dc motoremcimpedanceinput-impedance
Consider the following impedance frequency response for a DC motor :
Its equivalent circuit is :
Clearly, from the response, there exists a capacitor, inductor, and resistor, but my problem is how to determine that the resistor is connected in series with the inductor and all of these elements are in parallel with the capacitor.
Your equation has a sign error.
Applying KCL at the \$U_{in}\$ node, see that the current through \$R\$ is from right to left which implies
$$U_{in} - I_{in} R = V_1$$
In fact, and more generally, recognize that the parallel resistor and controlled current source can be replaced with a resistor of resistance \$-R\$.
To see this, look at a similar pair in isolation driven by a current test source:
simulate this circuit – Schematic created using CircuitLab
The test source "sees", in this case, a \$-1\Omega\$ resistance.
To find the input impedance, use a similar approach to find the equivalent impedance of the circuit to the right of the negative resistor and then you will have reduced the circuit to a simple circuit with two series impedances. The input impedance will then be obvious.
When there is an inductor impedance and a resistor impedance in series the total value of the impedance is: -
\$Z_{total} = \sqrt{R^2 + X_L^2}\$
Where, in your example R = 10 ohms and XL is 30 ohms (reactive) making the total impedance 31.623 ohms
Best Answer
Two possible RLC equivalent circuits accommodate the 100 ohm resistance at 0 Hz (below).
The second circuit of the two has a resonant dip at 1 MHz, while the first circuit of the two has a resonant peak at 1 MHz. Furthermore, the first circuit has -20dB slope above 1 MHz, while the second circuit has +20dB slope above 1 MHz.
First circuit meets all the criteria (100 ohms @ DC, resonant peak).
simulate this circuit – Schematic created using CircuitLab