ET103-121807-01
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TOPIC
DESCRIPTION
Basic Formulas (d.c.)
Ohm's Law for d.c. circuits (voltage, current, resistance, power)
Basic Formulas (a.c.)
Ohm's Law for a.c. circuits (voltage, current, impedance, power, power factor)
Basic Series Circuits
Calculation of resistance, voltage, current and power for series circuits
Basic Parallel Circuits
Calculation of resistance, voltage, current and power for parallel circuits
Networks
Kirchhoff's Voltage and Current Laws, Superposition, Thevenin, Norton and Millman Theorems
Alternating Current/Voltage
Calculation of rms, peak, peak-to peak, average voltage/current, frequency, period, wavelength
Inductance
Capacitance
Time Constants
Resonance
Inductance, energy stored in an inductor, inductive reactance, phase shift, inductive coupling
Capacitance, charge (Coulomb's Law), energy stored in a capacitor, capacitive reactance, phase shift
Calculation of RC and L/R time constants
Series/parallel resonance, resonant frequency, inductive/capacitive reactance, Q-factor, bandwidth
Coil Winding (air core)
Calculation of inductance, capacitance, resonant frequency, no. of turns for air core single/multi-layer coils
Coil Winding (toroids)
Calculation of inductance, capacitance, resonant frequency, no. of turns for toroid core single layer coils
Filters
Complex Math for A.C.
Basic Antennas
Component Data
Magnetic Circuits
Decibels
Transmission Lines
Basic Units & Conversions
Legal Notice
Low pass, high pass, band pass (constant-k, m-derived), resonant filter
Rectangular coordinates, polar coordinates, rectangular-to-polar conversion, polar-to-rectangular conversion
Half-wave dipole, quarter-wave vertical, folded dipole, 3-element yagi, range calculations
Resistor/capacitor color codes, wire chart, toroid data, resistance of cylindrical conductors, T.C. of resistance
Magnetic flux, magnetic field intensity, permeability, series magnetic circuit, hysteresis
Calculation of power, voltage, and current gain/loss
Impedance, inductance, capacitance, attenuation for coax and ladder transmission lines
Units, symbols, and definitions for electric, magnetic, and electromagnetic variables
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3. Version 1.0.3 01-01-2008
© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserve
Coulomb (C) - The basic unit of electric charge is the coulomb (C) named after Charles A. Coulomb. When a current of one ampere is maintained for one second, a
charge of one coulomb flows past a given point. It is equivalent to a charge of 6.25x10 18 electrons.
Ohm's Law - In 1827, Dr. George S. Ohm discovered that the current through a conductor is directly proportional to the difference of potential (voltage) across the
circuit. According to ohm's Law, a potential difference of one volt across a one ohm resistance will cause a current of one amp to flow through the resistance. Stated
as a formula, the ratio of volts to amps is a constant called resistance (R) and is measured in ohms (Ω).
Voltage (E or V) - The voltage between two points in a circuit is called the potential difference or electromotive force (emf) and is measured in volts (V) (named after
Count Alessandro Volta).
Current (I) - The current through a circuit is the rate of flow of electric charge and is measured in amperes (A) (named after Andre-Marie Ampere).
Resistance (R) - Resistance impedes the flow of current and is measured in ohms (Ω).
Power (P) - Power is the rate at which work is done (work per unit time) or energy produced (or consumed) in watts (W). The power consumed in a
circuit device is the work (or charge) multiplied by the charge/time or P=V*I watts. (For d.c. circuits, volt-amps and watts are equivalent in magnitude).
Note: In d.c. circuit diagrams and calculations, conventional (positive to negative) current flow is assumed.
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Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
NOTES
Ohm's Law - Calculate Resistance
Voltage, E
Current, I
Resistance, R
1.00
1.00
1.00
V
A
ohms
R=
E
I
Power, P
Current, I
Resistance, R
1.00
1.00
1.00
W
A
ohms
R=
P
I2
Voltage, E
Power, P
Resistance, R
1.00
1.00
1.00
V
W
ohms
R=
E2
P
Ohm's Law - Calculate Voltage
Current, I
Resistance, R
Voltage, E
1.00
1.00
1.00
A
ohms
V
Power, P
1.00
W
Resistance, R
Voltage, E
1.00
1.00
ohms
V
Current, I
Power, P
Voltage, E
1.00
1.00
1.00
A
W
V
E = IR
E = PR
E =
Practical Units and Conversions:
Coulomb = 6.25 x 1018 electrons.
Ampere = coulomb/second
Volt = joule/coulomb
Watt = joule/second
Ohm = volt/ampere
Siemens* = ampere/volt
*Originally the 'mho' for conductance.
P
I
Ohm's Law - Calculate Current
Voltage, E
Resistance, R
Current, I
1.00
1.00
1.00
V
ohms
A
I
Power, P
Resistance, R
Current, I
1.00
1.00
1.00
W
ohms
A
I=
Voltage, E
Power, P
Current, I
1.00
1.00
1.00
V
W
A
I
=
=
E
R
P
R
P
E
Ohm's Law - Calculate Power
Voltage, E
Resistance, R
Power, P
1.00
1.00
1.00
V
ohms
W
Voltage, E
Current
Power, P
1.00
1.00
1.00
V
A
W
P = EI
Current, I
Resistance, R
Power, P
1.00
1.00
1.00
A
ohms
W
P = I 2R
P =
© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
E2
R
Space For User Notes:
DEFINITIONS:
Voltage (E or V) - Generally, the voltage in a.c. circuits is the 'root mean squared' (RMS) or 'effective' voltage, measured involts (V).
Current (I) - Similarly, the current in a.c. circuits is the RMS value or effective value (equivalent d.c. value), measured inamperes (A).
Impedance (Z) - Impedance is the total opposition to the flow of an alternating current and it may consist of any combination of resistance, inductive reactance, and
capacitive reactance. Like resistance in d.c. circuits, it is measured in ohms (Ω).
Power (P) - Real Power (as opposed to apparent or reactive) is the power in watts (W) dissipated in heat through resistance.
Power Factor (PF) - PF is the ratio of the true power (watts) to the apparent power (volts x amps). It is expressed as the cosine of the phase angle (cos θ) or in a.c.
power applications, the cos θ is multiplied by 100 and expressed as a percentage.
Phase Angle (θ) - This is the angular difference in time between corresponding values in the cycles of two wave forms of the same frequency (i.e. voltage and current in
an a.c. circuit containing inductance, resistance and capacitance).
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
Ohm's Law - Calculate Impedance
Voltage, E
1.00
V
Current, I
1.00
A
Impedance, Z
1.00
ohms
PF, cos θ
Power, P
Current, I
Impedance, Z
1.00
1.00
1.00
1.00
W
A
ohms
PF, cos θ
Voltage, E
Power, P
Impedance, Z
1.00
1.00
1.00
1.00
V
W
ohms
NOTES
=
E
I
(no units)
Z =
P
I cos θ
2
(no units)
Ohm's Law - Calculate Voltage
Current, I
1.00
A
Impedance, Z
1.00
ohms
Voltage, E
1.00
V
PF, cos θ
Power, P
Impedance, Z
Voltage, E
1.00
1.00
1.00
1.00
W
ohms
V
PF, cos θ
Current, I
Power, P
Voltage, E
1.00
1.00
1.00
1.00
A
W
V
Z=
E2 cos θ
P
E = IZ
(no units)
PZ
E=
cos θ
Resistance, R = Z cos θ
cos θ = R/Z
Phase Angle, θ = cos-1(R/Z)
Reactance, X = Z sin θ
sin θ = X/Z
Phase Angle, θ = sin-1 (X/Z)
(no units)
Ohm's Law - Calculate Current
Voltage, E
1.00
V
Impedance, Z
1.00
ohms
Current, I
1.00
A
PF, cos θ
Power, P
Impedance, Z
Current, I
1.00
1.00
1.00
1.00
W
ohms
A
PF, cos θ
Voltage, E
Power, P
Current, I
1.00
1.00
1.00
1.00
V
W
A
E=
I
P
I cos θ
=
E
Z
Note: See Series and Parallel Circuits work sheets
to calculate values for a.c. impedance, Z. and the
phase angle, θ.
