ET103-121807-01 Click on topic below to jump to desired worksheet. 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 Please do not make illegal copies - Each CD contains a unique, hidden serial number. 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 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. RETURN TO INDEX 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). RETURN TO INDEX 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). RETURN TO INDEX 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.