Question1: FIG 1 shows an RLC network
Show that this network has the transfer function represented by:
The impedance of the inductor is given by
The impedance of the capacitor is given by
Comparing the circuit in FIG 1 with a resistive divider where the output voltage at the junction of the 2 resistors is given by
The transfer function of FIG 1 is given by
Multiplying top and bottom by sC gives
And dividing by LC gives
Question 2: In FIG 1, if the inductor is 2.2uH, the capacitor is 4.7uF, what value of R gives critical damping?
Critical damping is when
The undamped natural frequency of the circuit is given by
Which is 310.985krads-1 (which is equal to 49.495kHz).
From the general characteristic polynomial for a second order system
so R = 1.368Ω.
Question 3: If FIG 1 is excited with a step input, what will be the damped frequency (in Hertz) for a resistance of 0.5Ω?
We know that
Which is equal to 46.07kHz.
All of the above can be tested in LTspice.
Beware: the inductors in LTspice have an ‘invisible’ series resistance of 1mOhm. This can be set to 0mOhm by right clicking over the component and changing the ESR value