RLC Circuit Calculations Apologies for not showing much effort on my part, but I'm rather short on time as this is part of an exam I have coming up in two days, and I don't really know where to start on this problem. The course is not in English, and my English physics terminology is rather poor, but here's the translated problem description:
The question is always the same, with the integer before $R$ differing from $1$ to $5$, but the equation may come in 4 different forms.
"Draw a diagram of an electrical circuit, the complex impedance of which is in the following form: (j is the imaginary unit)
$Ž = \frac{2R}{1 + jωC 2R} + jωL$
Let $R$ be some unit active resistance, through which we express all the other resistances. Assuming $ωL = 3R$ and $\frac{1}{ωC}=5R$, draw the phasor diagram of this circuit, from which you can see the phase angle between the total voltage and the total current. Calculate this phase angle. Calculate all of the voltages present in the circuit diagram in units $U$, which is the voltage of the power supply. Calculate the strengths of all the currents in units $\frac{U}{R}$, with three-digit accuracy. Calculate the emitted power on the $2R$ resistor in percentages from the value $\frac{U^2}{R}$"
The other 3 possible equations are:
$Ž = \frac{1}{\frac{1}{3R + jωL} + jωC}$
$Ž = \frac{1}{\frac{1}{4R} - j\frac{1}{ωL}} + \frac{1}{jωC}$
$Ž = \frac{1}{\frac{ωC}{ωC 4R - j} - j\frac{1}{ωL}}$
What it seems like is that each of these circuits contains one of each of the elements: A resistor, an inductor, and a capacitor, and I have to figure out based on the equation which are connected in series, and which in parallel.
I remember that these kinds of circuits perform harmonic oscillation, so I'm guessing the phasor diagram will be some sort of sine wave. Two of them, one for the current voltage, one for the current current.
I haven't much of an idea on how to find the rest. I'd be happy with just a method of finding these answers, even if a good explanation can not be given.  
 A: You need to learn to recognize the form of the impedance of two elements in parallel. You do this by remembering the complex impedance of each:
$$\begin{align}Z_C &= \frac{1}{j\omega C}\\
Z_L &= j\omega L\\
Z_R &= R\end{align}$$
Now when you have R and L in parallel, the impedance will be
$$Z_{RL} = \frac{1}{\frac{1}{R} + \frac{1}{j\omega L}} $$
and you can do the same thing for each of the 3 combinations: RL, RC, LC
Once you do that, you will start to see "patterns" in the expressions for Z - and when you see patterns, you can replace that pattern with the appropriate parallel circuit. For example, the pattern for RL can be seen in the third of your four expressions. 
Similarly, if you have any two elements in series, you can recognize their pattern. I see a $3R+j\omega L$ in one of the equations: that's a 3R resistor in series with a L inductor.
By finding these patterns and grouping, it should be straightforward to find out what the circuits are.
Let me know if that is sufficient help - you will learn more from using a hint and working from there, than from being fed a fully formed solution.
The complex impedance tells us the amplitude and phase relationship between the voltage and current: that is, if you consider the voltage to be of the form
$$V = V_0 e^{j\omega t}$$
Then the current is simply
$$I =\frac{V}{Z}$$
If you write $Z = |Z| e^{j\theta}$ where $\theta$ is the phase angle, then you can see that the current can be written as
$$I = \frac{V_0}{|Z|} e^{j(\omega t-\theta)}$$
Which shows clearly the phase relationship between current and voltage. Now you can calculate $|Z|$ and $\theta$ as a function of $\omega$, and this will allow you to plot the relationship. The following simple program (simulating a small resistor in series with an inductor, and the two in parallel with a capacitor) show what I am talking about:
# plot phasor for RLC circuit
import numpy as np
import matplotlib.pyplot as plt

R=0.2
L=1
C=1
omega=np.linspace(0.0,10,10000)

def Z(R,L,C,w):
    return 1./(1./(R + 1j*w*L) + 1j*w*C)

z = Z(R,L,C,omega)
plt.figure()
plt.plot(np.real(z), np.imag(z))
plt.title('phasor diagram')
plt.xlabel('real part')
plt.ylabel('imaginary part')
plt.show()

Which generates this plot:

As you can see, with increasing frequency the phase first leads (because the RL part of the circuit dominates), then lags (when the capacitor dominates). Resonance is defined at the point where the impedance of the circuit is real.
In the problem as given, they actually specify the relationship between frequency, inductance, capcitance and resistance. That means that instead of an entire plot, they are only asking for a single point in the complex plane. You can substitute the values given in the expression for Z (without even having to know what the circuit looks like). Since
$$Z = \frac{2R}{1+j\omega C 2R}+j\omega L$$
and you are given
$$\omega L = 3R\\
\frac{1}{\omega C} = 5R$$
You can substitute in those values and end up with
$$Z = \frac{2R}{1+\frac25 j}+j3R$$
To get this in a form you can plot, you need to multiply top and bottom by $\left(1-\frac25j\right)$ to get
$$\begin{align}Z &= \frac{2R(1-\frac25 j)}{1+\frac{4}{25}}+j3R\\
&=\left(\frac{50}{29} + j\left(3-\frac{20}{29}\right)\right)\rm{R}\\
&=(1.72+2.31j)~R\end{align}$$
This is a complex number with a magnitude and phase you should know how to plot in the complex plane.
