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1996 -
2016
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Kirt
Blattenberger,
BSEE - KB3UON
RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The Internet was still largely an unknown entity at the time and not much was available in the form of WYSIWYG ...
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Here is an extensive table of impedance, admittance, magnitude, and phase angle equations (formulas) for fundamental series and parallel combinations of resistors, inductors, and capacitors. All schematics and equations assume ideal components, where resistors exhibit only resistance, capacitors exhibit only capacitance, and inductors exhibit only inductance.
For those unfamiliar with complex numbers, the "±j" operator signifies a phase of ±90°. Voltage across a capacitor lags the current through it by 90°, so -j is used along with its capacitive reactance (-j/ωC). Voltage across an inductor leads the current through it by 90°, so +j is used along with inductive reactance (jωL).
"M" is the mutual inductance between inductors.
"ω" is frequency in radians/second, and is equal to 2π times frequency in cycles/second.
This is probably one of the most comprehensive collections you will find on the Internet.
Z = R + jX |Z| = (R^{2} + X^{2})^{½} ϕ = tan^{-1}(X/R) Y = 1/Z
Circuit Configuration |
Impedance Z = R + jX |
Magnitude {Z} = (R^{2} + X^{2})^{½} |
Phase Angle ϕ = tan^{-1}(X/R) |
Admittance Y = 1/Z |
R | R | 0 | 1/R | |
jωL | ωL | +π/2 | -j/ωL | |
-j/ωC | 1/ωC | -π/2 | jωC | |
jω(L_{1}+L_{2}±2M) | ω(L_{1}+L_{2}±2M) | +π/2 | -j/[ω(L_{1}+L_{2}±2M)] | |
-(j/ω)(1/C_{1}+1/C_{2}) | (1/ω)(1/C_{1}+1/C_{2}) | -π/2 | jωC_{1}C_{2}/(C_{1}+C_{2}) | |
R+jωL | (R^{2}+ω^{2}L^{2})^{½} | tan^{-1}(ωL/R) | (R-jωL)/(R^{2}+ω^{2}L^{2}) | |
R-j/ωC | (1/ωC)(1+ω^{2}C^{2}R^{2})^{½} | -tan^{-1}(1/ωCR) | (R+j/ωC)/(R^{2}+1/ω^{2}C^{2}) | |
j(ωL-1/ωC) | (ωL-1/ωC) | ±π/2 | jωC/(1-ω^{2}LC) | |
R+j(ωL-1/ωC) | [R^{2}+(ωL-1/ωC)^{2}]^{½} | tan^{-1}[(ωL-1/ωC)/R] | ||
R_{1}R_{2}/(R_{1}+R_{2}) | R_{1}R_{2}/(R_{1}+R_{2}) | 0 | 1/R_{1}+1/R_{2} | |
+π/2 | ||||
-j/ω(C_{1}+C_{2}) | 1/ω(C_{1}+C_{2}) | -π/2 | jω(C_{1}+C_{2}) | |
ωLR/(R^{2}+ω^{2}L^{2})^{½} | tan^{-1}(R/ωL) | 1/R-j/ωL | ||
R(1-jωCR)/(1+ω^{2}C^{2}R^{2}) | R/(1+ω^{2}C^{2}R^{2})^{½} | -tan^{-1}(ωCR) | 1/R+jωC | |
jωL/(1-ω^{2}LC) | ωL/(1-ω^{2}LC) | ±π/2 | j(ωC-1/ωL) | |
[(1/R)^{2}+(ωC-1/ωL)^{2}]^{-½} | tan^{-1}[R(1/ωL-ωC)] | 1/R+j(ωC-1/ωL) | ||
Impedance Z | ||||
Magnitude |Z| | ||||
Phase Angle ϕ | ||||
Admittance | ||||
Impedance Z | ||||
Magnitude |Z| | ||||
Phase Angle ϕ | ||||
Admittance | ||||
Impedance Z | ||||
Magnitude |Z| | ||||
Phase Angle ϕ | ||||
Admittance | ||||
Impedance Z | ||||
Magnitude |Z| | ||||
Phase Angle ϕ | ||||
Admittance | ||||
Impedance Z | ||||
Magnitude |Z| | ||||
Phase Angle ϕ | tan^{-1}(X_{1}/R_{1})+tan^{-1}(X_{2}/R_{2})-tan^{-1}[(X_{1}+X_{2})/(R_{1}+R_{2})] | |||
Admittance | 1/(R_{1}+jX_{1})+1/(R_{2}+jX_{2}) |
Note: Corrections made to RLC Magnitude and Admittance formulas, and to RL||R Admittance formula on 7/3/2014. Thanks to Bob N. for catching the errors.
(source: Reference Data for Engineers, 1993)