RF CircuitsInductance

• Inductance depends on the physical configuration of the conductor. Ifa 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 bywinding the coil wire on a non-magnetic material (paper)

• At high frequencies a straight piece of wire can have significantinductance

• The approximate inductance of a single-layer air-core coil may becalculated as follows

L Hd n

d l

l o d

( )

.

µ =+

≥

2 2

18 40

4

L = Inductance

d = Diameter

l = Length

n = Number of Turns

( ) ( )[ ]

Example:

A 10 H inductor is required. The form on which the coil is wound is

one inch diameter and 1.25 inches long.

turns

µ

d l L

nx x

= = =

=+

=

1 1 25 10

10 18 1 40 1 25

126 1

, . ,

..

• A 26 - turn coil would be close enough. Since the coil is 1.25 incheslong 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 anythingsmaller) can be used

• When winding the coil the spacing between turns should be madeuniform

• Most inductance formulas lose accuracy when applied to small coilsbecause conductor thickness is no longer negligible. The figure belowshows the measured inductance of VHF coils and may be used as thebasis for circuit design

• Machine-wound coils with the diameters and turns per inch given intables 4 and 5 below are commonly available

• Forming a wire into a solenoid increases its inductance, and alsointroduces distributed capacitance between each turn which is at aslightly different potential

• At some frequency the effective capacitance will have a reactance(impedance) equal to that of the inductance and the inductor will showself-resonance

• Above self-resonance, a coil takes on the reactive properties of acapacitor

• At low frequencies the inductance of a straight, round, non-magnetic wire in free space is given by

L bb

a=

−

0 00022

0 75. ln .x

L = Inductance in H

a = Wire radius in mm

b = Length in mm

µ

• Skin effect reduces the inductance at VHF and above. As thefrequency approaches infinity the 0.75 constant approaches 1.0. Asa practical matter the effect is only a few percent

• A VHF or UHF tank circuit can be fabricated from a wire parallel to aground plane with one end grounded. A formula for the inductance isa follows:

• Another conductor configuration that is frequently used forinductors is the flat strip. This arrangement has lower skin effectloss because it has a higher surface area to volume ratio

L bb

w hw h

b=

+

+ ++

0 00508

20 5 0 2235. ln . .x

L = Inductance in H, b = Length in inches

w = Width in inches, h = Thickness in inches

µ

• For RF circuits the losses in solid iron core coils can be reduced to auseable level by grinding iron into a powder and mixing it with aninsulating binder

• The core is usually shaped in the form of a cylinder (slug) to fitinside the form on which the inductor is wound

• By pushing the slug in and out of the coil the inductance can bevaried over a considerable range

• When two coils are araranged with their axis on the same line asshown below current in coil 1 creates a magnetic field that cuts coil2 which creates an EMF in coil 2. This EMF is a result of themutual inductance between the two coils

• Maximum coupling is achieved when one coil is wound over theother. The coupling is least when two coils are far apart and at rightangles.

Resonance• The frequency at which a series circuit is resonant is that at which XL = XC

fL C

=1

2π

• A number of plots of current versus frequency for different valuesof R can be seen below

• The shape of the curve is determined by the ratio of reactance toresistance

• A “sharp” circuit will respond a great deal more to the resonantfrequency than other frequencies. A “broad” circuit will respondalmost equally to a band of frequencies centered around theresonant frequency. Both types of circuits are useful

• Most diagrams of resonant circuits show only inductance andcapacitance; no resistance is indicated. However, resistance ispresent in the wire of the coil and due to dielectric losses in thecapacitor at higher frequencies

• The value of reactance of either the inductor or capacitor at theresonant frequency divided by the series resistance in the circuit iscalled the Q (Quality factor) of the circuit

Q

QXR

X

R

=

= =

=

Quality factor

Reactance of coil or capacitor

Series resistance

• The unloaded Q of a circuit is determined by the inherent resistanceassociated with the components

• When a voltage of the resonant frequency is inserted in series in aresonant circuit the voltage that appears across either the inductor orcapacitor is higher than the applied voltage. The large currentthrough the reactance of the inductor and capacitor causes largevoltage drops. The voltage across either element is QE where E isthe applied voltage

• The -3 dB bandwidth (bandwidth at 0.707 relative response) isgiven by

BandwidthfQdB

o− =3

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 computersimulations

• The designs are easily scaled for different frequencies and impedancelevels

• The relationship between load resistance (RA), load reactance (XA),line impedance (Zo) (assuming negligible line resistance) andreflection coefficient is as follows

( )( )