(no units)
I=
P
Z cos θ
I =
P
E cos θ
P =
E2 cos θ
Z
(no units)
Ohm's Law - Calculate Power
PF, cos θ
1.00
(no units)
Voltage, E
1.00
V
Impedance, Z
1.00
ohms
Power, P
1.00
W
©
Z
RETURN TO INDEX
PF, cos θ
Voltage, E
Current, I
Power, P
1.00
1.00
1.00
1.00
V
A
W
PF, cos θ
Current, I
Impedance, Z
Power, P
1.00
1.00
1.00
1.00
A
ohms
W
(no units)
P = EI cos θ
(no units)
P = I 2Z cos θ
Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
Apparent Power, Papp = EI (volt-amps)
Real Power, P real = EI cos θ (watts)
Reactive Power, P reactive=EI sin θ (VAR)
Power factor, PF = cos θ = P real/Papp
Phase Angle, θ = cos-1(Preal/Papp)
If the series circuit consists of series capacitors only, the impedance, Z, is equal to the sum of the individual
capacitive reactances. The phase angle, θ, is equal to -90 0
(The voltage lags the current by 90 0).
SERIES CIRCUITS
L is the inductance in Henries
XL is the inductive reactance in Ohms
F is the frequency in Hertz
XC is the capacitive reactance in Ohms
Z is the impedance in Ohms
θ is the phase angle in degrees
R is the resistance in Ohms
If the series circuit consists of series inductors only, the impedance, Z, is equal to the sum of the individual
inductive reactances. The phase angle, θ, is equal to +90 0
(The voltage leads the current by 90 0).
An easy way to remember the phase relationship of voltage/current in inductive and capacitive circuits is: "eLi
the iCe man". (i.e. voltage leads in inductive circuits and current leads in capacitive circuits).
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Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
R & L in Series
Resistance, R
100.0
Reactance, X
100.0
Impedance, Z
141.4
Phase Angle
45.00
ohms
ohms
ohms
degrees
R & C in Series
Resistance, R
10.0
Reactance, X
10.0
Impedance, Z
14.1
Phase Angle
45.00
ohms
ohms
ohms
degrees
L & C in Series
Reactance, XL
30.0
Reactance, XC
31.0
Impedance, Z
-1.0
Phase Angle
-90.00
ohms
ohms
ohms
degrees
R, L, & C in Series
Resistance, R
20.0
ohms
Reactance, XL
20.0
ohms
Reactance, XC
20.0
ohms
Impedance, Z
20.0
ohms
Phase Angle
0.00
degrees
Inductive Reactance
Inductance
Frequency
Reactance
643.06
11.130
44.97
uH
kHz
ohms
Capacitive Reactance
Capacitance
Frequency
Reactance
0.32
11.130
44.97
uF
Hz
kilohms
NOTES
Z = R2 + X L2
θ = arctan
XL
R
Z = R 2 + X C2
θ = arctan
XC
R
Z = X L − XC
θ=0 when XL = XC (resonance)
Z = R2 + ( X L − X C )2
θ = arctan
X L − XC
R
XL = 2π fL
XC =
1
2π fC
Series Resistance
Resistance 1
Resistance 2
Resistance 3
Resistance 4
Resistance 5
Resistance 6
Total
2.000
2.000
2.000
2.000
2.000
2.000
12.000
ohms
ohms
ohms
ohms
ohms
ohms
ohms
RT = R1 + R2 + R3 + ...Rn
Note: If the series circuit contains less
than six resistors, enter 0 for the
remaining resistances.
© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
If XL - XC is positive, the circuit is inductive.
If XL - XC is negative, the circuit is capacitive.
PARALLEL CIRCUITS
L is the inductance in Henries
XL is the inductive reactance in Ohms
F is the frequency in Hertz
XC is the capacitive reactance in Ohms
Z is the impedance in Ohms
θ is the phase angle in degrees
R is the resistance in Ohms
An easy way to remember the phase relationship of voltage/current in inductive and capacitive circuits is:eLi
" the iCe
man". (i.e. voltage leads current in inductive circuits and current leads voltage in capacitive circuits).
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Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
R & L in Parallel
Resistance, R
6800.0
Reactance, X
8640.0
Impedance, Z
5343.5
Phase Angle
38.20
ohms
ohms
ohms
degrees
R & C in Parallel
Resistance, R
3300.0
Reactance
2530.0
Impedance, Z
2007.8
Phase Angle
52.52
ohms
ohms
ohms
degrees
L & C in Parallel
Reactance, XL
365.0
Reactance, XC
365.0
Impedance, Z
MAX
Phase Angle
0.00
ohms
ohms
ohms
degrees
R, L, & C in Parallel
Resistance, R
2200.0
Reactance, XL
770.0
Reactance, XC
535.0
Impedance, Z
1371.0
Phase Angle
51.45
ohms
ohms
ohms
ohms
degrees
R1&L in Parallel with R2&C - Case (A)
Resistance, R 1
100.0
ohms
Resistance, R 2
100.0
ohms
Reactance, XL
100.0
ohms
Reactance, XC
500.0
ohms
Impedance, Z
161.2
ohms
Phase Angle
29.74
degrees
R1 & L in Parallel with C - Case (B)
Impedance, Z
ohms
Phase Angle
degrees
R2 & C in Parallel with L - Case (C)
Impedance, Z
ohms
Phase Angle
degrees
643.06
11.130
44.97
uH
kHz
ohms
θ = arctan
0.32
11.130
0.04
R
XL
R * XC
R2 + X C2
Z =
θ = arctan
R
XC
XL * XC
XL − XC
Z =
0
θ=0 when XL = XC (resonance)
Z =
R * X L * XC
X L2X C 2 + R2 ( X L − X C )2
⎛ R ( X L − XC ) ⎞
⎟
⎝ X L * XC
⎠
θ = arctan ⎜
Z =
(R12 + X L2 )(R22 + X C2 )
(R1 + R2 )2 + ( X L − X C )2
θ = tan−1
R12 + X L2
R12 + ( X L − X C )2
Z = XL
R22 + X C2
R + ( X L − X C )2
1.000
1.000
1.000
0.000
2.000
2.000
0.250
ohms
ohms
ohms
ohms
ohms
ohms
ohms
θ = tan−1
X L XC − X C2 − R22
R2X L2
XL = 2π fL
XC =
RT =
© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
(C)
2
2
1
2π fC
Parallel Resistance
Resistance 1
Resistance 2
Resistance 3
Resistance 4
Resistance 5
Resistance 6
Total
(B)
XL XC − X L2 − R12
R1XC2
θ = tan−1
F
Hz
ohms
(A)
X L (R22 + XC2 ) − XC (R12 + XL2 )
R1(R22 + XC2 ) + R2(R12 + X L2 )
Z = XC
Capacitive Reactance
Capacitance, C
Frequency, f
Reactance, XC
R * XL
R2 + X L2
Z =
Inductive Reactance
Inductance, L
Frequency, f
Reactance, XL
NOTES
1
1 1
1
+
+ ...
R1 R2
Rn
Note: Diagrams (B) & (C) above are
special cases of (A). For (B), enter "0"
for Resistance R 2. For (C), enter "0"
for Resistance R 1.
Note: Due to the infinite number of circuit configurations, no calculations are presented, only the prinicples and methods of network solutions are
presented. Calculations from other worksheets may be used to reduce networks to equivalent values.
RETURN TO INDEX
DEFINITIONS
Kirchhoff's Voltage Law
The algebraic sum (for d.c. circuits) or the phasor
sum (for a.c. circuits) of the source voltages and
voltage drops around a closed electric circuit (loop) is
zero.
∑E
1
+ E2 + E3 + ...En = 0
Kirchhoff's Current Law
The algebraic sum (for d.c. circuits) or the phasor
sum (for a.c. circuits) of the currents in and out of a
node (point) is zero.
