electronics toolkit v103 - digilab corfu - start...

ET103-121807-01

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Calculation of resistance, voltage, current and power for parallel circuits

Kirchhoff's Voltage and Current Laws, Superposition, Thevenin, Norton and Millman Theorems

Calculation of resistance, voltage, current and power for series circuits

Ohm's Law for a.c. circuits (voltage, current, impedance, power, power factor)

Calculation of RC and L/R time constants

Capacitance

Time Constants

Calculation of rms, peak, peak-to peak, average voltage/current, frequency, period, wavelength

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

Low pass, high pass, band pass (constant-k, m-derived), resonant filter

Rectangular coordinates, polar coordinates, rectangular-to-polar conversion, polar-to-rectangular conversion

Filters

Complex Math for A.C.

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

Basic Antennas

Component Data

Magnetic flux, magnetic field intensity, permeability, series magnetic circuit, hysteresis

Calculation of power, voltage, and current gain/loss

Magnetic Circuits

Decibels

Impedance, inductance, capacitance, attenuation for coax and ladder transmission lines

Units, symbols, and definitions for electric, magnetic, and electromagnetic variables

Transmission Lines

Basic Units & Conversions

Coil Winding (air core)

Basic Formulas (d.c.)

Basic Formulas (a.c.)

Basic Series Circuits

Basic Parallel Circuits

Networks

Alternating Current/Voltage

Inductance

General Notes:1. The Toolkit worksheets are set to a default screen resolution of 800x600 pixels. For other screen resolutions, click on 'View' and set 'Zoom' at the desired percentage for best viewing.2. For best results when printing worksheets, set printer resolution at 600dpi if available on your printer. For draft quality, set printer resolution to 300dpi.3. Version 1.0.3 01-01-2008

© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserve

TOPIC DESCRIPTION

Ohm's Law for d.c. circuits (voltage, current, resistance, power)

Coil Winding (toroids) Calculation of inductance, capacitance, resonant frequency, no. of turns for toroid core single layer coils

Series/parallel resonance, resonant frequency, inductive/capacitive reactance, Q-factor, bandwidth

Calculation of inductance, capacitance, resonant frequency, no. of turns for air core single/multi-layer coils

Resonance

Voltage, E 1.00 VCurrent, I 1.00 A

Resistance, R 1.00 ohms

Power, P 1.00 WCurrent, I 1.00 A


Voltage, E 1.00 VPower, P 1.00 W


Current, I 1.00 AResistance, R 1.00 ohms

Voltage, E 1.00 V

Power, P 1.00 WResistance, R 1.00 ohms

Voltage, E 1.00 V

Current, I 1.00 APower, P 1.00 W

Voltage, E 1.00 V

Voltage, E 1.00 VResistance, R 1.00 ohms

Current, I 1.00 A

Power, P 1.00 WResistance, R 1.00 ohms

Current, I 1.00 A

Voltage, E 1.00 VPower, P 1.00 W

Current, I 1.00 A

Voltage, E 1.00 VResistance, R 1.00 ohms

Power, P 1.00 W

Voltage, E 1.00 VCurrent 1.00 A

Power, P 1.00 W

Current, I 1.00 AResistance, R 1.00 ohms

Power, P 1.00 W

© Copyright 2003-2008 XL Technologies, Inc. All Rights Reserved.

Space For User Notes:

RETURN TO INDEXEnter values and units of measurement in gray cells. Calculated results are displayed in yellow cells.

Ohm's Law - Calculate Power

NOTES

Ohm's Law - Calculate Resistance

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.

CALCULATIONS FORMULAS

Ohm's Law - Calculate Voltage

Ohm's Law - Calculate Current

Practical Units and Conversions:Coulomb = 6.25 x 1018 electrons.Ampere = coulomb/secondVolt = joule/coulombWatt = joule/secondOhm = volt/ampereSiemens* = ampere/volt*Originally the 'mho' for conductance.

ER

I=

2

PR

I=

2ER

P=

E IR=

E PR=

PE

I=

EI

R=

PI

R=

PI

E=

2EP

R=

P EI=

2P I R=

Voltage, E 1.00 VCurrent, I 1.00 A

Impedance, Z 1.00 ohms

PF, cos θ 1.00 (no units)Power, P 1.00 WCurrent, I 1.00 A


PF, cos θ 1.00 (no units)Voltage, E 1.00 VPower, P 1.00 W


Current, I 1.00 AImpedance, Z 1.00 ohms

Voltage, E 1.00 VPF, cos θ 1.00 (no units)Power, P 1.00 W

Impedance, Z 1.00 ohmsVoltage, E 1.00 V

PF, cos θ 1.00 (no units)Current, I 1.00 APower, P 1.00 W

Voltage, E 1.00 V

Voltage, E 1.00 VImpedance, Z 1.00 ohms

Current, I 1.00 A

PF, cos θ 1.00 (no units)Power, P 1.00 W

Impedance, Z 1.00 ohmsCurrent, I 1.00 A

PF, cos θ 1.00 (no units)Voltage, E 1.00 VPower, P 1.00 W

Current, I 1.00 A

PF, cos θ 1.00 (no units)Voltage, E 1.00 V

Impedance, Z 1.00 ohmsPower, P 1.00 W

PF, cos θ 1.00 (no units)Voltage, E 1.00 VCurrent, I 1.00 APower, P 1.00 W

PF, cos θ 1.00 (no units)Current, I 1.00 A

Impedance, Z 1.00 ohmsPower, P 1.00 W


Resistance, R = Z cos θcos θ = R/Z

Phase Angle, θ = cos-1(R/Z)

Reactance, X = Z sin θsin θ = X/Z

Phase Angle, θ = sin-1 (X/Z)

Note: See Series and Parallel Circuits work sheets to calculate values for a.c. impedance, Z. and the phase angle, θ.

