Hi Abhishekdas!
Abhishekdas said:
Is V always equal to iZ or is it true only for the rms and maximum values...
Abhishekdas said:
...SO all along are we always dealing with rms values? and so if V=iZ assuming rms values of i and Z or for instantaneous values also(which is my orginal question)...? it doesn't seem true for instantaneous values though...
Complex V always equals
complex I times Z.
Here's some notes I was making on impedance …
Complex voltage and current:
In a steady sinusoidal (AC) circuit of frequency \omega, the (instantaneous) voltage and current V\text{ and }I can always be written:
V =\ V_x\cos\omega t + V_y\sin\omega t and I =\ I_x\cos\omega t + I_y\sin\omega t
Then the complex voltage and complex current between any two points are the
constants defined as \bold{V} =\ V_x+jV_y\text{ and }\bold{I} =\ I_x+jI_y.
The complex number Z =\ \bold{V}/\bold{I} is called the impedance between those two points.
Similarly, dV/dt\text{ and }dI/dt can always be written:
dV/dt =\ V'_x\cos\omega t + V'_y\sin\omega t and dI/dt =\ I'_x\cos\omega t + I'_y\sin\omega t
Then the complex voltage derivative and complex current derivative between any two points are
constants defined as \bold{V}' =\ V'_x + jV'_y\text{ and }\bold{I}' =\ I'_x + jI'_y.
Obviously, V'_x\ =\ \omega V_y\text{ and }V'_y\ =\ -\omega V_x, and so \bold{V}' = j\omega\bold{V}. Similarly \bold{I}' = j\omega\bold{I}.
Resistors capacitors and inductors:
For ordinary voltage and current, Ohm's Law, and the capacitor and inductor laws, state:
V =\ RI,\ \ \ dV/dt =\ I/C,\ \ \ V =\ LdI/dt
For complex voltage and current, these become:
\bold{V} =\ R\bold{I},\ \ \ \bold{V}' =\ \bold{I}/C,\ \ \ \bold{V} =\ L\bold{I}'
which can be rewritten
without the derivatives as the fundamental complex rules:
\bold{V} =\ R\bold{I},\ \ \ \bold{V} =\ \bold{I}/j\omega C,\ \ \ \bold{V} =\ j\omega L\bold{I}
In other words: the impedance across a resistor capacitor and inductor are:
Z =\ R,\ \ \ Z =\ 1/j\omega C,\ \ \ Z =\ j\omega L
In a varying sinusoidal (AC) circuit of (fixed) frequency \omega, the coefficients V_x\ V'_x\ V_y\ V'_y\ I_x\ I'_x\ I_y\text{ and }I'_y are not constants, and the fundamental rules become:
\bold{V} =\ R\bold{I},\ \ \ j\omega C\bold{V} + Cd\bold{V}/dt =\ \bold{I},\ \ \ \bold{V} =\ j\omega L\bold{I} + Ld\bold{I}/dt
This can be dealt with by replacing the fixed real frequency \omega by a complex "s-plane" (Laplace transform) frequency s
Complex power:
Power = work per time = voltage times charge per time = voltage times current:
P = VI =\ V_{max}I_{max}\cos(\omega t + \phi/2)\cos(\omega t - \phi/2)
=\ V_{max}I_{max}(\cos\phi + \cos2\omega t)/2
=\ V_{rms}I_{rms}(\cos\phi + \cos2\omega t)
So the average power is the constant part, V_{rms}I_{rms}\cos\phi, to which is added a component varying with double the circuit frequency, V_{rms}I_{rms}\cos2\omega t (so a graph of the whole power is a sine wave shifted by a ratio \cos\phi above the x-axis).