∑
I 1 + I 2 + I 3 + ... I n = 0
Thevenin's Theorem for d.c (or a.c.) Circuits
Any two terminal network of resistors (or impedances)
and voltage sources is equivalent to a single resistor
(or impedance) in series with a single constant
voltage source.
Norton's Theorem for d.c. (or a.c.) Circuits
Any two terminal network of resistors (or impedances)
and current sources is equivalent to a single resistor
(or impedance) in parallel with a single constant
current source.
Millman's Theorem
Any number of constant current sources that are
directly connected in parallel can be converted to a
single current source whose total output is the
algebraic sum (for d.c.) or the phasor sum (for a.c.) of
the individual source currents, and whose total
internal resistance (or impedance) is the result of
combining the individual source resistances (or
impedances) in parallel.
Superposition Theorem
In a network of linear resistances (or impedances)
containing more than one source, the resultant
current flow at any one point is the algebraic sum (for
d.c.) or the phasor sum (for a.c.) of the current that
would flow at that point if each source is considered
separately, and all other sources are temporarily
replaced by their equivalent internal resistances (or
impedances). This would involve replacing each
voltage source by a short-circuit and each current
source with an open circuit.
©
Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
NOTES
Amplitude - The amplitude of a periodic curve (in electronics, typically a sinusoidal wave) is taken as the maximum displacement or value of the curve.
Frequency - The number of complete cycles occurring in a periodic curve in a unit of time is called the frequency (f) of the curve.
Period - The time (T) required for a periodic function, or curve, to complete one cycle is called the period.
Phase Angle - The angular difference (Θ) between two curves or waves is called the phase angle.
RMS - The effective value of a sine wave of current can be calculated by taking equally space samplings and extracting the the square root of their mean, or
average, values.
Peak - The maximum instantaneous value of an alternating quantity such as voltage or current.
Peak-Peak - The amplitude of an alternating quantity measured from positive peak to negative peak.
Average Value - The average of many instantaneous amplitude values taken at equal intevals of time during a half cycle of alternating current. The average value
of a pure sine wave during one half cycle is 0.637 times its maximum or peak value.
RETURN TO INDEX
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
Frequency
Period, t
1
Frequency, f
1
Frequency, f
Period, t
f =
mSec
kHz
Period
1
0.001
NOTES
1
t
t =
kHz
Sec
f is the frequency in Hertz
t is the period in seconds
1
f
f is the frequency in Hertz
t is the period in seconds
λ is the wavelength I meters
Wavelength
Frequency, f
3.75
80
Wavelength, λ
λ =
Mhz
Meters
C
f
Note: Conversion factors are for
sinewaves only
a.c. Voltage or Current
Avg
Peak
Peak-Peak
RMS
123.000
193.233
386.712
136.653
V
V
V
V
*
*
*
*
peak = 1.571*avg
peak-peak = 3.144*avg
rms = 1.111*avg
Peak
Peak-Peak
RMS
Avg
120.000
240.000
84.840
76.440
uA
uA
uA
uA
*
*
*
*
peak-peak = 2.000*peak
rms = 0.707*peak
avg = 0.637*peak
Peak-Peak
RMS
Avg
Peak
240.000
84.720
76.320
120.000
mV
mV
mV
mV
*
*
*
*
rms = 0.353*peak-peak
avg = 0.318*peak-peak
peak = 0.500*peak-peak
RMS
Avg
Peak
Peak-Peak
84.720
76.163
119.794
239.588
mA
mA
mA
mA
*
*
*
*
avg = 0.899*rms
peak = 1.414*rms
peak-peak = 2.828*rms
Calculate Power
Phase Angle
10.00
Voltage, E
120.00
Current, I
10.00
Power, P REAL
1181.769
Apparent Power
1200.000
Reactive Power
208.378
PF, cos θ
0.985
C is the velocity of light (3x10 8 m/sec)
f is the frequency in Hertz
Degrees
V
A
W
VA
VAR
(no units)
Sine Wave Characteristics
Degrees
Rad
SinΘ Voltage
0
0
0
0.0%
0
0
0.707
70.7% rms
π/4
45
0
0.866
86.6%
π/3
60
0
1
100.0% peak
π/2
90
0
0
0.0%
π
180
Primary Relationships
Vavg =
2
π
Vrms =
V peak = 0.637V peak
V peak
2
= 0.707V peak
PREAL = EI cos θ
PAPPARENT = EI
PREACTIVE = EI sinθ
PF = cos θ
* For consistency, only the units in the top (gray) cells may be changed. All other cells correspond to units of top (gray) cell.
© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
DEFINITIONS:
Inductance, L - Inductance is the ability of a conductor to produce an induced voltage as the current in the conductor is varied. Typically inductors take the form
of a coil of wire that concentrates the magnetic flux lines thereby increasing the inductance. The unit of inductance is the Henry - the amount of inductance which
will induce a counter EMF of one volt when the inducing current is varied at the rate of one ampere per second.
Inductive Reactance, XL - This is the characteristic of an inductor to impede the flow of a.c. current. The higher the inductive reactance, the more the a.c. curent
is impeded (just as resistance impedes the flow of current in a d.c. circuit). An important characteristic of inductive reactance is that it increases as the frequency
is increased (just the opposite of capacitive reactance).
Energy Stored, W - An inductor stores energy in the electric field, since an electric current is induced back into the conductor by the decaying magnetic field.
The amount of energy stored in an inductor (Joules) is directly proportional to the inductance and the square of the current.
RETURN TO INDEX
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
Inductance
Frequency, f
Reactance, XL
Inductance, L
25.00
44.97
286.288
kHz
ohms
uH
Inductive Reactance
Inductance, L
Frequency, f
Reactance, XL
0.00
800.000
10.05
H
Hz
ohms
Frequency
Inductance, L
Reactance, XL
Frequency, f
107.86
2.640
3895.50
uH
kilohms
kHz
Energy Stored
Inductance, L
Current, I
Energy Stored
10.00
2.00
20.00
H
Amps
Joules
Parallel Inductance
Inductance 1
Inductance 2
Inductance 3
Inductance 4
Inductance 5
Inductance 6
Total
1.000
1.000
1.000
1.000
1.000
1.000
0.167
mH
mH
mH
mH
mH
mH
mH
Series Inductance
Inductance 1
Inductance 2
Inductance 3
Inductance 4
Inductance 5
Inductance 6
Total
1.000
1.000
1.000
1.000
1.000
1.000
6.000
mH
mH
mH
mH
mH
mH
mH
Series Inductive Reactance
Reactance 1
Reactance 2
Reactance 3
Reactance 4
Reactance 5
Reactance 6
Total
1.000
1.000
1.000
1.000
1.000
1.000
6.000
ohms
ohms
ohms
ohms
ohms
ohms
ohms
Parallel Inductive Reactance
Reactance 1
Reactance 2
Reactance 3
Reactance 4
Reactance 5
Reactance 6
Total
1.000
1.000
1.000
1.000
1.000
1.000
0.167
ohms
ohms
ohms
ohms
ohms
ohms
ohms
NOTES
XL
L=
2π f
INDUCTIVE REACTANCE
XL = 2π fL
f =
XL
2π L
W = (1/2)LI2
LT =
1
L1
1
1
1
+
+ ...
L2
Ln
LT = L1 + L2 + ...Ln
X T = X 1 + X 2 + ... X n
XT =
© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
1
X1
+
1
1
1
+ ...
X2
Xn
Formula Variables:
L is the inductance in Henries
XL is the inductive reactance in Ohms
f is the frequency in Hertz
W is the energy stored in Joules
Z is the impedance in Ohms
V is the voltage in Volts
I is the current in Amps
R is the resistance in Ohms
DEFINITIONS:
Capacitance, C - This is the ability of a dielectric to store an electric charge which is measured in Farads (after Michael Faraday). Physically, a capacitor
consists of a dielectric material between two conductors. In operation, d.c. voltages are blocked while a.c. voltages pass through.