Ohm's Law - Calculate Current

Ohm's Law - Calculate Impedance

Ohm's Law - Calculate Voltage

RETURN TO INDEX

NOTESCALCULATIONS FORMULAS

Apparent Power, Papp = EI (volt-amps)Real Power, Preal = EI cos θ (watts)Reactive Power, Preactive=EI sin θ (VAR)Power factor, PF = cos θ = Preal/Papp

Phase Angle, θ = cos-1(Preal/Papp)

Ohm's Law - Calculate Power

DEFINITIONS:Voltage (E or V) - Generally, the voltage in a.c. circuits is the 'root mean squared' (RMS) or 'effective' voltage, measured in volts (V).Current (I) - Similarly, the current in a.c. circuits is the RMS value or effective value (equivalent d.c. value), measured in amperes (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.

EZ

I=

2cos

PZ

I θ=

2cosE

ZP

θ=

E IZ=

cos

PZE

θ=

cos

PE

I θ=

EI

Z=

cos

PI

Z θ=

cos

PI

E θ=

2cosE

PZ

θ=

cosP EI θ=

2cosP I Z θ=

Resistance, R 100.0 ohmsReactance, X 100.0 ohms

Impedance, Z 141.4 ohmsPhase Angle 45.00 degrees



Reactance, XL 30.0 ohmsReactance, XC 31.0 ohmsImpedance, Z -1.0 ohmsPhase Angle -90.00 degrees

Resistance, R 20.0 ohmsReactance, XL 20.0 ohmsReactance, XC 20.0 ohmsImpedance, Z 20.0 ohmsPhase Angle 0.00 degrees

Inductance 643.06 uHFrequency 11.130 kHzReactance 44.97 ohms

Capacitance 0.32 uFFrequency 11.130 HzReactance 44.97 kilohms

Resistance 1 2.000 ohmsResistance 2 2.000 ohmsResistance 3 2.000 ohmsResistance 4 2.000 ohmsResistance 5 2.000 ohmsResistance 6 2.000 ohms

Total 12.000 ohms


θ=0 when XL = XC (resonance)

Note: If the series circuit contains less than six resistors, enter 0 for the remaining resistances.

SERIES CIRCUITSL is the inductance in HenriesXL is the inductive reactance in OhmsF is the frequency in HertzXC is the capacitive reactance in OhmsZ is the impedance in Ohmsθ is the phase angle in degreesR is the resistance in Ohms

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).

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).

CALCULATIONS FORMULAS NOTES

L & C in Series

R, L, & C in Series

Inductive Reactance

Enter values and units of measurement in gray cells. Calculated results are displayed in yellow cells. RETURN TO INDEX

R & L in Series

R & C in Series

Capacitive Reactance

Series Resistance

2 2

LZ R X= +

L CZ X X= −

2 2( )L CZ R X X= + −

2 2

CZ R X= +

arctan LX

Rθ =

arctan CX

Rθ =

arctan L CX X

Rθ −

=

2LX fLπ=

1

2CX

fCπ=

1 2 3 ...TR R R R Rn= + + +



Resistance, R 3300.0 ohmsReactance 2530.0 ohms


Reactance, XL 365.0 ohmsReactance, XC 365.0 ohmsImpedance, Z MAX ohmsPhase Angle 0.00 degrees

Resistance, R 2200.0 ohmsReactance, XL 770.0 ohmsReactance, XC 535.0 ohmsImpedance, Z 1371.0 ohmsPhase Angle 51.45 degrees

(A)Resistance, R1 100.0 ohmsResistance, R2 100.0 ohmsReactance, XL 100.0 ohmsReactance, XC 500.0 ohmsImpedance, Z 161.2 ohmsPhase Angle 29.74 degrees (B)

Impedance, Z ohmsPhase Angle degrees

Impedance, Z ohmsPhase Angle degrees (C)

Inductance, L 643.06 uHFrequency, f 11.130 kHz

Reactance, XL 44.97 ohms

Capacitance, C 0.32 FFrequency, f 11.130 Hz

Reactance, XC 0.04 ohms

Resistance 1 1.000 ohmsResistance 2 1.000 ohmsResistance 3 1.000 ohmsResistance 4 0.000 ohmsResistance 5 2.000 ohmsResistance 6 2.000 ohms

Total 0.250 ohms


R2 & C in Parallel with L - Case (C)

PARALLEL CIRCUITSL is the inductance in HenriesXL is the inductive reactance in OhmsF is the frequency in HertzXC is the capacitive reactance in OhmsZ is the impedance in Ohmsθ is the phase angle in degreesR is the resistance in Ohms

If XL - XC is positive, the circuit is inductive.If XL - XC is negative, the circuit is capacitive.

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).



Note: Diagrams (B) & (C) above are special cases of (A). For (B), enter "0" for Resistance R2. For (C), enter "0" for Resistance R1.

