becomes critical for RF circuit operation
• Additional matching networks can be seen below
Transmission Lines
• A transmission line is the means by which RF energy is conveyed
from one point to another. Common examples include coaxial (coax)
cable and TV parallel - wire line. At high frequencies and power
levels special wave guides can be used
• In transmission lines the propagation delay from one end to the other
must be taken into account
• In a practical transmission line the energy travels from 65 - 97 % of
the speed of light. This line characteristic is called the velocity factor
(VF) of the line
• The transmission line may be thought of as being composed of a
whole series of small inductors and capacitors connected as shown
below
• Each inductor represents the inductance of a short section of one wire
and each capacitor represents the capacitance between two such short
sections
• All the small inductors have the same value and all the small
capacitors have the same value. If an impulse is applied on one end,
the combination of inductors and capacitors has a characteristic
impedance ( ZO ). Its value is approximately equal to L / C
where L and C are the inductance and capacitance per unit length
• In the transmission line above there are no resistors and thus no power
is lost in the line. However, as far as the source is concerned the
impedance ZO is exactly the same as if the line were replaced by a pure
resistance. This is because the energy leaves the source and travels out
along the line. The characteristic impedance determines the amount of
current that can flow when a given voltage is applied to the line
• All practical transmission lines exhibit some power loss due to the
inherent resistance in the conductors that make up the line and
dielectric leakage between conductors. Generally these losses can be
ignored
• The inductance and capacitance per unit length depend upon the size
of the conductors and the spacing between them. A line with closely
spaced large conductors will have low impedance
Matched Lines
• Transmission lines are connected to, or terminated in a load at the
output end of the line. If the load is a pure resistance of value equal to
the characteristic impedance of the line, the line is said to be matched
• Such a line acts just as if the line was infinitely long. Energy travels
outward along the line from the source until it reaches the load, where
it is completely absorbed
• If a very short burst of power is emitted from a source this is
represented by a vertical line at the left of the series of lines below
• As the pulse appears across the load all the energy may be absorbed or
part of it may be reflected. The reflected wave is represented by the
second in the series. As the second line reaches the source the process
is repeated. After a few reflections the intensity of the traveling wave
becomes very small
• The ratio of the voltage in the reflected wave to the voltage in the
incident wave is defined as the voltage reflection coefficient ( ρ )
Q of Loaded Circuits
•
When a circuit delivers power to a load (as in the case of a
transmitter) the power consumed in the circuit is usually negligible
compared to that in the load. The parallel impedance of the
resonant circuit will be so high compared to the load that the
impedance of the combined circuit is equal to the load resistance.
Under these conditions the Q of a parallel resonant circuit loaded by
a resistive impedance is
Q=
•
R
X
The effective Q of a circuit loaded by a parallel resistance increases
when the reactances are decreased. A circuit loaded with a
relatively low resitance must have low reactance elements (large
capacitance and small inductance) to have a high Q
Impedance Transformation
• An important application of the parallel resonant circuit is an
impedance matching device in the output circuit of an RF power
amplifier
• There is an optimum value of load resistance for each type of transistor
and set of operating conditions, however, the resistance of the load is
usually much lower than the value required for proper device operation
• To transform the actual load resistance to the desired value, the load
may be tapped across part of the coil. This is equivalent to connecting
a higher value of load resistance across the whole circuit
• At high frequency the magnetic flux lines do not cut every turn of the
coil. A desired reflected impedance usually must be obtained by
experimental adjustment
• When the load resistance has a very low value (less than 100 Ω) it may
be connected in series in the resonant circuit as shown in Figure A
below. In which case it is transformed into an equivalent parallel
impedance
•
If the Q is at least 10, the parallel impedance is
X2
ZR =
R
Z R = Resistive parallel impedance at
resonance
X = Reactance of coil or capacitor
R = Load resistance inserted in series
•
•
•
•
If the Q is below 10 the reactance will have to be adjusted to obtain
a resistive impedance of desired value
