September 2020 StudentZone Quiz Solution

Question1:   FIG 1 shows an RLC network

  

Show that this network has the transfer function represented by:

 

Answer 1:

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?

 

Answer 2:

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

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?

 

Answer 3:

We know that

so

so

We know that

so

Which is equal to 46.07kHz.

Note:

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