θ=00 when XL = XC (resonance)

Parallel Resistance


R1&L in Parallel with R2&C - Case (A)

Inductive Reactance

R1 & L in Parallel with C - Case (B)

R & L in Parallel

R & C in Parallel

L & C in Parallel

R, L, & C in Parallel

2 2

* C

C

R XZ

R X=

+

2 2

* L

L

R XZ

R X=

+

*L C

L C

X XZ

X X=

−

arctanL

R

Xθ =

arctanC

R

Xθ =

( )arctan

*

L C

L C

R X X

X Xθ

⎛ ⎞−= ⎜ ⎟

⎝ ⎠

2LX fLπ=1

2CX

fCπ=

1 2

1

1 1 1...

T

n

R

R R R

=+ +

2 2 2 2

1 2

2 2

1 2

( )( )

( ) ( )

L C

L C

R X R XZ

R R X X

+ +=

+ + −

2 2 2 21 2 1

2 2 2 2

1 2 2 1

( ) ( )tan

( ) ( )

L C C L

C L

X R X X R X

R R X R R Xθ − + − +

=+ + +

2 2

1

2 2

1( )

LC

L C

R XZ X

R X X

+=

+ −

2 21 1

2

1

tan L C L

C

X X X R

R Xθ − − −

=

2 2

2

2 2

2 ( )

CL

L C

R XZ X

R X X

+=

+ −2 2

1 2

2

2

tan L C C

L

X X X R

R Xθ − − −

=

2 2 2 2

* *

( )

L C

L C L C

R X XZ

X X R X X=

+ −


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

Kirchhoff's Voltage LawThe 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.

DEFINITIONS NOTES

Kirchhoff's Current LawThe 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.

Thevenin's Theorem for d.c (or a.c.) CircuitsAny 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.) CircuitsAny 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 TheoremAny 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 TheoremIn 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.

∑ =+++ 0...321 nEEEE

∑ =+++ 0...321 nIIII

Period, t 1 mSecFrequency, f 1 kHz

Frequency, f 1 kHzPeriod, t 0.001 Sec

Frequency, f 3.75 MhzWavelength, λ 80 Meters

Avg 123.000 V *Peak 193.233 V *

Peak-Peak 386.712 V *RMS 136.653 V * Degrees Rad SinΘ Voltage

00 0 0 0.0%Peak 120.000 uA * 450 π/4 0.707 70.7% rms

Peak-Peak 240.000 uA * 600 π/3 0.866 86.6%RMS 84.840 uA * 900 π/2 1 100.0% peakAvg 76.440 uA * 1800 π 0 0.0%

Peak-Peak 240.000 mV *RMS 84.720 mV *Avg 76.320 mV *

Peak 120.000 mV *

RMS 84.720 mA *Avg 76.163 mA *

Peak 119.794 mA *Peak-Peak 239.588 mA *

Phase Angle 10.00 DegreesVoltage, E 120.00 VCurrent, I 10.00 A

Power, PREAL 1181.769 WApparent Power 1200.000 VAReactive Power 208.378 VAR

PF, cos θ 0.985 (no units)

* For consistency, only the units in the top (gray) cells may be changed. All other cells correspond to units of top (gray) cell.


Calculate Power

Sine Wave Characteristics

Primary Relationships

peak = 0.500*peak-peak

avg = 0.899*rmspeak = 1.414*rms

peak-peak = 2.828*rms

rms = 0.707*peakavg = 0.637*peak

rms = 0.353*peak-peakavg = 0.318*peak-peak

rms = 1.111*avg

peak = 1.571*avgpeak-peak = 3.144*avg

peak-peak = 2.000*peak

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.



a.c. Voltage or Current

Wavelength

f is the frequency in Hertzt is the period in seconds

f is the frequency in Hertzt is the period in seconds

Frequency

Period

λ is the wavelength I metersC is the velocity of light (3x108 m/sec)f is the frequency in Hertz

Note: Conversion factors are for sinewaves only

1f

t=

1t

f=

C

fλ =

APPARENTP EI=

sinREACTIVEP EI θ=

cosREALP EI θ=

cosPF θ=

peakpeak V

VVrms 707.0

2==

peakpeakavg VVV 637.02==

π


Frequency, f 25.00 kHzReactance, XL 44.97 ohmsInductance, L 286.288 uH

Inductance, L 0.00 HFrequency, f 800.000 Hz


Inductance, L 107.86 uHReactance, XL 2.640 kilohmsFrequency, f 3895.50 kHz

Inductance, L 10.00 HCurrent, I 2.00 Amps

Energy Stored 20.00 Joules

Inductance 1 1.000 mHInductance 2 1.000 mHInductance 3 1.000 mHInductance 4 1.000 mHInductance 5 1.000 mHInductance 6 1.000 mH

Total 0.167 mH

Inductance 1 1.000 mHInductance 2 1.000 mHInductance 3 1.000 mHInductance 4 1.000 mHInductance 5 1.000 mHInductance 6 1.000 mH

Total 6.000 mH

Reactance 1 1.000 ohmsReactance 2 1.000 ohmsReactance 3 1.000 ohmsReactance 4 1.000 ohmsReactance 5 1.000 ohmsReactance 6 1.000 ohms

Total 6.000 ohms


Total 0.167 ohms


INDUCTIVE REACTANCE

Inductive Reactance

RETURN TO INDEX

Energy Stored Formula Variables:L is the inductance in HenriesXL is the inductive reactance in Ohmsf is the frequency in HertzW is the energy stored in JoulesZ is the impedance in OhmsV is the voltage in VoltsI is the current in AmpsR is the resistance in Ohms

Parallel Inductance

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.