Capacitive Reactance, Xc - This is the characteristic of a capacitor to impede the flow of a.c. current. The higher the capacitive reactance , the more the a.c.
curent is impeded (just as resistance impedes the flow of current in a d.c. circuit). An important characteristic of capacitive reactance is that it increases as the
frequency is decreased (just the opposite of inductive reactance).
Charge, Q - When a voltage is applied to opposing plates of the capacitor, negative and positive electric charges build up creating a field that stresses the
dielectric. The higher the voltage, the more the dielectric is stressed and the higher the charge (in Coulombs).
Energy Stored, W - The amount of energy stored in a capacitor (Joules) is directly proportional to the capacitance and the square of the voltage.
RETURN TO INDEX
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
NOTES
Capacitance
Frequency, f
Reactance, XC
Capacitance, C
11.13
44.97
317.982
MHz
ohms
pF
Capacitive Reactance
Capacitance, C
Frequency, f
Reactance, XC
317.98
11.130
44.97
pF
MHz
ohms
Frequency
Capacitance, C
Reactance, XC
Frequency, f
317.98
44.970
11.13
pF
ohms
MHz
Charge & Energy Stored
Capacitance, C
Voltage, E
Energy Stored
Charge, Q
5.00
100.00
25000.00
500.00
F
Volts
Joules
1.000
1.000
1.000
1.000
1.000
1.000
0.167
uF
uF
uF
uF
uF
uF
uF
Parallel Capacitance
Capacitance 1
Capacitance 2
Capacitance 3
Capacitance 4
Capacitance 5
Capacitance 6
Total
1.000
1.000
1.000
1.000
1.000
1.000
6.000
pF
pF
pF
pF
pF
pF
pF
Series Capacitive Reactance
Reactance 1
Reactance 2
Reactance 3
Reactance 4
Reactance 5
Reactance 6
Total
1.000
1.000
1.000
1.000
1.000
1.000
6.000
ohms
ohms
ohms
ohms
ohms
ohms
ohms
Parallel Capacitive Reactance
Reactance 1
Reactance 2
Reactance 3
Reactance 4
Reactance 5
Reactance 6
Total
1.000
1.000
1.000
1.000
1.000
1.000
0.167
2π fX C
XC =
f =
CAPACITIVE REACTANCE
1
1
2π fC
1
2π CX C
Q = CE
W = (1/2)CE2
Coulombs
Series Capacitance
Capacitance 1
Capacitance 2
Capacitance 3
Capacitance 4
Capacitance 5
Capacitance 6
Total
C =
ohms
ohms
ohms
ohms
ohms
ohms
ohms
© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
CT =
1
C1
1
1
1
+
+ ...
C2
Cn
CT = C1 + C2 + ...Cn
XT = X1 + X2 + ...Xn
XT =
1
X1
+
1
1
1
+ ...
X2
Xn
Formula Variables:
C is the capacitance in Farads
Xc is the capacitive reactance in Ohms
f is the frequency in Hertz
Q is the electric charge in Coulombs
W is the energy stored in Joules
Z is the impedance in Ohms
E is the voltage in Volts
I is the current in Amps
R is the resistance in Ohms
RC & L/R TIME CONSTANTS
t - The time constant in seconds
L - the inductance in henries
C - The capacitance in farads
R - The resistance in ohms
The time constant is the time, in seconds, that it takes a voltage across a capacitor or for the current through an
inductor to build up to 63.2% of its final value.
The Time Constant is also the time, in seconds, that it takes the voltage across a capacitor or the current through an
inductor to discharge to 36.8% of its initial value.
A long time constant takes approximately 5 time constants to build up to 99% of its final value.
A short time constant is defined as one-fifth or less the pulse width, in time, for the applied voltage.
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
RC Time Constant
Resistance, R
Capacitance, C
Time Const, τ
5
1
0.005
kilohms
pF
uSec
Time Const, τ
Capacitance, C
Resistance, R
0.005
1
5
uSec
pF
kilohms
Time Const, τ
Resistance, R
Capacitance, C
1
1
1
uSec
Ohms
uF
C =
τ
R
τ = R *C
R=
τ
C
L/R Time Constant
©
Resistance, R
Inductance, L
Time Const, τ
1
1
1
Ohms
uH
uSec
τ=
L
R
Time Const, τ
Inductance, L
Resistance, R
1
1
1
uSec
uH
Ohms
R=
L
τ
Time Const, τ
Resistance, R
Inductance, L
1
1
1
uSec
Ohms
uH
L =τ *R
Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
NOTES
RETURN TO INDEX
DEFINITIONS:
Resonant Frequency - In an LC circuit, the resonant frequency occurs when the inductive and capacitive reactances are equal and opposite, such that X c = XL.
Resonance - In an LC circuit, as the frequency is increased, the inductive reactance increases and the capacitive reactance decreases. Due to these opposing
characteristics, there is a frequency where the inductive and capacitive reactances are equal to each other. This condition is called resonance and the circuit is
called a resonant circuit .
Q Factor - The ratio of the reactance (capacitive or inductance) to the device's resistance is known as the Q Factor or figure of merit.
Bandwidth - The width of the resonant band of frequencies with a response of 70.7% of the magnitude and centered around the resonant frequency (f R) is called the
bandwidth of the tuned circuit.
RETURN TO INDEX
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
NOTES
Frequency
Inductance, L
Capacitance, C
Frequency, f
10.00
100.00
5.033
1
uH
pF
MHz
f =
uH
MHz
pF
C=
pF
MHz
uH
L=
2π
Capacitance
Inductance, L
Frequency, f
Capacitance, C
11.13
7.112
45.00
1
4π 2f 2L
Inductance
Capacitance, C
Frequency, f
Inductance, L
45.00
7.112
11.13
Inductive Reactance
Inductance, L
Frequency, f
Reactance, XL
11.13
7.112
497.36
45.00
7.112
497.30
pF
MHz
ohms
1.00
10.00
0.10
10.00
(no units)
Frequency, f1
Frequency, f2
(series circuits)
150.00
0.047
7.088
7.136
1.00
10.00
0.10
(parallel circuits)
fr
= f1 − f2
Q
Δf
f1 = fr −
2
Δf
f2 = fr +
Δf =
2
Q Factor (Resonant Circuit)
Frequency, fr
Bandwidth, Δf
Q-Factor
2π fC
(no units)
MHz
ohms
MHz
MHz
MHz
ohms
ohms
(no units)
© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
Q =
Parallel RLC Circuit @ Resonance:
Z=R
Xc = XL
Phase Angle = 0
Power Factor = 1
Z = Max
I = Min.
Vo = Max.
1
X
R
Q = LorC =
R
X LorC
ohms
ohms
Bandwidth
Resonant Freq., f R
7.112
Q-Factor
Delta f
4π 2f 2C
XC =
Q Factor (Components)
Reactance, X
Resistance, R
Series Q
Parallel Q
1
XL = 2π fL
uH
MHz
ohms
Capacitive Reactance
Capacitance, C
Frequency, f
Reactance, XC
LC
Formula Variables:
L is the inductance, Henries
C is the capacitance, Farads
R is the resistance, Ohms
X is the reactance (X L or XC), Ohms
f is the frequency, Hertz
Q is the ratio of X to R (no units)
Z is the impedance, Ohms
fR
Δf
Series RLC Circuit @ Resonance:
Z=R
Xc = XL
Phase Angle = 0
Power Factor = 1
Z = Min
I = Max
Vo = Min
COIL WINDING (AIR CORE)
DEFINITIONS:
Filter - A network that is designed to attenuate certain frequencies, but pass other frequencies, is called a filter.
Bands - A filter possesses at least one pass band and at least one stop band.
Stop Band - A band of frequencies for which the attenuation is theoretically infinite.
Pass Band - A band of frequencies for which the attenuation is theoretically zero.
Cutoff Frequency - The frequencies that separate the various pass and stop bands are called cutoff frequencies.
