What is Impedence? AC Circuits Explained

[Abhishekdas]

WHat is impedence...?

Homework Statement

This is really confusing me...In my book it is mentioned that in Ac circuits V_rms=i_rmsZ and V_max=i_maxZ where i and V are current and Voltage respectively...And it is also mentioned that in LCR ac circuits impedence act as ohmic resistors in dc circuits...Now my doubt is very simple...Is V always equal to iZ or is it true only for the rms and maximum values...

Looking forward to some help...thank you...

Homework Equations

The Attempt at a Solution

 

[Abhishekdas]

When i flipped a few pages back for a recap i saw that the impedence derivations for different circuits and for eg in an LR cicuits they are saying that V=V_r + jV_L...Here V_L and V_R stands for PD across Inductor and Resistor...NNow jV_l has been replaced by jiwL...but V isn't always equal to iwl...because in inductor circuits in ac the voltage leads current by 90 degrees...SO all along are we always dealing with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...

 

[jambaugh]

Just as when a circuit settles into steady state you have DC equations involving resistance
$$V=iR$$
when a circuit has settled into a sinusoidal periodic state you have AC equations involving impedance.
$$V= iZ$$
Here you are expressing complex voltage and complex current. The complex value is simply a means of combining in phase and out of phase components:
$$V = V_0 e^{j\omega t} = V_0[\cos(\omega t) + j\sin(\omega t)]$$
(note $V_0$ is also complex valued!)
The actual voltage at a given instant is the real component, but that initial voltage is also complex so this will be:
$$\Re(V)=\Re(V_0) \cos(\omega t) -\Im(V_0)\sin(\omega t)$$
Likewise with complex current.

Now understanding this complex format you can express the cyclic AC relationship between current and voltage using complex Impedance in place of DC's real Resistance.
$$Z=R+jX$$
So yes, it is not just the r.m.s. value but the complex value at any instant which satisfies:
$$V = iZ$$

To get the physical voltage and current take real components... but notice that with complex impedance the real voltage will be proportional to both real and imaginary components of current, and vice versa.

The r.m.s. values come from averaging the square of the real component over a cycle and taking the square root, "root mean square" but will in the sinusoidal AC case simply be sqrt(2)/2 times the magnitude of the complex value.

So the r.m.s. equation should read:
$$V_{rms} =i_{rms}|Z| =i_{rms} \sqrt{R^2 + X^2}$$
where |Z| is the magnitude of the complex impedance. The difference between using Z and |Z| should be glaring since Z is complex while the r.m.s. values are real values.

Finally, note that once you have the behavior of circuits for sinusoidal periodic cases of various frequencies, you can then analyze transient behavior using Fourier analysis.

 

[tiny-tim]

Hi Abhishekdas!

Is V always equal to iZ or is it true only for the rms and maximum values...

...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).

 

[Abhishekdas]

Thank you guys ...both of you...
So basically in complex form V=IZ always...ok...
But you know what i am not excatly getting whatever you guys wrote involving complex numbers and stuff and in my book they have given a very brief part on the complex equations... I mean i am pretty much and amature in this chapter...So please don't mind about the face that i did not get it that much...but you when i go furthur into the chapter i guess i will get back to you...But before that i need to refer to some good physics books to get these things...