Experiment 9: AC circuits, Lab Reports of Experimental Physics

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Experiment 9:

AC circuits

INTRO TO EXPERIMENTAL PHYS-

LAB 1493/1494/

Introduction

● Last week (RC circuit):

● (^) Constant Voltage power source (constant over time)

● This week:

● (^) A new component: the inductor

● (^) Alternating Current (AC) circuits

➢ (^) Time dependent voltage source ● (^) Leads to:

➢ Time dependent currents (alternating currents) ➢ (^) Phase shifts in voltage and currents in components with respect to one another ➢ (^) Resonance

PHYS 1493/1494/2699: Exp. 9 – AC circuits

Introducing the inductor

● It stores energy in form of magnetic

fields (analogous to the capacitor)

● From Faraday’s law one deduces the

expression for the potential difference

at the two ends of the inductor:

● The inductor is only sensitive to the

change in current! No change = no

voltage

● Negative sign indicates that the inductor

opposes any change in current ( Lenz's

Law )

Change Voltage

PHYS 1493/1494/2699: Exp. 9 – AC circuits

Why AC circuits?

● Sensitive to input frequency

( i.e. function generator

frequency)

● Serve as signal frequency

filters:

● (^) High-frequency filters

● (^) Low-frequency filters

● (^) Band-pass filters

● Transformers

● (^) Induction effects - Ability to raise or lower the voltage amplitude.

● Generators and Motors

e.g. Radios

e.g. Speakers

PHYS 1493/1494/2699: Exp. 9 – AC circuits

AC circuits: capacitors

● More interesting case: connect a

capacitor to the AC voltage source

● Last time we saw that the voltage

across a capacitor is given by:

● Therefore, when the current is

sinusoidal the voltage is given by:

~

PHYS 1493/1494/2699: Exp. 9 – AC circuits

AC circuits: capacitors

● The voltage is sinusoidal:

● The extra π/2 in the expression

is the phase of the voltage.

● Voltage across the capacitor

lags behind the current by:

~

PHYS 1493/1494/2699: Exp. 9 – AC circuits

● Given our expression for VR , the maximum value of the

voltage across the resistor is just given by Ohm’s Law:

● The maximum voltage across the capacitor is a function of ω:

● But, the maximum voltage across the inductor is also a function

of the driving frequency:

Voltage maxima: a closer look

Inductive reactance

Capacitive reactance

Given an oscillating input current the capacitor voltage is higher for small frequencies and lower for high frequencies

The inductor voltage is instead higher for large frequencies and lower for small ones

PHYS 1493/1494/2699: Exp. 9 – AC circuits

Physical explanation: capacitors

● (^) Question: Why does the capacitor resist low-frequency signals more

than high-frequency ones?

● (^) Last time: when charging/discharging the capacitor, the current – the

rate at which you can charge it – decreases exponentially. It becomes harder and harder to push in more charge as the capacitor fills up.

Easy to charge = low reactance ( X (^) C )

Hard to charge = high reactance ( X (^) C )

PHYS 1493/1494/2699: Exp. 9 – AC circuits

Physical explanation: inductors

● (^) Question: Why does the inductor resist high-frequency signals more than low-frequency ones?

● (^) Think about the nature of an inductor: it is a coil of wire. If the current in the wire changes, then the magnetic flux through the coil changes → induction!

● (^) Lenz’s Law: a coil will oppose changes in magnetic flux. Self-induced EMF is:

● (^) Rapidly varying signals strongly change the flux, so the inductor “pushes back” harder against the flow of current! ● (^) Voltage is maximum (and opposing) when I changes most rapidly (high frequency) ● (^) Voltage = 0 when I is constant (low frequency)

A life spent after a minus sign… and everyone always forgets about it!

(Heinrich Lenz)

PHYS 1493/1494/2699: Exp. 9 – AC circuits

RLC circuits

● Let’s see what happens when we combine all these three

components in a series:

● From Kirchhoff’s first law (loops):

Use what we learned about inductors and capacitors a few slides ago sin(ωt ±^ π/2) = ± cos(ωt) PHYS 1493/1494/2699: Exp. 9 – AC circuits

RLC circuits: phase shift

● What about the maximum amplitude for the

voltage?

● Let take again:

● Let’s now square both equations and add them together:

● The quantity Z is called the impedance of the RLC circuit

● NOTE: the previous equation resembles very closely Ohm’s law

for resistors!

● This procedure can actually be generalized introducing the so-

called phasor formalism

PHYS 1493/1494/2699: Exp. 9 – AC circuits

Resonant frequency

● (^) So the whole RLC system has this peculiar

frequency dependent “effective resistance”. In particular: ● (^) High-frequencies: killed by the inductor ● (^) Low-frequencies: killed by the capacitor

● (^) We therefore expect to have a particular

frequency ( ω 0 ) in the middle range that goes through the system almost untouched

● (^) For a given input voltage, the current in the circuit is maximum when Z

is minimum i.e. when XL = XC.

● (^) The resonant frequency is given by:

High Resonant Low

L

C

PHYS 1493/1494/2699: Exp. 9 – AC circuits

The Experiment

PHYS 1493/1494/2699: Exp. 9 – AC circuits

Main goals

● Resonance of RLC circuit:

● (^) Measure the resonant frequencies and FWHM for three known circuits ● (^) Compute the unknown inductance of a copper coil by finding the resonant frequency of the whole system

● Observe the phase shift, φ , between the driving signal and

the three components (R, L and C) of the circuit

● Compare with expected value

PHYS 1493/1494/2699: Exp. 9 – AC circuits