RETURN TO INDEX
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
Low Pass Filters - Cutoff Frequency
Inductance, L
Capacitance,C
Frequency, f
11.13
45.00
14.223
uH
pF
MHz
fcutoff =
High Pass Filters - Cutoff Frequency
Inductance, L
Capacitance,C
Frequency, f
11.13
45.00
3.556
uH
pF
MHz
Band Pass Filters - Center Frequency
Inductance, L
Capacitance,C
Frequency, f
11.13
45.00
7.112
uH
pF
MHz
fcutoff =
fcenter =
Half-Wave Filter Design (5-Pole)
Frequency, f
Load
Cutoff Freq.
13.5
50
15.255
MHz
ohms
MHz
Inductance, L1
Inductance, L2
0.52
0.52
uH
uH
Capacitance, C1
208.66
417.32
208.66
pF
pF
pF
Capacitance, C2
Capacitance, C3
NOTES
© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
1
π LC
1
4π LC
1
2π LC
The calculations on this worksheet are based on air core coils (ferrite, iron core, and toroids are addressed in a separate worksheet). Two calculations are presented for
single layer coils: one based on the radius, and the other based on the diameter of the coil. The wire tables are based on an average as the dimensions of wire products
vary slightly among manufacturers.
For convenience, a calculation is included for determining the resonant frequency of an LC circuit. The resonant frequency for an inductor and capacitor is the same
whether they are connected in series or parallel. As an example, if you have a known capacitor, the required inductance can be determined for a desired resonant
frequency. Using the calculated inductance, determine the number of turns required based on the diameter of available coil forms. Or, using the inductance formula, the
inductance of an existing coil can be determined by entering its diameter, length, and number of turns in the appropriate calculator. Formulas assume short coils (length <
10x diameter).
RETURN TO INDEX
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
NOTES
Coil Inductance (based on radius)
Coil Radius, r
1
inches
(no units)
No. of Turns, N
40
Coil Length, l
1
inches
Inductance, L
84.21
uH
Spacing
40
TPI
Typ. Wire Size
22
AWG
Inductance of a coil based on radius ,
length, and number of turns.
Coil Inductance (based on dia.)
Coil Dia., d
2
inches
(no units)
No. of Turns, N
40
Length of Coil, l
inches
1
Inductance, L
uH
84.21
Spacing
40
TPI
Typ. Wire Size
22
AWG
Inductance of a coil based on diameter ,
length, and number of turns.
r 2N2
L=
9r + 10l
L=
d 2N 2
18d + 40 l
Number of Turns (based on radius)
inches
Coil Radius, r
0.25
Length of Coil, l
1
inches
Inductance, L
uH
8.16
(no units)
No. of Turns, N
39.99
Spacing
40.0
TPI
Wire Size
22
AWG
Number of turns required for a coil
based on radius, length, and
inductance.
Number of Turns (based on dia.)
Coil Dia., d
0.5
inches
Length of Coil, l
1
inches
Inductance, L
uH
8.16
(no units)
No. of Turns, N
39.99
Spacing
40.0
TPI
Wire Size
22
AWG
Number of turns required for a coil
based on diameter, length, and
inductance.
Resonant Frequency
107.85
6.77
5.890
Inductance, L
Capacitance, C
Frequency, f
uH
pF
MHz
L(9r + 10 l )
N=
r
N=
L(18d + 40 l )
d
f =
1
2π
LC
Formula Variables:
L is the inductance, Henries
r is the coil radius, inches
d is the coil diameter, inches
l is the coil length, inches
N is the number of turns
b is the depth of coil winding for multi-layer coils*
TPI is the number of turns per inch
AWG is the American Wire Gauge standard
C is the Capacitance
f is the Frequency
* These formulas are based on short coils
(i.e. length < 10x diameter of coil).
Copper Wire Table
AWG
10
12
14
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
TPI
enameled
9.6
12.0
15.0
18.9
21.2
23.6
26.4
29.4
33.1
37.0
41.3
46.3
51.7
58.0
64.9
72.7
81.6
90.5
Inductance, Straight Wire
Dia. Of Wire, d
0.001
cm
Length of Wire, l
200
cm
Induct. L (low freq)
2.061
uH
Induct. L (high freq)
1.961
uH
©
Inductance of a multi-layer coil based on
radius, number of turns, length, and
depth of coil.
0.8(rN 2 )
L=
6r + 9l + 10b
2l
− 0.75⎤⎥
Llowfreq = 0.002l ⎡⎢log
d /2
⎣
⎦
2l
− 1.00 ⎤⎥
Lhighfreq = 0.002l ⎡⎢log
d /2
⎣
⎦
Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
mm
0.1019
0.0808
0.0641
0.0508
0.0453
0.0403
0.0359
0.0320
0.0285
0.0254
0.0226
0.0201
0.0179
0.0159
0.0142
0.0126
0.0113
0.0100
2.59
2.05
1.63
1.29
1.15
1.02
0.91
0.81
0.72
0.64
0.57
0.51
0.45
0.40
0.36
0.32
0.29
0.25
TPI*
insulated
*Depends on type of insulation
1 inch = 2.54 cm
Multi-Layer Coil (based on radius)
Coil Radius, r
0.55
inches
No. of Turns, N
40
Length of Coil, l
1
inches
Depth of Coil, b
0.1
inches
Inductance, L
29.113
uH
Diameter
inches
1 cm = 0.3937 in.
1 meter = 39.37 in
Iron Powder Toroid Cores:
Iron powder toroids are suitable for tuned tank circuits, filters, network inductors, and any applicationrequiring a high Q inductor.
Iron powder toroids are more stable than ferrites and do not saturate as easily. For best Q, use the mix specified for the applications frequency range.
Toroid cores are assigned a core size and mix model number by the manufacturer to identify them as shown in the chart below.
For example, a T-12-0 core (tan, phenolic) would exhibit 3.0 uH (microHenrys) per 100 turns; a T-12-1 (blue, carbonyl) would exhibit 48 uH per 100 turns, etc.
RETURN TO INDEX
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
IRON POWDER TOROID CORES, uH PER 100 TURNS
Mix
0
1
2
3
6
7
10
12
15
17
26
Frequency
MHz
50 - 250
0.15 - 2
0.25 - 10
0.02 - 1
2 - 30
3 - 35
10 - 100
20 - 200
0.1 - 3
40 - 180
DC - 1
Color
Tan
Blue
Red
Gray
Yellow
White
Black
Green/white
Red/White
Blue/Yellow
Yellow/White
Material
Phenolic
Carbonyl
C
Carbonyl
E
Carbonyl
HP
Carbonyl
SF
Carbonyl
TH
Carbonyl
W
Synthetic
Oxide
Carbonyl
GS6
Carbonyl
Hydrogen
Reduced
u
1
20
10
35
8.5
9
6
4
25
4
75
Temp
Stability
ppm/0C
0
280
95
370
35
30
150
170
190
50
825
Core Size/Mix
T-12
T-16
T-20
T-25
T-30
T-37
T-44
T-50
T-68
T-80
T-94
T-106
T-130
T-157
T-184
T-200
T-225
T-225A
T-300
T-400
T-400A
0
1
2
3
6
7
10
12
15
17
26
3.0
3.0
3.5
4.5
6.0
4.9
6.5
6.4
7.5
8.5
10.6
19.0
15.0
-
48
44
52
70
85
80
105
100
115
115
160
325
200
320
500
250
-
20
22
27
34
43
40
52
49
57
55
84
135
110
140
240
120
120
215
115
185
360
60
61
76
100
140
120
180
175
195
180
248
450
350
420
720
425
425
-
17
19
22
27
36
30
42
40
47
45
70
116
96
115
195
100
100
-
18
24
29
37
32
46
43
52
50
133
103
105
-
12
13
16
19
25
25
33
31
32
32
58
-
7.5
8
10
12
16
15
18.5
18
21
22
32
-
50
55
55
85
93
90
160
135
180
170
200
345
250
360
-
7.5
8
10
12
16
15
18.5
18
21
22
-
145
180
235
325
275
360
320
420
450
590
900
785
870
1640
895
-
CALCULATIONS
FORMULAS
Number of Turns
8.16
300.00
16.492
Inductance, L
uH/100Turns, AL
No. Turns, N
Inductance
16.49
No. of Turns, N
uH/100Turns, AL
Inductance, L
300.00
8.160
Resonant Frequency
8.16
200.00
3.940
Inductance, L
Capacitance, C
Frequency, f
NOTES
uH
*
*
N = 100
*
*
uH
L = AL ⎜
uH
pF
mHz
* no units
© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
L
AL
⎛ N2 ⎞
4 ⎟
⎝ 10 ⎠
f =
1
2π
LC
Formula Variables:
L is the inductance, Henries
N is the number of turns
AL is the inductance in uH per 100 turns (See table)
AWG is the American Wire Gauge standard
C is the capacitance in Farads
f is the frequency in Hertz
Antennas
Hertz Antenna - Type of antenna that is complete in itself and capable of self-oscillation (i.e. half or full wavelength dipole).