ρ =− +

− +

R Z X

Z R XA o A

o A A

2 2

2 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 asthe 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

SWRSWRSWR

=+−

=−+

11

11

ρρ

ρ

• Power loss in a transmission line varies logrithmically with thelength. It is customary to express line loss in dB per unit length

• Addition loss occurs when the transmission line is not matchedproperly. 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 flowover the entire 360o of the input signal cycle. Class A amplifier islinear, however the power supply efficiency is poor (< 50 % typically25 %)

• Class B amplifier is chosen to permit power supply current to flowover most of the 360o of the input signal cycle. Class B amplifier hassome 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 flowonly in narrow pulses corresponding to the peaks of the input signal.Class C amplifiers are extremely non-linear . High Q tank circuits arerequired to suppress unwanted frequency components. Principle assetis high efficiency (typ. 85 %). Can also be used as a frequencymultiplier

• Even higher frequency can be achieved with specialized Class D andClass 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 100respectively, and divide all C and L values by the same number. Thegain and SWR data remains the same

• To scale to 1 - 10 KHz, 10 - 100 KHz, or 0.1 - 1.0 MHz range, dividefrequencies by 1000, 100, or 10 and multiply component values by thesame number

Impedance Scaling

• If the desired new impedance level differs from 50 Ω by a factor of0.1, 10 or 100, the 50 Ω designs are scaled by shifting the decimalpoints of the component values. For example, if the impedance levelis increased by ten (to 500 Ω) the decimal point of the capacitors isshifted to the left one place and the decimal point of the inductors isshifted to the right one place

• A simple algorithm can be used for scaling by a factor of 1.2, 1.5 or1.86

• The standing wave ratio (SWR) indicates the level of signal reflection.For RF applications, SWR values less than 1.2 are recommended tominimize undesired reflections

Coupled Resonators

• Coupled resonators are frequently encountered in RF circuits.Applications include simple filters, oscillator tuned circuits andantennas

• This circuit can be applied when it is desired to match a low-value loadresistance (such as found in a mobile whip antenna) to a more practicalvalue

• As the frequency of operation is increased, discrete components mustbecome physically smaller, eventually a point is reached where otherforms of networks must be used. Also, the Q of the devicesthemselves 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 conveyedfrom one point to another. Common examples include coaxial (coax)cable and TV parallel - wire line. At high frequencies and powerlevels special wave guides can be used

• In transmission lines the propagation delay from one end to the othermust be taken into account

• In a practical transmission line the energy travels from 65 - 97 % ofthe 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 awhole series of small inductors and capacitors connected as shownbelow

• Each inductor represents the inductance of a short section of one wireand each capacitor represents the capacitance between two such shortsections

• All the small inductors have the same value and all the smallcapacitors have the same value. If an impulse is applied on one end,the combination of inductors and capacitors has a characteristicimpedance ( ZO ). Its value is approximately equal to

where L and C are the inductance and capacitance per unit length

• In the transmission line above there are no resistors and thus no poweris lost in the line. However, as far as the source is concerned theimpedance ZO is exactly the same as if the line were replaced by a pureresistance. This is because the energy leaves the source and travels outalong the line. The characteristic impedance determines the amount ofcurrent that can flow when a given voltage is applied to the line

• All practical transmission lines exhibit some power loss due to theinherent resistance in the conductors that make up the line anddielectric leakage between conductors. Generally these losses can beignored

• The inductance and capacitance per unit length depend upon the sizeof the conductors and the spacing between them. A line with closelyspaced large conductors will have low impedance

L C/

Matched Lines

• Transmission lines are connected to, or terminated in a load at theoutput end of the line. If the load is a pure resistance of value equal tothe 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 travelsoutward along the line from the source until it reaches the load, whereit is completely absorbed

• If a very short burst of power is emitted from a source this isrepresented 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 orpart of it may be reflected. The reflected wave is represented by thesecond in the series. As the second line reaches the source the processis repeated. After a few reflections the intensity of the traveling wavebecomes very small

• The ratio of the voltage in the reflected wave to the voltage in theincident 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 atransmitter) the power consumed in the circuit is usually negligiblecompared to that in the load. The parallel impedance of theresonant circuit will be so high compared to the load that theimpedance of the combined circuit is equal to the load resistance.Under these conditions the Q of a parallel resonant circuit loaded bya resistive impedance is

QRX

=

• The effective Q of a circuit loaded by a parallel resistance increaseswhen the reactances are decreased. A circuit loaded with arelatively low resitance must have low reactance elements (largecapacitance and small inductance) to have a high Q

