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 ( ) tan1 (?) . 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 (ω0) and very high frequencies (ωinfinity)? 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.