Marconi Antenna - Type of antenna that relies on the ground (earth) as part of antenna (i.e. 1/4 wavelength vertical ground plane).
Permittivity of Free Space, εo - 8.85 x 10 -12 farads/meter
Permeability of Free Space, μo - 4π x 10-7 henrys/meter or 1.257 x 10-6 henrys/meter.
Velocity of Light (E-M Radiation), C - C=1/SQRT(μoεo) = 3x108 meters/sec
Radiation Resistance of Free Space, η0 = SQRT(μo/εo) = 377 Ω
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
NOTES
Antenna Calculator(s)
Dimensions in Feet
1/4 Wave Vertical Antenna
7.100
Frequency, f
MHz
42.25
Wavelength
Meters
32.96
1/4 Wavelength
Feet
10.05
1/4 Wavelength
Meters
234
f
Dimensions In Meters
1/ 4 λ =
71.34
f
1/2 Wave Dipole Antenna
7.040
Frequency, f
MHz
42.61
Wavelength
Meters
32.96
Length Per Side
Feet
65.92
1/2 Wavelength
Feet
20.27
1/2 Wavelength
Meters
Dimensions in Feet
3-Element Beam Antenna
14.020
Frequency, f
MHz
21.40
Wavelength
Meters
66.76
Wavelength
Feet
Director, DI, 0.45λ
30.04
Feet
Spacing, DD, 0.10λ
6.68
Feet
Driven El., DE, 0.5λ
33.38
Feet
Spacing, DR, 0.15λ
10.01
Feet
Reflector, RF, 0.55λ
36.72
Feet
Director: DI
Spacing: DD
Driven Element: DE
Spacing: DR
Reflector: RF
Matching Transformer
7.040
Frequency, f
MHz
0.98
Velocity Factor, V
34.07
Total Length, L
Feet
Line of Sight Propagation
328.0
Height @ XMTR
feet
29.5
Height @ RCVR
feet
33.19
Total Range
miles
©
1/ 4 λ =
468
f
Dimensions In Meters
1/2λ =
1/2λ =
142.68
f
DI = 0.45* λ
DD = 0.10 * λ
DE = 0.5* λ
DR = 0.15* λ
RF = 0.55* λ
L=
246 * V
f
Velocity Factors:
Air Insulated Coax -0.85
Ladder Line - 0.975
Twin Lead - 0.82
Polyethylene Coax-0.66
D = 1.41*( HT + HR )
D = 3.6*( HT + HR )
Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
RETURN TO INDEX
Transmission Line - A transmission line is the connecting link between a source of r.f. power (transmitter) and the load (antenna). The main purpse of the transmission
line is to transfer maximum power to the antenna with minimum losses. The two main types of transmission lines are the 'parallel-conductor' (i.e. open-wire, ladder line,
or two-wire) and the 'coaxial line' (or 'coax' for short).
Velocity of Propagation - The presence of dielectrics in a coaxial line reduces the velocity of propagation of an electromagnetic wave through the transmission line. Fo
this reason, transmission line specfications will include the velocity factor for the line.
Characteristic Impedance, ZO - Due to the physical characteristics of a transmission line, it will exhibit distributed capacitance and impedance and therefore exhibits a
characteristic or surge impedance.
Standing Wave Ratio - The ratio of maximum voltage along the line to the minimum volatage along the line is called the voltage standing wave ration (v.s.w.r.) or the
standing wave ratio (s.w.r.). The lower the ratio, the better is the match with the lowest s.w.r. representing the maximum power transfer.
Attenuation - The is the measure of losses along a transmission line and is usually specified as dB per foot (dB/ft).
RETURN TO INDEX
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
Two Parallel Lines - Impedance
0.1305
inches
Ctr-Ctr Distance, D
10.00
inches
1.00
Rel Permittivity, ε
Impedance, Z 603.176
ohms
Dia. Of Conductors, d
Z=
Two Parallel Lines - Capacitance
0.1305
inches
Ctr-Ctr Distance, D
10.00
inches
Length, l
1.00
feet
1.00
Rel Permittivity, ε
Capacitance, C
1.684
pF
Two Parallel Lines - Inductance
0.1305
inches
Ctr-Ctr Distance, D
10.00
inches
Length, l
1.00
feet
1.00
Rel. Permeability, μ
Inductance, L
0.612
uH
Dia. Of Conductors, d
Two Parallel Lines - Attenuation
0.1305
inches
Ctr-Ctr Distance, D
10.00
inches
Frequency, f
400.00
mHz
Length, l
100.00
feet
Attenuation
0.220
dB
Dia. Of Conductors, d
Coax - Impedance
0.108
Dia. Of Outer Cond., D
0.41
2.30
Rel Permittivity, ε
Impedance, Z
52.234
Dia. of Inner Cond., d
Coax - Capacitance
0.108
Dia. Of Outer Cond., D
0.41
Length, l
1.00
2.30
Rel Permittivity, ε
Capacitance, C
29.490
Dia. of Inner Cond., d
Coax - Inductance
0.108
Dia. Of Outer Cond., D
0.41
Length, l
1.00
2.30
Rel. Permeability, μ
Inductance, L
0.185
Dia. of Inner Cond., d
Coax - Attenuation
0.108
Dia. Of Outer Cond., D
0.41
Frequency, F
400.00
Length, l
100.00
Attenuation
0.188
Dia. of Inner Cond., d
©
inches
inches
276
ε
C=
Dia. Of Conductors, d
NOTES
log
2D
d
3.68ε l
2D
log
d
L = 0.281μ l log
dB =
2D
d
(
3.14 f
l ) 10-5
(
2D
( d ) log
d
Z =
138
ε
Permittivity (Dielectric Const),
air=1.0
teflon=2.1
glass=7.6
mica=7.5
plexiglas=2.6 - 3.5
polystyrene=2.4 - 3.0
Permeability, μ
non-ferrous=1.0
log
ε
)
D
d
ohms
inches
inches
feet
C=
log
pF
inches
inches
feet
7.36ε l
D
d
L = 0.140μ l log
D
d
uH
inches
inches
mHz
feet
dB
dB =
Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
4.6 F (D + d )
* l *10−6
D
(D * d )log
d
Capacitance, C is in pF
Inductance, L is in uH
Frequency, F is in MHz
Length, l is in feet
Distance, Diameter are in inches
Impedance, Z is in ohms
BEL -The Bel (B) is the unit of measurement used to express a ratio between two quantities, typically power, current or voltage.
Decibel - A dimensionless unit for expressing the ratio of two values. It is equal to 10 times the log10 of a power ratio or 20 times the log10 of the voltage or current
ratio.
dBm - This is an absolute measurement of the power level compared to a reference of 1mW. For RF, 0 dBm = 1mW into 50 ohms or -30 dBw.
dBi - The absolute measurement of the gain (or loss) of an antenna as compared to an isotropic antenna reference.
dBd - The absolute measurement of gain (or loss) of an antenna as compared to a half wave dipole reference antenna. If the isotropic antenna is assumed to be
unity gain, then the gain of a dipole is 2.14 dBi. Stated another way, dBd = dBi - 2.14.
dBw - The absolute measurement of gain (or loss) compared to a reference of 1 watt. For RF, 0 dBw = 1 watt into 50 ohms or 600 ohms for AF. Stated another
way, 0 dBw = +30 dBm.
dBμV - The absolute measurement of gain (or loss) compared to areference of 1 μVolt into 50 ohms. 0 dBuV = 1 μVolt into 50 ohms for RF.