Impedance Transformation

• An important application of the parallel resonant circuit is animpedance matching device in the output circuit of an RF poweramplifier

• There is an optimum value of load resistance for each type of transistorand set of operating conditions, however, the resistance of the load isusually much lower than the value required for proper device operation

• To transform the actual load resistance to the desired value, the loadmay be tapped across part of the coil. This is equivalent to connectinga higher value of load resistance across the whole circuit

• At high frequency the magnetic flux lines do not cut every turn of thecoil. A desired reflected impedance usually must be obtained byexperimental adjustment

• When the load resistance has a very low value (less than 100 Ω) it maybe connected in series in the resonant circuit as shown in Figure Abelow. In which case it is transformed into an equivalent parallelimpedance

• If the Q is at least 10, the parallel impedance is

ZXRR = =

2

Z Resistive parallel impedance at

resonance

X = Reactance of coil or capacitor

R = Load resistance inserted in series

R

• If the Q is below 10 the reactance will have to be adjusted to obtaina resistive impedance of desired value

• While the circuits above provide an impedance step up, the circuitshave some disadvantages such as a common connection with no DCisolation and a common ground with potentially troublesomeground loop currents

• As a result a circuit with only mutual magnetic coupling is oftenpreferential

• Networks involving reactive elements are usually narrow band innature. As we shall see ferites allow us to construct impedancetransformers 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 iscalled the secondary

• Electrical energy can be transferred from one circuit to another withoutdirect 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

E ES PS

P

=

ηη

ηη

E = Secondary voltage

E = Primary voltage

= Number of turns in secondary

= Number of turns in primary

S

P

S

P

• Note that the above equation is ideal and does not take into accountlosses

• A transformer cannot create power. Hence, the power taken fromthe secondary cannot exceed that taken by the primary from thesource

• In an ideal transformer the following relationship is true

Z ZP SP

S

=

ηη

2

Z = Impedance looking into primary

terminals from source of power

Z = Impedance of load connected to

secondary

P

S

• A load of any given impedance connected to the secondary of atransformer will be transformed to a different value looking into theprimary (impedance matching)

• The transformed (reflected) impedance has the same phase angle as theload impedance

• For use in RF circuits a suitable core type must be chosen to providethe required Q. The wrong core material destroys the Q of an inductoror 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 UHFspectrum. The principal advantage of this type of core is the self-shielding characteristic

• Ferrite beads are small toroidal inductors that are typically less than0.25 inches in diameter

• They are commonly used as parasitic suppressers at the input andoutput terminals of amplifiers. Another common application is indecoupling networks that are used to prevent unwanted migration ofRF energy from one section of circuitry to another

• In some circuits it is necessary only to place one or more beads over ashort length of wire to obtain ample inductive reactance to create anRF choke

Common RF Circuits

Ladder Networks

• Any two circuits that are coupled can be drawn schematically asshown below. The circuit in the box of Figure A could consist of aninfinite variety of resistors, capacitors, inductors and even transmissionlines. However, it will be assumed that the network can be reduced toa combination of series and shunt elements consisting of onlyinductors and capacitors.

• The circuit in Figure B is often called a ladder network. If no resistiveelements are present or can be neglected, the network is said to belossless (i.e. It will consume no power). Note that this assumption isusually not valid for transmitting circuits

• The most important consideration in coupled networks is the amountof power delivered to the load

• Common to use standard source resistances (RP) of 50, 75, 300 and600 ohms. The value of RP might be considered as the impedancelevel associated with a complex combination of sources,transmission lines, coupled networks and even antennas

• The maximum available power is given by

PE

RMAXAC

P

=2

4

• If the coupling network is lossless the power deliverd to the load is

P I RO IN IN= 2

• The effective attenuation is defined as the ratio of the power deliverdto the load in terms of the maximum available power

ATTNP

PO

MAX

= −

10 log

• In the special case where XP and XS are either zero or can be combinedinto a coupling network and where RP is equal to RS the effectiveattenuation is equal to the insertion loss

• The insertion loss is the ratio of the power delivered to the load (withthe coupling network present) to the power delivered to the load withthe 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 availablepower does not occur with the coupling network out of the circuitbecause of unequal source and load resistances and non-zeroreactances. With the coupling network present the resistances arematched and the reactances are “tuned out”

• The action of the coupling network is to maximize the power deliveredto the load. They are commonly referred to as matching networks

• In many cases it is desirable to deliver the greatest amount of power toa load at specific frequencies. A device which accomplishes this iscalled a filter. It is often possible to combine the processes of filteringand matching into one network