Impedance is the opposition to an ac current.
In a series circuit, impedance is the vector sum of resistance and reactance
Z be the impedance,
V the peak or root mean square (RMS)
voltage as a vector,
I be the peak or RMS current as a vector.
Z = V/I
As in the notes for LCR series circuit, let
the peak or RMS voltage through a resistor be
the peak or RMS voltage across an inductor be
and the peak or RMS voltage across the capacitor be
The total peak or RMS voltage,
VT, can calculated as shown below
|VT| = √(|VR|2+|VL-VC|2) |VT|/|I| = √(|VR|2/|I|2+|VL/I-VC/I|2)
From the notes on capacitive reactance,
Ohm's law, we have
XC = VC/I XL = VL/I R = VR/I
|VT|/|I| = √(|VR|2/|I|2+|VL/I-VC/I|2) = √(R2+|XL-XC|2) |Z| = √(R2+|XL-XC|2)
Note that using complex numbers, we can write:
V = VR+j(VL-VC) Z = V/I = VR/I+j(VL-VC)/I = R+j(XL-XC) = R +j(ωL-1/(ωC))
For the phase, i.e. difference between the maximum current and voltage, we have:
tan θ = (XL-XC)/R
In a RC series circuit,
tan θ = -XC/R = -1/(RωC)
In a RL series circuit,
tan θ = XL/R = ωL/R
A LCR circuit is
Fischer-Cripps. A.C., The Electronics Companion. Institute of Physics, 2005.
Copyright © 2014 Barry Watson. All rights reserved.