Frequency


Inductance

Parallel Inductive Reactance

Series Inductive Reactance

Series Inductance

π= 2LX fL

π=

2

LXf

L

= 2(1/2)W LI

1 2 ...T nL L L L= + +

1 2

1

1 1 1...

T

n

L

L L L

=+ +

nT XXXX ...21 ++=

1 2

1

1 1 1...

T

n

X

X X X

=+ +

π=

2

LXL

f


Frequency, f 11.13 MHzReactance, XC 44.97 ohms

Capacitance, C 317.982 pF

Capacitance, C 317.98 pFFrequency, f 11.130 MHz


Capacitance, C 317.98 pFReactance, XC 44.970 ohmsFrequency, f 11.13 MHz

Capacitance, C 5.00 FVoltage, E 100.00 Volts

Energy Stored 25000.00 JoulesCharge, Q 500.00 Coulombs

Capacitance 1 1.000 uFCapacitance 2 1.000 uFCapacitance 3 1.000 uFCapacitance 4 1.000 uFCapacitance 5 1.000 uFCapacitance 6 1.000 uF

Total 0.167 uF

Capacitance 1 1.000 pFCapacitance 2 1.000 pFCapacitance 3 1.000 pFCapacitance 4 1.000 pFCapacitance 5 1.000 pFCapacitance 6 1.000 pF

Total 6.000 pF


Total 6.000 ohms


Total 0.167 ohms


RETURN TO INDEX

Series Capacitive Reactance

Parallel Capacitance

Formula Variables:C is the capacitance in FaradsXc is the capacitive reactance in Ohmsf is the frequency in HertzQ is the electric charge in CoulombsW is the energy stored in JoulesZ is the impedance in OhmsE is the voltage in VoltsI is the current in AmpsR is the resistance in Ohms

Series Capacitance

Charge & Energy Stored

Parallel Capacitive Reactance

Capacitance


CAPACITIVE REACTANCE


Frequency

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.

1 2

1

1 1 1...

T

n

X

X X X

=+ +

1 2 ...T nX X X X= + +

1 2...T nC C C C= + +

1 2

1

1 1 1...

T

n

C

C C C

=+ +

2(1/2)W CE=

Q CE=

1

2 C

fCXπ

=

1

2CX

fCπ=

1

2 C

CfXπ

=

Resistance, R 5 kilohmsCapacitance, C 1 pFTime Const, τ 0.005 uSec

Time Const, τ 0.005 uSecCapacitance, C 1 pFResistance, R 5 kilohms

Time Const, τ 1 uSecResistance, R 1 Ohms

Capacitance, C 1 uF

Resistance, R 1 OhmsInductance, L 1 uH

Time Const, τ 1 uSec

Time Const, τ 1 uSecInductance, L 1 uH

Resistance, R 1 Ohms

Time Const, τ 1 uSecResistance, R 1 OhmsInductance, L 1 uH


NOTES


RC & L/R TIME CONSTANTSt - The time constant in secondsL - the inductance in henriesC - The capacitance in faradsR - 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.

RC Time Constant

L/R Time Constant

CALCULATIONS FORMULAS

*R Cτ =

RC

τ=

CR

τ=

L

Rτ =

*L Rτ=

LR

τ=


Inductance, L 10.00 uHCapacitance, C 100.00 pF

Frequency, f 5.033 MHz

Inductance, L 11.13 uHFrequency, f 7.112 MHz

Capacitance, C 45.00 pF


Inductance, L 11.13 uH

Inductance, L 11.13 uHFrequency, f 7.112 MHz




Reactance, X 1.00 ohmsResistance, R 10.00 ohms

Series Q 0.10 (no units)Parallel Q 10.00 (no units)

Resonant Freq., fR 7.112 MHzQ-Factor 150.00 ohms

Delta f 0.047 MHzFrequency, f1 7.088 MHzFrequency, f2 7.136 MHz

Frequency, fr 1.00 ohmsBandwidth, Δf 10.00 ohms

Q-Factor 0.10 (no units)


Frequency

Inductance

Capacitance

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.


Formula Variables: L is the inductance, HenriesC is the capacitance, FaradsR is the resistance, OhmsX is the reactance (XL or XC), Ohmsf is the frequency, HertzQ is the ratio of X to R (no units)Z is the impedance, Ohms

Series RLC Circuit @ Resonance:Z = RXc = XLPhase Angle = 0Power Factor = 1Z = MinI = MaxVo = Min

Parallel RLC Circuit @ Resonance:Z = RXc = XLPhase Angle = 0Power Factor = 1Z = MaxI = Min.Vo = Max.

Bandwidth

Q Factor (Components)

(series circuits) (parallel circuits)

Q Factor (Resonant Circuit)


Inductive Reactance

1

2f

LCπ=

2 2

1

4C

f Lπ=

2 2

1

4L

f Cπ=

2LX fLπ=

1

2CX

fCπ=

LorC

LorC

X RQ

R X= =

1 2rf

f f fQ

Δ = = −

12

r

ff f

Δ= −

22

r

ff f

Δ= +

RfQ

f=

Δ


Inductance, L 11.13 uHCapacitance,C 45.00 pF






Frequency, f 13.5 MHzLoad 50 ohms

Cutoff Freq. 15.255 MHz

Inductance, L1 0.52 uHInductance, L2 0.52 uH

Capacitance, C1 208.66 pFCapacitance, C2 417.32 pFCapacitance, C3 208.66 pF


Half-Wave Filter Design (5-Pole)

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.

