Lab 5The AC Circuit, Impedance , High-Pass and

Low-Pass Filters

This experiment requires only a spreadsheet upload

Crystal radio (AM radio)Where is the crystal? No longer in there. Modern crystal radios use diodes instead.

Today: AC circuits with a capacitor or an inductor

AM operates from 535 to 1605 kHz.

Lab #5: RC and RL AC Circuits

• remember how AC circuits containing capacitors and resistors, as well as inductors and resistors, behave.

First, remember how to describe a sine (or cosine)

Capacitors and AC sourcesVoltage across cap is the same as the voltage from supply.

When the voltage is changing quickly, the charge also has to change quickly -> big current

Size of the current depends on the frequencyGet biggest currents at high frequencies

∆𝑉=𝑉 0sin (𝜔𝑡 )= 1𝐶𝑄

𝑄=𝐶𝑉 0sin (𝜔𝑡 )

𝐼=𝑑𝑄𝑑𝑡

=𝐶𝑉 0𝜔cos (𝜔𝑡 )=𝐶ω𝑉 0 sin (𝜔𝑡+𝜋 /2 )

𝐼=𝑉 0

1/ (ω𝐶)sin (𝜔𝑡+𝜋 /2 )

𝐼 0=𝑉 0

1/ (ω𝐶 )

• Presence of the capacitor affects the size of the current in the circuit in a frequency-dependent way.

• “phases” of signals across voltage source, resistor, and capacitor differ

• math is most easily done by modeling the voltage source as instead of and an imaginary reactive impedance for the capacitor (to shift its affect on the current by 90 degrees) and then taking the real part at the end.

AC RC CircuitsAs before, be careful with the grounds!

The MathWhat is the current?

Any complex number can be written as a magnitude and an angle in the complex plane.

Easy to read off mag of current.

Current (and thus voltage across the resistor) is shifted in phase from the voltage source by φ

What is the current at very large ω?

𝑉 0𝑒𝑖 𝜔𝑡=𝐼 (𝑡)∙𝑍

¿ 𝐼 (𝑡 ) ∙ (𝑅+ 1𝑖𝜔𝐶 )=𝐼 (𝑡) ∙(𝑅− 𝑖

𝜔𝐶 )

𝑉 0𝑒𝑖 𝜔𝑡=𝐼 (𝑡 ) ∙𝑅√1+ 1

(𝜔𝑅𝐶 )2𝑒𝑖𝜑

𝐼 (𝑡)=𝑉 0/𝑅

√1+1

(𝜔𝑅𝐶 )2

𝑒𝑖 (𝜔𝑡−𝜑 )

Voltage across R and across C

𝑽 (𝒕 )=𝑽 𝟎𝐜𝐨𝐬 (𝝎𝒕 )𝑽 𝑪 (𝒕 )=−𝑽 𝟎 sin𝝋 sin (𝝎𝒕−𝝋 )𝑽 𝑹 (𝒕 )=𝑽 𝟎 cos𝝋 cos (𝝎𝒕−𝝋 )

VR(t) leads V(t) by VC(t) lags V(t) by p/2 -

Inductive Impedance

L XL = iwL

Z = [R2 + (wL)2]1/2 t = L/R tan = wL/R = wt

𝐼 (𝑡)=𝑉 0/𝑅

√1+1

(𝜔𝜏 )2

𝑒𝑖 (𝜔𝑡−𝜑 )

As with the RC circuit, the current can be written as

𝐼 (𝑡)=𝑉 0

𝑅cos𝜑𝑒𝑖 (𝜔𝑡 −𝜑 )

lab

• For a fixed frequency, measure the phase shift f two ways and “compare” (using a c2 test)

• Measure the phase shift versus frequency two ways and use to extract RC. Compare to RC calculated directly.

hints• Include systematic errors for R and C measurements, but not for t and V

measurements with scope.

• MAKE SURE DUTY CYCLE IS ALL THE WAY COUNTER CLOCKWISE!

• phase shift can not be greater than p

• remember “compare” is a mathematical operation involving a c2 test

• make sure the voltage oscillates around zero (using the offset knob). Make sure there is no dc offset.

• remember, VR leads VIN by f

• If your wave form looks funny, your amplitude is too big for the instrumentation amplifier. Make it smaller

Error Propagation

Error PropagationYou measure dt. Calculate y = cotan(2 p f dt). What is error in y?