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
RETURN TO INDEX
NOTES
Note: Input and Ouput values must be in the same units. Therefore, in the following calculations, the output units
are automatically adjusted based on the units selected for the input power, voltage, or current.
Power (gain/loss)
Power In
0.001
Power Out
100
Loss/Gain
50
W
W
dB
Voltage (gain/loss)
Voltage In
1
V
Voltage Out
50
V
Loss/Gain 33.9794
dB
Current (gain/loss)
Current In
100
A
Current Out
1
A
Loss/Gain
-40
dB
dB is the power gain or loss in decibels
Pout is the output power in Watts
Pin is the input power in watts
log is the logarithm to the base 10
Vout
Vin
dB is the voltage gain or loss in decibels
Vout is the output voltage in volts
Vin is the input voltage in volts
log is the logarithm to the base 10
I
dB = 20log out
Iin
dB is the current gain or loss in decibels
Iout is the output current in amps
Iin is the input current in amps
log is the logarithm to the base 10
dB = 20log
Power (gain/loss) 1mW
Power In
1
mW
Power Out
100
mW
Loss/Gain
20
dBm
P
dBm = 10log out
1mW
Power (gain/loss) 1 Watt
W
Power In
1
Power Out 100000
W
Loss/Gain
50
dBw
P
dBW = 10log out
1W
Voltage (gain/loss) 1 μV
Voltage In
1
uV
Voltage Out
50
uV
Loss/Gain 33.9794
dBμV
©
P
dB = 10log out
Pin
V
dB μV = 20log out
1μV
Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
dBm is the power gain or loss in decibels
referenced to 1 mW at 600 ohms for audio or
50 ohms for radio frequencies.
Pout is the output power in Watts
Pin is the input power @ 1 mWatt
log is the logarithm to the base 10
dBw is the power gain or loss in decibels
referenced to 1 W at 600 ohms for audio or 50
ohms for radio frequencies.
Pout is the output power in Watts
Pin is the input power @ 1 Watt
log is the logarithm to the base 10
dBμV is the voltage gain or loss in decibels
referenced to 1 μV at 600 ohms for audio or 50
ohms for radio frequencies.
Vout is the output voltage in μVolts
Vin is the input voltage @ 1 μV
log is the logarithm to the base 10
Definitions:
Weber - The Weber (Φ) is the magnetic flux which induces an emf of one volt when a conductor cuts through the field in one second.
Reluctance, R - The opposition by a circuit to the establishment of a magnetic field in amp-turns per weber.
Mutual Inductance - The measure of the magnetic flux linkage between two coils, measured in Henrys. The mutual inductance is one henry when the current of one
coil is changing at the rate of one amp per second induces a voltage of one volt in the second coil.
RETURN TO INDEX
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
Magnetic Field Intensity
Current
20.0
amps
No. of Turns
10.0
Length, l
0.5
meters
Mag. Field Intensity, H
400.00
amp-t/meter
Magnetic Flux Density
20.0
Flux, Φ
Area
10.0
Mag. Flux Density, B
2.00
webers
meters
teslas
Magnetic Flux, Φ
Mag. Flux Density,B
20.0
Area
10.0
20.0
Flux, Φ
teslas
meters
webers
Permeability
Mag. Flux Density,B
20.0
Mag. Field Intensity,H
10.0
2.00
Permeability, μ
=
H
Mutual Inductance
Inductance, L1
10.0
Inductance, L2
1.0
Coupling Factor, k
0.5
Mutual Inductance
1.58
LT, Series Aiding
14.16
LT, Series Oppose
7.84
LT, Parallel Aiding
0.96
LT, Parallel Oppose
0.53
©
*N
l
φ
A
φ = BA
μ =
Transformer Voltage Ratio
Pri. Voltage
20.0
volts
Pri. Turns
10.0
turns
Sec. Voltage
10.0
volts
Sec. Turns
5.00
turns
Transformer Impedance Ratio
Pri. Imped.
20.0
ohms
Pri. Turns
10.0
turns
Sec. Imped.
10.0
ohms
Sec. Turns
50.00
turns
I
B =
teslas
amp-t/meter
tesla-m/amp
Transformer Current Ratio
Pri. Current
20.0
amps
Pri. Turns
10.0
turns
Sec. Current
10.0
amps
Sec. Turns
5.00
turns
NOTES
B
H
V1 N1
=
V2 N2
Henrys
Henrys
Henrys
Henrys
Henrys
Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
B is the magnetic flux density in teslas
(webers/meter2)
A is the cross sectional area in meters 2
Φ is the magnetic flux in webers (volt-secs)
B is the magnetic flux density in teslas
(webers/meter2)
A is the cross sectional area in meters2
Φ is the magnetic flux in webers (volt-secs)
B is the magnetic flux density in teslas
(webers/meter2)
H is the magnetic field intensity,
amp-turns/meter
μ is the Permeability in tesla-meter/amp
V1 is the voltage on the transformer primary
V2 is the voltage on the transformer secondary
N1 is the number of turns on the primary
N2 is the number of turns on the secondary
I1 N1
=
I2 N2
I1 is the current on the transformer primary
I2 is the current on the transformer secondary
N1 is the number of turns on the primary
N2 is the number of turns on the secondary
Z1 N12
=
Z 2 N22
Z1 is the transformer primary impedance
Z2 is the transformer secondary impedance
N1 is the number of turns on the primary
N2 is the number of turns on the secondary
Series Aiding
Henrys
Henrys
H is the magnetic field intensity, amp-turns/meter
I is the current, amps
l is the length, meters
N is the number of turns
Note: For magnetic field intensity in oersteds,
multiply amp-turms/meter by 0.01257
LT = L1 + L2 + 2M
Series Opposing
LT = L1 + L2 − 2M
Parallel Aiding
LT =
L1L2 − M2
L1L2 + 2M
Parallel Opposing
LT =
L1L2 − M2
L1L2 − 2M
L1 is inductance of first coil in Henrys
L2 is inductance of second coil in Henrys
LT is total inductance in Henrys
M is the mutual inductance in Henrys
Real and Imaginary Number - In a.c. calculations, it is generally more practical to represent real and reactive values in terms of complex numbers. Thus the square root of (R2+
X2) becomes R + jX where R is the real part and X is the imaginary (reactive) part.
Phase - in the complex number, R + jX, R is the in-phase of the complex number and X is the out-of-phase portion.
Rectangular Form - The expression R = jX is referred to as the rectangualr for or rectangular coordinates.
Polar Form - When the rectangualr components of R + jX are resolved into a single magnitude of Z rotated through an angle of Θ, the expression is referred to as the polar form
or polar coordinate. So that R + jX = Z/Θ, where R=ZcosΘ, X=ZsinΘ, Θ=arctan(X/R), Z=R/cosΘ, and Z=X/sinΘ.
RETURN TO INDEX
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
Rectangular to Polar
Resistance, R
20.0
ohms
Reactance, X
-20.0
ohms
Impedance, Z
28.3
ohms
-45.00
degrees
Phase Angle, θ
FORMULAS
R + jX = Z ∠θ
Z =
R2 + X 2
X
θ = tan−1 ⎛⎜ ⎞⎟
⎝R ⎠
Enter inductive reactance as positive and
capacitive reactance as negative.
Polar to Rectangular
Impedance, Z
28.3
ohms
-45.0
degrees
Phase Angle, θ
Resistance, R
20.0
ohms
Reactance, X
-20.0
ohms
A positive reactance indicates
inductance and a negative reactance
indicates capacitance.