Low Pass Filters - Cutoff Frequency

Band Pass Filters - Center Frequency

High Pass Filters - Cutoff Frequency

1cutofff

LCπ=

1

4cutofff

LCπ=

1

2centerf

LCπ=

Coil Radius, r 1 inchesNo. of Turns, N 40 (no units)

Coil Length, l 1 inchesInductance, L 84.21 uH

Spacing 40 TPITyp. Wire Size 22 AWG

Coil Dia., d 2 inchesNo. of Turns, N 40 (no units)

Length of Coil, l 1 inchesInductance, L 84.21 uH

Spacing 40 TPITyp. Wire Size 22 AWG

Coil Radius, r 0.25 inches TPI TPI*Length of Coil, l 1 inches AWG enameled inches mm insulated

Inductance, L 8.16 uH 10 9.6 0.1019 2.59No. of Turns, N 39.99 (no units) 12 12.0 0.0808 2.05

Spacing 40.0 TPI 14 15.0 0.0641 1.63Wire Size 22 AWG 16 18.9 0.0508 1.29

17 21.2 0.0453 1.1518 23.6 0.0403 1.0219 26.4 0.0359 0.91

Coil Dia., d 0.5 inches 20 29.4 0.0320 0.81Length of Coil, l 1 inches 21 33.1 0.0285 0.72

Inductance, L 8.16 uH 22 37.0 0.0254 0.64No. of Turns, N 39.99 (no units) 23 41.3 0.0226 0.57

Spacing 40.0 TPI 24 46.3 0.0201 0.51Wire Size 22 AWG 25 51.7 0.0179 0.45

26 58.0 0.0159 0.4027 64.9 0.0142 0.3628 72.7 0.0126 0.32

Inductance, L 107.85 uH 29 81.6 0.0113 0.29Capacitance, C 6.77 pF 30 90.5 0.0100 0.25

Frequency, f 5.890 MHz *Depends on type of insulation

Coil Radius, r 0.55 inchesNo. of Turns, N 40Length of Coil, l 1 inchesDepth of Coil, b 0.1 inchesInductance, L 29.113 uH

Dia. Of Wire, d 0.001 cmLength of Wire, l 200 cmInduct. L (low freq) 2.061 uH

Induct. L (high freq) 1.961 uH


Diameter

Formula Variables:L is the inductance, Henries r is the coil radius, inchesd is the coil diameter, inchesl is the coil length, inchesN is the number of turnsb is the depth of coil winding for multi-layer coils*TPI is the number of turns per inchAWG is the American Wire Gauge standardC is the Capacitancef is the Frequency

* These formulas are based on short coils (i.e. length < 10x diameter of coil).

Number of turns required for a coil based on radius, length, and inductance.

Number of turns required for a coil based on diameter, length, and inductance.

Inductance of a coil based on diameter , length, and number of turns.

Inductance of a coil based on radius , length, and number of turns.

Copper Wire Table

Inductance, Straight Wire

Multi-Layer Coil (based on radius) Inductance of a multi-layer coil based on radius, number of turns, length, and depth of coil.

1 inch = 2.54 cm 1 cm = 0.3937 in. 1 meter = 39.37 in

Coil Inductance (based on dia.)

Resonant Frequency

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).

Number of Turns (based on dia.)

Coil Inductance (based on radius)

Number of Turns (based on radius)



2 2

18 40

d NL

d=

+ l

2 2

9 10

r NL

r=

+ l

(9 10 )L rN

r

+=

l

(18 40 )L dN

d

+=

l

1

2f

LCπ=

20.8( )

6 9 10

rNL

r b=

+ +l

20.002 log 0.75

/2lowfreqL

d

⎡ ⎤= −⎢ ⎥⎣ ⎦l

l

20.002 log 1.00

/2highfreqL

d

⎡ ⎤= −⎢ ⎥⎣ ⎦l

l

Mix 0 1 2 3 6 7 10 12 15 17 26Frequency

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/WhiteMaterial Phenolic Carbonyl

CCarbonyl

ECarbonyl

HPCarbonyl

SFCarbonyl

THCarbonyl

WSynthetic

OxideCarbonyl

GS6 Carbonyl Hydrogen Reduced

u 1 20 10 35 8.5 9 6 4 25 4 75Temp

Stability ppm/0C

0 280 95 370 35 30 150 170 190 50 825

Core Size/Mix 0 1 2 3 6 7 10 12 15 17 26T-12 3.0 48 20 60 17 18 12 7.5 50 7.5 -T-16 3.0 44 22 61 19 - 13 8 55 8 145T-20 3.5 52 27 76 22 24 16 10 55 10 180T-25 4.5 70 34 100 27 29 19 12 85 12 235T-30 6.0 85 43 140 36 37 25 16 93 16 325T-37 4.9 80 40 120 30 32 25 15 90 15 275T-44 6.5 105 52 180 42 46 33 18.5 160 18.5 360T-50 6.4 100 49 175 40 43 31 18 135 18 320T-68 7.5 115 57 195 47 52 32 21 180 21 420T-80 8.5 115 55 180 45 50 32 22 170 22 450T-94 10.6 160 84 248 70 - 58 32 200 - 590T-106 19.0 325 135 450 116 133 - - 345 - 900T-130 15.0 200 110 350 96 103 - - 250 - 785T-157 - 320 140 420 115 - - - 360 - 870T-184 - 500 240 720 195 - - - - - 1640T-200 - 250 120 425 100 105 - - - - 895T-225 - - 120 425 100 - - - - - -

