Radio Frequency by Steve Winder and Joe Carr - HTML preview

26.8 Useful formulae

Boolean Algebra (laws of)

Absorption: A+(A.B) = A A.(A+ B) = A

Annulment: A+ 1 = 1
A.0 = 0
Association: (A+ B)+ C = A+(B+ C)
(A.B).C = A.(B.C)
Commutation: A+ B = B+ A
A.B = B.A
Complements: A+ ¯A = 1
A.¯A = 0

De Morgan’s:

 

(

 

A

 

+

 

B

 

)

 

= ¯

 

¯

 

A.B

 

(A.B) = ¯A+ ¯B

 

Distributive: A.(B+ C) =(A.B)+(A.C) A+(B.C) =(A+ B).(A+ C)

 

¯

Double negation: A = A
Identity: A+ O = A
A.1 = A
Tautology: A.A = A
A+ A = A

Capacitance

The capacitance of a parallel plate capacitor can be found from 0.885KA_{C = d}

 

C is in picofarads,K is the dielectric constant(air = 1).A is the area of the plate in square cm andd the thickness of the dielectric. Calculation of overall capacitance with:

 

Parallel capacitors −C = C₁ + C₂+···

 

Series capacitors − ¹ =¹ +¹ +···_{C C1 C2}

Characteristic impedance

(

 

open wire

 

)Z

 

=

 

276 log 2

 

D

 

ohms_d

 

where

 

^{D= wire spacing in same units.}d = wire diameter

( coaxial)Z =¹³⁸ log^do ohms_{√(K) di} whereK = dielectric constant,d_o = outside diameter of inner conductor,d_i = inside diameter of outer conductor.

Dynamic resistance

In a parallel-tuned circuit at resonance the dynamic resistance is

 

Rd = ^L =QωL = ^Q ohms_{Cr ωC}

where L = inductance (henries), C = capacitance (farads), r = effective series resistance (ohms).Q =Q-value of coil, andω = 2π × frequency (hertz).

Frequency – wavelength – velocity

(See also Resonance)

 

The velocity of propagation of a wave is

 

v =fλ metres per second

 

wheref = frequency (hertz) andλ = wavelength (metres).

For electromagnetic waves in free space the velocity of propagationv is approximately 3× 10⁸ m/sec, and iff is expressed in kilohertz andλ in metres

f ^{300 000} kilohertz f = ³⁰⁰ megahertz =_{λ λ}

 

or

 

λ ^{300 000} metres λ = ³⁰⁰ metres =_{f f}

 

f in kilohertz f in megahertz

Impedance

The impedance of a circuit comprising inductance, capacitance and resistance in series is

 

Z

 

=

 

R

 

² + ωL−^{1 2} ωC

 

where R = resistance (ohms), ω = 2π × frequency (hertz). L = inductance (henries), andC = capacitance (farads).

Inductance

Single layer coils

 

L(

 

in microhenries

 

)

 

=

 

a²N²_{9a + 10l} approximately

 

If the desired inductance is known, the number of turns required may be determined by the formula

 

î ù

 

N

 

=

 

5L_{ð1+ 1+} 0.36n²a³

 

na² L^û

where N = number of turns,a = radius of coil in inches,n = number of turns per inch.L = inductance in microhenries (µH) andl = length of coil in inches.

Calculation of overall inductance with:

 

Series inductors −L = L₁ + L₂+···

 

Parallel inductors −¹ =¹ +¹ +···_{L L1 L2}

Meter conversions

Increasing range of ammeters or milliammeters
Current range of meter can be increased by connecting a shunt resistance across meter terminals. IfR_m is the resistance of the meter;R_s the value of the shunt resistance andn the number of times it is wished to multiply the scale reading, then

R

 

s

 

=

 

R_m (n− 1)

Increasing range of voltmeters
Voltage range of meter can be increased by connecting resistance in series with it. If this series resistance isR_s andR_m andn as before, thenR_s =R_m ×(n− 1).

Negative feedback

Voltage feedback

 

Gain with feedback

 

=

A 1+Ab whereA is the original gain of the amplifier section over which feedback is applied (including the output transformer if included) andb is the fraction of the output voltage fed back.

Distortion with feedback ^d approximately =_1+Ab

 

whered is the original distortion of the amplifier.

Ohm’s Law

^{V V}I = RV =IR R =_I

 

whereI = current (amperes),V = voltage (volts), andR = resistance (ohms).

Power

In a d.c. circuit the power developed is given by

 

2

 

W²R watts =VI =^V =I_R

 

whereV = voltage (volts),I = current (amperes), andR = resistance (ohms).

Power ratio

P

 

=

 

10 log

 

P₁ P₂ whereP = ratio in decibels,P₁ andP₂ are the two power levels.

Q

TheQ value of an inductance is given by

 

ωL_{Q = R}

Radio horizon distance

The radio horizon at VHF/UHF and up is approximately 15% further than the optical horizon. Several equations are used in calculating the distance. IfD is the distance to the radio horizon, andH is the antenna height, then:

D^√H =k

 

1. WhenD is in statute miles (5280 feet) andH in feet, thenK =

1 .42.
2. WhenD is in nautical miles (6000 feet) andH in feet, thenK =
1.23.
3. WhenD is in kilometres andH is in metres, thenK = 4.12.

Reactance

The reactance of an inductor and a capacitor respectively is given by

 

X_L =ωL ohms X_C =¹ ohms_ωC

 

where ω = 2π × frequency (hertz), L = inductance (henries), and C = capacitance (farads).

 

The total resistance of an inductance and a capacitance in series isX_L −X_C.

Resistance

Calculation of overall resistance with: Series resistors −R = R₁ + R₂+··· Parallel resistors − ¹ =¹ +¹ +···_{R R1 R2}

Resonance

The resonant frequency of a tuned circuit is given by

 

1 f =_2π√_LC hertz

 

whereL = inductance (henries), andC = capacitance (farads). IfL is in microhenries (µH) andC is picofarads, this becomes

 

10

 

6

 

f =_2π√LC kilohertz

 

The basic formula can be rearranged

 

1 1 L =_{4π2f 2C} henries C =_4π2fL farads Since 2πf is commonly represented byω, these expressions can be written

 

1 1 L =_ω2C henries C =_ω2L farads

Time constant

For a combination of inductance and resistance in series the time constant (i.e. the time required for the current to reach 63% of its final value) is given by

τ =^L seconds_R

 

whereL = inductance (henries), andR = resistance (ohms).

For a combination of capacitance and resistance in series the time constant (i.e. the time required for the voltage across the capacitance to reach 63% of its final value) is given by

τ =CR seconds whereC = capacitance (farads), andR = resistance (ohms).

Transformer ratios

The ratio of a transformer refers to the ratio of the number of turns in one winding to the number of turns in the other winding. To avoid confusion it is always desirable to state in which sense the ratio is being expressed: e.g. the ‘primary-to-secondary’ ration_p/n_s. The turns ratio is related to the impedance ratio thus

n_p Z_p n_s Z_s

where n_p = number of primary turns,n_s = number of secondary turns, Z_p = impedance of primary (ohms), andZ_s = impedance of secondary (ohms).

Wattage rating

If resistance and current values are known,

 

W =I²R whenI is in amperes ^or milliamps^{2 W =} 1 000 000 ^×R If wattage rating and value of resistance are known, the safe current for the resistor can be calculated from

 

milliampers

 

=

 

1

 

.

 

000

×
watts
ohms

Wavelength of tuned circuit

Formula for the wavelength in metres of a tuned oscillatory circuit is: 1885^√LC,whereL = inductance in microhenries andC = capacitance in microfarads.