# ω i ω ϕ ω γ ω

```PHYS 5061
Homework set 1
Due: Feb 9.
1) Consider a resistor, capacitor, and inductor, all in series:
Find the complex impedance of this combination as a function of (angular) frequency.
Express this in the form Z  f ( )ei ( ) , and express  ( )  tan1 (?) .
These pieces can be written in a more compact form if the following are introduced: the
1
characteristic frequency o 
and the parameter   R / L to simplify the notation.
LC
Try to do this for your expressions for f ( ) and the phase. Try re-writing them in
terms of these.
(b) What’s the minimum impedance of the circuit? At what frequency does it occur.
What happens to the impedance at very low (ω0) and very high frequencies
(ωinfinity)?
2) Consider the op-amp circuit below, which does something physicists often do with
pencil and paper: solves a differential equation. Op-amps can be used as “analog
computers” and once were a popular tool. Note that beyond the usual feedback for each
op-amp, there is overall feedback of the “output” back to the “input,” so that Vin = Vout
What mathematical operation does this circuit perform on the signal present at the point
labeled Vin ? What will Vout look like if you hook the circuit up to an oscilloscope?
Don’t settle for the trivial answer of Vout =0. What’s the differential equation this circuit
simulates?
3) Find the gain ( G() = |Vout / Vin| ) as a function of frequency of the inverting
amplifier shown. Plot the general shape. (b) With R1 = 5 kohms, R2 = 50 kohms, and C
= 470 pF, what is the gain at low frequencies? What is this gain expressed in dB? At
what frequency is the gain reduced by 3 dB from its low frequency value? Give your
answer in Hz, not just as  in rad/s. (c) What’s the phase shift at this frequency?
4) Complete the timing diagram for the following circuit.
```