T-225A - - 215 - - - - - - - -T-300 - - 115 - - - - - - - -T-400 - - 185 - - - - - - - -

T-400A - - 360 - - - - - - - -

Inductance, L 8.16 uHuH/100Turns, AL 300.00 *

No. Turns, N 16.492 *

No. of Turns, N 16.49 *uH/100Turns, AL 300.00 *


Inductance, L 8.16 uHCapacitance, C 200.00 pF

Frequency, f 3.940 mHz

* no units


CALCULATIONS NOTESFORMULAS

Resonant Frequency

Number of Turns

Inductance

Formula Variables:L is the inductance, Henries N is the number of turnsAL is the inductance in uH per 100 turns (See table)AWG is the American Wire Gauge standardC is the capacitance in Faradsf is the frequency in Hertz

IRON POWDER TOROID CORES, uH PER 100 TURNS


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.

1

2f

LCπ=

100L

LN

A=

2

410

L

NL A

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Frequency, f 7.100 MHzWavelength 42.25 Meters

1/4 Wavelength 32.96 Feet1/4 Wavelength 10.05 Meters

Frequency, f 7.040 MHzWavelength 42.61 Meters

Length Per Side 32.96 Feet1/2 Wavelength 65.92 Feet1/2 Wavelength 20.27 Meters

Frequency, f 14.020 MHzWavelength 21.40 MetersWavelength 66.76 Feet

Director, DI, 0.45λ 30.04 FeetSpacing, DD, 0.10λ 6.68 FeetDriven El., DE, 0.5λ 33.38 FeetSpacing, DR, 0.15λ 10.01 Feet

Reflector, RF, 0.55λ 36.72 Feet

Frequency, f 7.040 MHzVelocity Factor, V 0.98

Total Length, L 34.07 Feet

Height @ XMTR 328.0 feetHeight @ RCVR 29.5 feet

Total Range 33.19 miles


Line of Sight Propagation

Matching Transformer

Velocity Factors:Air Insulated Coax -0.85Ladder Line - 0.975Twin Lead - 0.82Polyethylene Coax-0.66

AntennasHertz 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/meterPermeability 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/secRadiation Resistance of Free Space, η0 = SQRT(μo/εo) = 377 Ω

CALCULATIONS NOTES


FORMULAS

3-Element Beam Antenna

Antenna Calculator(s)

1/4 Wave Vertical Antenna

1/2 Wave Dipole Antenna

Dimensions in Feet

Dimensions In Meters

Director: DISpacing: DDDriven Element: DESpacing: DRReflector: RF

Dimensions in Feet

Dimensions In Meters

1.41*( )T RD H H= +

3.6 *( )T RD H H= +

246 *VL

f=

0.45*DI λ=

2341/ 4

fλ =

71.341/ 4

fλ =

4681/2

fλ =

142.681/2

fλ =

0.10 *DD λ=0.5*DE λ=0.15*DR λ=0.55*RF λ=

Dia. Of Conductors, d 0.1305 inchesCtr-Ctr Distance, D 10.00 inches

Rel Permittivity, ε 1.00Impedance, Z 603.176 ohms


Length, l 1.00 feetRel Permittivity, ε 1.00Capacitance, C 1.684 pF


Length, l 1.00 feetRel. Permeability, μ 1.00



Frequency, f 400.00 mHzLength, l 100.00 feet

Attenuation 0.220 dB

Dia. of Inner Cond., d 0.108 inchesDia. Of Outer Cond., D 0.41 inches

Rel Permittivity, ε 2.30Impedance, Z 52.234 ohms


Length, l 1.00 feetRel Permittivity, ε 2.30Capacitance, C 29.490 pF


Length, l 1.00 feetRel. Permeability, μ 2.30



Frequency, F 400.00 mHzLength, l 100.00 feet

Attenuation 0.188 dB


Capacitance, C is in pFInductance, L is in uHFrequency, F is in MHzLength, l is in feetDistance, Diameter are in inchesImpedance, Z is in ohms

Coax - Impedance

Coax - Capacitance

Coax - Inductance

Coax - Attenuation

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. Fothis 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).



Permittivity (Dielectric Const), εair=1.0teflon=2.1glass=7.6mica=7.5plexiglas=2.6 - 3.5polystyrene=2.4 - 3.0Permeability, μnon-ferrous=1.0Two Parallel Lines - Attenuation

Two Parallel Lines - Impedance

Two Parallel Lines - Capacitance

Two Parallel Lines - Inductance

ε=

276 2log

DZ

d

ε=

3.68

2log

CD

d

l

μ=2

0.281 logD

Ld

l

ε=

138log

DZ

d

ε=

7.36

log

CD

d

l

μ= 0.140 logD

Ld

l

−+= 64.6 ( )

* *10

( * )log

F D ddB

DD d

d

l

( )( )( )-53.14 f

dB = 102D

d logd

l

Power In 0.001 WPower Out 100 WLoss/Gain 50 dB

Voltage In 1 VVoltage Out 50 VLoss/Gain 33.9794 dB

Current In 100 ACurrent Out 1 ALoss/Gain -40 dB

Power In 1 mWPower Out 100 mWLoss/Gain 20 dBm

Power In 1 WPower Out 100000 WLoss/Gain 50 dBw

Voltage In 1 uVVoltage Out 50 uVLoss/Gain 33.9794 dBμV


Current (gain/loss) dB is the current gain or loss in decibelsIout is the output current in ampsIin is the input current in ampslog is the logarithm to the base 10

Power (gain/loss) 1mW 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 WattsPin is the input power @ 1 mWattlog is the logarithm to the base 10

Voltage (gain/loss) dB is the voltage gain or loss in decibelsVout is the output voltage in voltsVin is the input voltage in voltslog is the logarithm to the base 10

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.



