The easiest way of solving such problems is to use complex phasors. The total complex impedance is
$$Z_T=R_1+R_2||j\omega L=R_1+\frac{j\omega R_2L}{R_2+j\omega L}=877.18\cdot e^{30.67^{\circ}}\Omega$$
with \$\omega=2\pi\cdot 10,000 Hz\$, \$R_1=470\Omega\$, \$R_2=988\Omega\$, and \$L=10mH\$ (as shown in your circuit diagram).
This gives for the total current
$$I_s=\frac{V_0}{Z_T}=9.12\cdot e^{-30.67^{\circ}} mA$$
with \$V_0=8V\$.
The voltage across \$R_1\$ is
$$V_1=I_sR_1=4.28\cdot e^{-30.67^{\circ}} V$$
The voltage across \$R_2\$ and \$L\$ is
$$V_2=I_s(R_2||j\omega L)=I_s\frac{j\omega R_2L}{R_2+j\omega L}=4.83\cdot e^{26.88^{\circ}} V\quad(=V_0-V_1)$$
The current through \$R_2\$ is
$$I_2=\frac{V_2}{R_2}=4.89\cdot e^{26.88^{\circ}} mA$$
The current through the inductor is
$$I_L=\frac{V_2}{j\omega L}=7.70\cdot e^{-63.12^{\circ}} mA\quad(=I_s-I_2)$$
I realize that there are quite a few differences between these results and your calculations and measurements. I just used the values you gave in the circuit diagram, and I did my best to avoid any errors.
You have drawn a differential choke not a common-mode choke - a CM choke has dots at the same end and doesn't easily saturate because current flowing sets up cancelling flux thus nearly entirely preventing saturation. The differential choke in your circuit very easily saturates because the fluxes from the two coils are additive.
The CM choke blocks only common mode currents i.e. currents of the same direction trying to enter and pass through the choke - this is a good example of transformer action. For differential currents (as per the signal you want to pass) the windings are in effect anti-phase and have a low impedance: -
impedance = \$j\omega (L - M)\$ where L is the inductance of a winding and M is the mutual inductance between windings. M is usually very close in value to L therefore the net inductance for differential signals is lower than for common mode signals. For common-mode currents, the impedance is \$j\omega (L + M)\$ i.e. significantly more than for differential currents.
I get the feeling that you are expecting a CM choke to deal with high frequency differential currents differently to low frequency currents. This is theoretically not the case but, in reality, because leakage inductances exist (not all the flux is shared between the two coils), there is a tendency for higher frequencies to receive more attenuation.
For more info read this on page 6
Best Answer
Here's a hand-waving answer with no maths.
An inductor is just a coil of wire. It may have a ferrite or iron core, but it doesn't have to.
A coil of wire is an electromagnet. Pass a current through it, and it generates a magnetic field. A DC current gives a steady magnetic field, while an AC current gives an alternating field.
If you apply a varying magnetic field to a coil of wire, you get a generator. Most practical generators move the coil while keeping the magnets fixed, because that's easier. But it works the other way round as well. The voltage you get out of a generator depends on how fast the magnetic field is changing. So if you spin a generator really fast, you get more voltage than if you spin it slowly.
Putting the two together, if you pass a DC current through an inductor, you get a steady magnetic field and nothing else happens.
If you pass AC through it, you generate an alternating magnetic field. But an alternating magnetic field turns it into a generator, which generates a voltage to oppose your current.
If it's low frequency AC, then the magnetic field is only changing slowly, so the voltage it produces is quite weak. But for high frequency AC, the field is changing faster, and the reverse voltage is stronger.
So inductors don't block DC, but they do block AC, and they block high frequencies more strongly than low frequencies.