While the circuits above provide an impedance step up, the circuits
have some disadvantages such as a common connection with no DC
isolation and a common ground with potentially troublesome
ground loop currents
As a result a circuit with only mutual magnetic coupling is often
preferential
Networks involving reactive elements are usually narrow band in
nature. As we shall see ferites allow us to construct impedance
transformers that are both broad band and high frequency
Transformers
• Two coils having mutual inductance constitute a transformer. The coil
connected to the source of energy is called the primary and the other is
called the secondary
• Electrical energy can be transferred from one circuit to another without
direct connection. In the process voltage levels can be changed
•
•
A transformer can only be used with AC
The induced voltage in the secondary is proportional to the number
of tuns in each coil
η 
ES = E P  S 
ηP 
E S = Secondary voltage
E P = Primary voltage
η S = Number of turns in secondary
•
•
•
η P = Number of turns in primary
Note that the above equation is ideal and does not take into account
losses
A transformer cannot create power. Hence, the power taken from
the secondary cannot exceed that taken by the primary from the
source
In an ideal transformer the following relationship is true
η 
Z P = ZS  P 
 ηS 
2
Z P = Impedance looking into primary
terminals from source of power
ZS = Impedance of load connected to
secondary
• A load of any given impedance connected to the secondary of a
transformer will be transformed to a different value looking into the
primary (impedance matching)
• The transformed (reflected) impedance has the same phase angle as the
load impedance
• For use in RF circuits a suitable core type must be chosen to provide
the required Q. The wrong core material destroys the Q of an inductor
or transformer at RF
• Ferrite or powdered iron cores are commonly used for RF
• Toroid cores are useful from a few hundred hertz well into the UHF
spectrum. The principal advantage of this type of core is the selfshielding characteristic
• Ferrite beads are small toroidal inductors that are typically less than
0.25 inches in diameter
• They are commonly used as parasitic suppressers at the input and
output terminals of amplifiers. Another common application is in
decoupling networks that are used to prevent unwanted migration of
RF energy from one section of circuitry to another
• In some circuits it is necessary only to place one or more beads over a
short length of wire to obtain ample inductive reactance to create an
RF choke
Common RF Circuits
Ladder Networks
• Any two circuits that are coupled can be drawn schematically as
shown below. The circuit in the box of Figure A could consist of an
infinite variety of resistors, capacitors, inductors and even transmission
lines. However, it will be assumed that the network can be reduced to
a combination of series and shunt elements consisting of only
inductors and capacitors.
• The circuit in Figure B is often called a ladder network. If no resistive
elements are present or can be neglected, the network is said to be
lossless (i.e. It will consume no power). Note that this assumption is
usually not valid for transmitting circuits
•
•
•
The most important consideration in coupled networks is the amount
of power delivered to the load
Common to use standard source resistances (RP) of 50, 75, 300 and
600 ohms. The value of RP might be considered as the impedance
level associated with a complex combination of sources,
transmission lines, coupled networks and even antennas
The maximum available power is given by
PMAX
•
EAC 2
=
4 RP
If the coupling network is lossless the power deliverd to the load is
PO = I IN 2 RIN
•
The effective attenuation is defined as the ratio of the power deliverd
to the load in terms of the maximum available power
 PO 
ATTN = − 10 log 

 PMAX 
• In the special case where XP and XS are either zero or can be combined
into a coupling network and where RP is equal to RS the effective
attenuation is equal to the insertion loss
• The insertion loss is the ratio of the power delivered to the load (with
the coupling network present) to the power delivered to the load with
the coupling network absent
• Unlike the attenuation the insertion loss can take on a negative value
(i.e. power gain). This is due to the fact that the maximum available
power does not occur with the coupling network out of the circuit
because of unequal source and load resistances and non-zero
reactances. With the coupling network present the resistances are
matched and the reactances are “tuned out”
• The action of the coupling network is to maximize the power delivered
to the load. They are commonly referred to as matching networks
• In many cases it is desirable to deliver the greatest amount of power to
a load at specific frequencies. A device which accomplishes this is
called a filter. It is often possible to combine the processes of filtering
and matching into one network