Note: Input and Ouput values must be in the same units. Therefore, in the following calculations, the output unitsare automatically adjusted based on the units selected for the input power, voltage, or current.

Voltage (gain/loss) 1 μV

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 WattsPin is the input power @ 1 Wattlog 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 μVoltsVin is the input voltage @ 1 μVlog is the logarithm to the base 10

Power (gain/loss) 1 Watt

Power (gain/loss) dB is the power gain or loss in decibelsPout is the output power in WattsPin is the input power in wattslog is the logarithm to the base 10

10log out

in

PdB

P=

20log out

in

VdB

V=

20log out

in

IdB

I=

10log1

outm

PdB

mW=

10log1

outW

PdB

W=

20log1

outVdB V

Vμ

μ=

Current 20.0 ampsNo. of Turns 10.0

Length, l 0.5 metersMag. Field Intensity, H 400.00 amp-t/meter

Flux, Φ 20.0 webersArea 10.0 meters

Mag. Flux Density, B 2.00 teslas

Mag. Flux Density,B 20.0 teslasArea 10.0 meters

Flux, Φ 20.0 webers

Mag. Flux Density,B 20.0 teslasMag. Field Intensity,H 10.0 amp-t/meter

Permeability, μ 2.00 tesla-m/amp

Pri. Voltage 20.0 voltsPri. Turns 10.0 turns

Sec. Voltage 10.0 voltsSec. Turns 5.00 turns

Pri. Current 20.0 ampsPri. Turns 10.0 turns

Sec. Current 10.0 ampsSec. Turns 5.00 turns

Pri. Imped. 20.0 ohmsPri. Turns 10.0 turns

Sec. Imped. 10.0 ohmsSec. Turns 50.00 turns

Inductance, L1 10.0 HenrysInductance, L2 1.0 Henrys

Coupling Factor, k 0.5Mutual Inductance 1.58 Henrys

LT, Series Aiding 14.16 HenrysLT, Series Oppose 7.84 HenrysLT, Parallel Aiding 0.96 Henrys

LT, Parallel Oppose 0.53 Henrys


Series Aiding

Series Opposing

Parallel Aiding

Parallel Opposing

Magnetic Field Intensity

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)

Mutual Inductance L1 is inductance of first coil in HenrysL2 is inductance of second coil in HenrysLT is total inductance in HenrysM is the mutual inductance in Henrys

B is the magnetic flux density in teslas (webers/meter2)H is the magnetic field intensity, amp-turns/meterμ is the Permeability in tesla-meter/amp

Permeability

Magnetic Flux, Φ 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)

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.



H is the magnetic field intensity, amp-turns/meterI is the current, ampsl is the length, metersN is the number of turnsNote: For magnetic field intensity in oersteds, multiply amp-turms/meter by 0.01257

Magnetic Flux Density

Transformer Impedance Ratio Z1 is the transformer primary impedanceZ2 is the transformer secondary impedanceN1 is the number of turns on the primaryN2 is the number of turns on the secondary

Transformer Voltage RatioV1 is the voltage on the transformer primaryV2 is the voltage on the transformer secondaryN1 is the number of turns on the primaryN2 is the number of turns on the secondary

Transformer Current Ratio I1 is the current on the transformer primaryI2 is the current on the transformer secondaryN1 is the number of turns on the primaryN2 is the number of turns on the secondary

*I NH =

l

1 1

2 2

V N

V N=

1 1

2 2

I N

I N=

2

1 1

2

2 2

Z N

Z N=

BA

φ=

B

Hμ =

BAφ =

1 22TL L L M= + +

1 2 2TL L L M= + −

1 2 2

1 2 2T

L L ML

L L M

−=

+

1 2 2

1 2 2T

L L ML

L L M

−=

−

Resistance, R 20.0 ohmsReactance, X -20.0 ohms

Impedance, Z 28.3 ohmsPhase Angle, θ -45.00 degrees

Impedance, Z 28.3 ohmsPhase Angle, θ -45.0 degreesResistance, R 20.0 ohmsReactance, X -20.0 ohms


Polar to Rectangular

Enter inductive reactance as positive and capacitive reactance as negative.

A positive reactance indicates inductance and a negative reactance indicates capacitance. 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/hypcos θ = adj/hyptan θ = opp/adjcot θ = adj/oppsec θ = hyp/adjcsc θ = hyp/opp

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=Z1

2/2θ1

Square Root: Z1^0.5=Z1^0.5/θ1/2

Rectangular to Polar

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.



