*Impedance* is the opposition to an ac current.

In a series circuit, impedance is the vector sum of resistance and reactance
(capacitive reactance,
inductive reactance).
Let `Z`

be the impedance, `V`

the peak or root mean square (RMS)
voltage as a vector,
and `I`

be the peak or RMS current as a vector.
Then

` ````
Z = V/I
```

As in the notes for LCR series circuit, let
the peak or RMS voltage through a resistor be `V`

,
the peak or RMS voltage across an inductor be _{R}`V`

,
and the peak or RMS voltage across the capacitor be _{L}`V`

.
The total peak or RMS voltage, _{C}`V`

, can calculated as shown below
_{T}

` ````
|V
```_{T}| = √(|V_{R}|^{2}+|V_{L}-V_{C}|^{2})
|V_{T}|/|I| = √(|V_{R}|^{2}/|I|^{2}+|V_{L}/I-V_{C}/I|^{2})

From the notes on capacitive reactance, `X`

,
inductive reactance, _{C}`X`

, and
Ohm's law, we have
_{L}

` ````
X
```_{C} = V_{C}/I
X_{L} = V_{L}/I
R = V_{R}/I

So

` ````
|V
```_{T}|/|I| = √(|V_{R}|^{2}/|I|^{2}+|V_{L}/I-V_{C}/I|^{2})
= √(R^{2}+|X_{L}-X_{C}|^{2})
|Z| = √(R^{2}+|X_{L}-X_{C}|^{2})

Note that using complex numbers, we can write:

` ````
V = V
```_{R}+j(V_{L}-V_{C})
Z = V/I
= V_{R}/I+j(V_{L}-V_{C})/I
= R+j(X_{L}-X_{C})
= R +j(ωL-1/(ωC))

For the phase, i.e. difference between the maximum current and voltage, we have:

` ````
tan θ = (X
```_{L}-X_{C})/R

In a RC series circuit, `X`

, gives
_{L}=0

` ````
tan θ = -X
```_{C}/R
= -1/(RωC)

In a RL series circuit, `X`

, gives
_{C}=0

` ````
tan θ = X
```_{L}/R
= ωL/R

A LCR circuit is

*capacitively reactive*if`X`

_{C}>X_{L}*inductively reactive*if`X`

_{C}<X_{L}*resonant*if`X`

_{C}=X_{L}

Fischer-Cripps. A.C., *The Electronics Companion.* Institute of Physics, 2005.

Copyright © 2014 Barry Watson. All rights reserved.