Z ∠θ = R + jX
R = Z cos θ
X = Z sinθ
SERIES CIRCUIT:
θ = tan-1 (X/R)
R = Zcos θ = SQRT (Z2-X2)
X = Zsin θ = SQRT(Z2-R2)
Z = R/cos θ = X/sin θ
sin θ = opp/hyp
cos θ = adj/hyp
tan θ = opp/adj
cot θ = adj/opp
sec θ = hyp/adj
csc θ = hyp/opp
©
Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
NOTES
Rules For Complex Math:
If Y1=Z1/θ1 = R1+ jX1 and Y2=Z2/θ2=R2+jX2
When adding or subtracting, use the
Rectangular Form:
Addition: Y1+Y2 =(R1+R2) +j(X1+X2)
Subtraction: Y1-Y2=(R1-R2)+j(X1-X2)
When multiplying or dividing, use the
Polar Form:
Multiplication: Y1Y2=Z1Z2/θ1+θ2
Division: Y1/Y2=Z1/Z2/θ1-θ2
Square: Z12=Z12/2θ1
Square Root: Z1^0.5=Z1^0.5/θ1/2
COMPONENT DATA
Resistance of a Conductor - The resistance of a cylindrical conductor is directly proportional to the length of the conductor, inversely proportional to the
cross-sectional area and is dependent on the conductors material composition (expressed as its resistivity).
Temperature Coeficient - Most conducting materials exhibit an increase in resistance as the temperature rises (within certain ranges). Other materials
exhibit a negative temperature coefficience (carbon, germanium,and silicon). The change in resistance due to temperature is expressed as the temperature
coeficient of temperature, α (alpha).
RETURN TO INDEX
Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.
CALCULATIONS
FORMULAS
NOTES
Resistance of a Conductor
Length, l
Area, S
Resistivity, r
Resistance, R
1000
0.001
1.72E-08
0.017
meters
meters2
Ω-meters
R = ρ
ohms
Cross sectional Area of Conductor
Diameter, d
0.1
meters
Area, S
0.01
meters2
S =π
Thermal Resistance Changes
Initial Resistance
100
ohms
Initial Temp, T1
80
degrees
Final Temp, T2
120
degrees
Temp. Coef., α
Final Resistance
S
d2
4
Rfinal = Rinitial [1+ α (T2 − T1)]
0.00393
115.7
l
ohms
R is the conductor's resistance in ohms
ρ is the resistivity of the conductor in Ω-meters
l is the length of the conductor in meters
2
S is the cross sectional area in meters
2
S is the cross sectional area in meters
π is a constant 3.14
d is the diameter of the conductor in meters
Rfinal is the final resistance in ohms
Rinitial is the initial resistance in ohms
T1 is the initial temperature
T2 is the final temperature
α is the temperature coeficient
Temperature Characteristics
Resistor Color Code
1st
Black
Brown
Red
Orange
Yellow
Green
Blue
Violet
Gray
White
1
2
3
4
5
6
7
8
9
2nd
0
1
2
3
4
5
6
7
8
9
Gold
Silver
M
Tolerance
10
101
102
103
104
105
106
107
10-1
10-2
Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.
Resistivity, ρ Resistivity, ρ
ohm-cmil/ft @
ohm-m
@ 20 C
20 C
Temp.
Coeficient
α
constantan
1.46E-08
1.72E-08
2.83E-08
5.50E-08
7.80E-08
1.20E-07
4.90E-07
8.782E-14
1.037E-13
1.702E-13
3.308E-13
4.692E-13
7.218E-13
2.947E-12
3.80E-03
3.93E-03
3.90E-03
4.50E-03
6.00E-03
5.50E-03
8.00E-07
AWG
TPI (enam)
Conductor
0
0
In the Resistor Color Code Chart,
the values in Column "M" are multipliers
©
Failure
Rate
±1%
±2%
±0.5%
±0.25%
±0.1%
±0.05%
1
0.1
0.01
0.001
silver
copper
aluminum
tungsten
nickel
iron
±5%
±10%
0
Copper Wire Table
Dia (inches)
10
9.6
0.1019
12
12.0
0.0808
14
15.0
0.0641
16
18.9
0.0508
17
21.2
0.0453
18
23.6
0.0403
19
26.4
0.0359
20
29.4
0.0320
21
33.1
0.0285
22
37.0
0.0254
23
41.3
0.0226
24
46.3
0.0201
25
51.7
0.0179
26
58.0
0.0159
27
64.9
0.0142
28
72.7
0.0126
29
81.6
0.0113
30
90.5
0.0100
*Depends on type of insulation
Dia (mm)
2.59
2.05
1.63
1.29
1.15
1.02
0.91
0.81
0.72
0.64
0.57
0.51
0.45
0.40
0.36
0.32
0.29
0.25
TPI*
(insul)
RETURN TO INDEX
Electric Circuit
Ohm's Law: Resistance=EMF/Current
Comments
VARIABLE
SYMBOL
SI UNITS
*
EMF
V or E
I
volt
ampere
V
A
α
Ω
S
S/m
V/m
Current
2
Current Density
Resistance
Conductance
Conductivity
Electric Field Intensity
Susceptibility
Permittivity
Charge
Charge Quantity
Energy
Power
Resistivity
Capacitance
Inductance
Impedance
Admittance
Susceptance
Reactance
Resistivity
R
G
δ
E
η
ε
e
Q
E
P or W
ρ
C
L
Z
Y
B
X
ρ
amps/meter
ohm
Siemens (mho)
Siemens/meter
volts/meter
coulomb/volt-m
Farad/meter
electron volt
coulomb
joule
watt
ohm-meter
Farad
Henry
ohm
Siemen
Siemen
ohm
ohm-meter
Also equivalent to one joule/coulomb.
One amp represents 6.24x10 18 electrons past a point in one second.
The resistance that results in one amp to flow through a circuit device with a potential of one volt across it
The reciprocal of resistance.
Also referred to as Electric Field Strength
C/Vm
F/m
Ev The charge of one electron.
C
J Energy is the capacity for doing work.
W Power is the rate at which work is performed or energy expended. Also one joule/second.
Ω-m The resistivity is one ohm-meter when one amp flows through a one meter conductor with one volt applied
F Also one coulomb/volt
H Also one volt-sec/amp
Ω
S Reciprocal of Impedance
S If resistance is zero, susceptance is the reciprocal of reactance. Formerly mhos.
Ω
Ω-m
* Abbreviations
Magnetic Circuit
Rowland's Law: Reluctance=MMF/Flux
VARIABLE
SYMBOL
MMF
F
Φ
B
R
P
μ
H
Flux
Flux Density
Reluctance
Permeance
Permeability
Magnetic Field Intensity
Reluctivity
Comments
SI UNITS
*
amp-turn
weber
Wb
tesla
T
amp-turn/weber
weber/amp-turn
tesla-meter/amp
amps/meter
meters/henry
ν
F=H x L = (amps/meter) x meters = amps Also, F=N x I amp-turns. 1 Amp-turn=1.257 Gilberts.
flux,Φ, webers = B x A = (E) x (Time) Therefore, webers = volt-secs. 1 Weber=108 Maxwells.
B = Φ/area, teslas Therefore, teslas = webers/meter2 = 104 gauss
Reluctance is the magnetic analog of electrical resistance, but also changes with permeability. R=MMF/F
Reciprocal of Reluctance: P = 1/R
Absolute permeability, μ=B/H = Φ/HA Permeability of Free Space, μο= 1.257x10−6 henrys/meter
Actually, H = (N x I)/L N=# turns, I=amps, and L=length (amp-turns/meter). 1 A-T=0.01257 Oersteds.
Reciprocal of Permeability
* Abbreviations
Comments
Electromagnetic
VARIABLE
SYMBOL
Electric Field Intensity
E
H
volts/meter
amps/meter
*
V/m
A/m
f
λ
watts/meter2
Hertz
meters
W/m2
Hz
λ
Magnetic Field Intensity
EM Field Strength
Frequency
Wavelength
SI UNITS
* Abbreviations
© Copyright 2003-2008
XL Technologies, Inc. All Rights Reserved.