R jX Z θ+ = ∠

2 2Z R X= +

1tan

X

Rθ − ⎛ ⎞= ⎜ ⎟

⎝ ⎠

cosR Z θ=

sinX Z θ=

Z R jXθ∠ = +

Length, l 1000 metersArea, S 0.001 meters2

Resistivity, r 1.72E-08 Ω-metersResistance, R 0.017 ohms

Diameter, d 0.1 meters Area, S 0.01 meters2

Initial Resistance 100 ohmsInitial Temp, T1 80 degreesFinal Temp, T2 120 degrees

Temp. Coef., α 0.00393Final Resistance 115.7 ohms

1st 2nd M Tolerance FailureRate

Black 0 100

Brown 1 1 101 ±1% 1 silver 1.46E-08 8.782E-14 3.80E-03Red 2 2 102 ±2% 0.1 copper 1.72E-08 1.037E-13 3.93E-03

Orange 3 3 103 - 0.01 aluminum 2.83E-08 1.702E-13 3.90E-03Yellow 4 4 104 - 0.001 tungsten 5.50E-08 3.308E-13 4.50E-03Green 5 5 105 ±0.5% nickel 7.80E-08 4.692E-13 6.00E-03Blue 6 6 106 ±0.25% iron 1.20E-07 7.218E-13 5.50E-03

Violet 7 7 107 ±0.1% constantan 4.90E-07 2.947E-12 8.00E-07Gray 8 8 ±0.05%White 9 9Gold 10-1 ±5%Silver 10-2 ±10%

AWG TPI (enam) Dia (inches) Dia (mm) TPI* (insul)

10 9.6 0.1019 2.59In the Resistor Color Code Chart, 12 12.0 0.0808 2.05the values in Column "M" are multipliers 14 15.0 0.0641 1.63

16 18.9 0.0508 1.2917 21.2 0.0453 1.1518 23.6 0.0403 1.0219 26.4 0.0359 0.9120 29.4 0.0320 0.8121 33.1 0.0285 0.7222 37.0 0.0254 0.6423 41.3 0.0226 0.5724 46.3 0.0201 0.5125 51.7 0.0179 0.4526 58.0 0.0159 0.4027 64.9 0.0142 0.3628 72.7 0.0126 0.3229 81.6 0.0113 0.2930 90.5 0.0100 0.25

*Depends on type of insulation


Copper Wire Table

ConductorResistivity, ρ

ohm-m @ 200 C

Resistivity, ρ ohm-cmil/ft @

200 C

Temp.Coeficient

α

COMPONENT DATAResistance 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).



S is the cross sectional area in meters2

π is a constant 3.14d is the diameter of the conductor in meters

Resistance of a Conductor

Temperature Characteristics

Cross sectional Area of Conductor

R is the conductor's resistance in ohmsρ is the resistivity of the conductor in Ω-metersl is the length of the conductor in metersS is the cross sectional area in meters2

Thermal Resistance ChangesRfinal is the final resistance in ohmsRinitial is the initial resistance in ohmsT1 is the initial temperatureT2 is the final temperatureα is the temperature coeficient

Resistor Color Code

RS

ρ=l

2

4

dS π=

2 1[1 ( )]final initialR R T Tα= + −

VARIABLE SYMBOL SI UNITS *EMF V or E volt V

Current I ampere ACurrent Density amps/meter2 α

Resistance R ohm ΩConductance G Siemens (mho) SConductivity δ Siemens/meter S/m

Electric Field Intensity E volts/meter V/mSusceptibility η coulomb/volt-m C/VmPermittivity ε Farad/meter F/m

Charge e electron volt EvCharge Quantity Q coulomb C

Energy E joule JPower P or W watt W

Resistivity ρ ohm-meter Ω-mCapacitance C Farad FInductance L Henry HImpedance Z ohm ΩAdmittance Y Siemen S

Susceptance B Siemen SReactance X ohm ΩResistivity ρ ohm-meter Ω-m

*

VARIABLE SYMBOL SI UNITS *MMF F amp-turnFlux Φ weber Wb

Flux Density B tesla TReluctance R amp-turn/weberPermeance P weber/amp-turnPermeability μ tesla-meter/amp

Magnetic Field Intensity H amps/meterReluctivity meters/henry ν

*

VARIABLE SYMBOL SI UNITS *Electric Field Intensity E volts/meter V/m

Magnetic Field Intensity H amps/meter A/mEM Field Strength watts/meter2 W/m2

Frequency f Hertz HzWavelength λ meters λ

*


Energy is the capacity for doing work.Power is the rate at which work is performed or energy expended. Also one joule/second.The resistivity is one ohm-meter when one amp flows through a one meter conductor with one volt appliedAlso one coulomb/volt

The charge of one electron.

The resistance that results in one amp to flow through a circuit device with a potential of one volt across itThe reciprocal of resistance.

Also referred to as Electric Field Strength

Also equivalent to one joule/coulomb.One amp represents 6.24x1018 electrons past a point in one second.

Electric Circuit CommentsOhm's Law: Resistance=EMF/Current

Comments

Reciprocal of Permeability

Abbreviations

Abbreviations

RETURN TO INDEX

Electromagnetic

Also one volt-sec/amp

If resistance is zero, susceptance is the reciprocal of reactance. Formerly mhos.Reciprocal of Impedance

Magnetic CircuitRowland's Law: Reluctance=MMF/Flux Comments

Absolute permeability, μ=B/H = Φ/HA Permeability of Free Space, μο= 1.257x10−6 henrys/meter

Abbreviations

Actually, H = (N x I)/L N=# turns, I=amps, and L=length (amp-turns/meter). 1 A-T=0.01257 Oersteds.

Reluctance is the magnetic analog of electrical resistance, but also changes with permeability. R=MMF/FB = Φ/area, teslas Therefore, teslas = webers/meter2 = 104 gauss

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.

Reciprocal of Reluctance: P = 1/R

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