Experiment 4: AC Circuits

Intro to Experimental Phys-labPhysics UN 1494

Valerie Hsieh
vh2289@columbia.edu
Office hours: Fridays, 10-11am, Pupin 413

Announcements

  • This week: AC circuits in Pupin 528
  • Due this week: Experiment 3 (capacitance & the oscilloscope) full report
  • Looking ahead:
Week of Lecture (Monday) In lab Assignment due
Mar 3 AC circuits AC circuits data collection Capacitance report
Mar 10 Interferometer Interferometer data collection AC circuits report
Mar 17 Spring break (no lecture) Spring break (no lab) -
Mar 24 Polarization & interference
Polarization & interference
data collection
Polarization & interference data collection| Polarization & interference | | :--- | | data collection |
 Interferometer report
Week of Lecture (Monday) In lab Assignment due Mar 3 AC circuits AC circuits data collection Capacitance report Mar 10 Interferometer Interferometer data collection AC circuits report Mar 17 Spring break (no lecture) Spring break (no lab) - Mar 24 Polarization & interference "Polarization & interference data collection" Interferometer report| Week of | Lecture (Monday) | In lab | Assignment due | | :--- | :--- | :--- | :--- | | Mar 3 | AC circuits | AC circuits data collection | Capacitance report | | Mar 10 | Interferometer | Interferometer data collection | AC circuits report | | Mar 17 | Spring break (no lecture) | Spring break (no lab) | - | | Mar 24 | Polarization & interference | Polarization & interference <br> data collection | Interferometer report |

Announcements

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  • 2 or more 0 s will result in automatic failure of the course

Introduction

  • Last lab (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

Varying Electromagnetic Fields

  • The 2nd and 4th Maxwell equations in vacuum and with no charges read:
∇ → × E → = − ∂ B → ∂ t ∇ → × B → = 1 c 2 ∂ E → ∂ t ∇ → × E → = − ∂ B → ∂ t ∇ → × B → = 1 c 2 ∂ E → ∂ t {:[ vec(grad)xx vec(E)=-(del( vec(B)))/(del t)],[ vec(grad)xx vec(B)=(1)/(c^(2))(del( vec(E)))/(del t)]:}\begin{aligned} & \vec{\nabla} \times \vec{E}=-\frac{\partial \vec{B}}{\partial t} \\ & \vec{\nabla} \times \vec{B}=\frac{1}{c^{2}} \frac{\partial \vec{E}}{\partial t} \end{aligned}∇→×E→=−∂B→∂t∇→×B→=1c2∂E→∂t
  • Varying magnetic field generates electric field and vice versa!

Varying Electromagnetic Fields

  • We therefore expect a changing B B BBB field to give rise to a current or potential difference
  • This is summarized by Faraday's law:
E = − d Φ B d t E = − d Φ B d t E=-(dPhi_(B))/(dt)\mathcal{E}=-\frac{d \Phi_{B}}{d t}E=−dΦBdt
  • Change in magnetic flux creates an e.m.f.
  • Magnetic flux can be varied in two ways:
  1. Change the B field
  2. Change the surface

Introducing the Inductor

  • Capacitor: stores energy in the form of electric fields
  • Inductor: stores energy in form of magnetic fields
  • From Faraday's law one deduces the expression for the potential difference at the two ends of the inductor:
V L = − L d I d t V L = − L d I d t V_(L)=-L(dI)/(dt)V_{L}=-L \frac{d I}{d t}VL=−LdIdt

Introducing the Inductor

V L = − L d I d t V L = − L d I d t V_(L)=-L(dI)/(dt)V_{L}=-L \frac{d I}{d t}VL=−LdIdt
  • 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)

Why AC Circuits?

  • A C → A C → AC rarrA C \rightarrowAC→ Alternating current source
  • A few uses:
  • Sensitive to input frequency (i.e. function generator frequency)
  • Serve as signal frequency filters:
  • High-frequency filters
  • Low-frequency filters
  • Band-pass filters

Why AC Circuits?

  • AC → → rarr\rightarrow→ Alternating current source
  • A few uses:
  • Transformers
  • Induction effects - Ability to raise or lower the voltage amplitude.
  • Generators and Motors

Scientist Spotlight: Edith Clarke

  • First woman in US to be employed as an electrical engineer
  • First woman to graduate with an advanced degree in electrical engineering (in 1919)
  • Patented the first early graphing calculator, which helped solve equations involving V , L , I V , L , I V,L,I\mathrm{V}, \mathrm{L}, \mathrm{I}V,L,I in power transmission lines (related to her work at General Electric)
  • First woman to present at IEEE
CIRCUIT ANALYSIS OF A-C POWER SYSTEMS
VOLUME
Symmetrical and Related Components
  • Authored a power engineering textbook which contains principles that form the basis of the modern US electrical grid

AC Circuits: Sources

  • AC circuits have an enormous range of applications. Here we cover the most important aspects
  • For this lab:
  • Consider only sources that vary sinusoidally:
I ( t ) = I max sin ⁡ ( ω t ) I ( t ) = I max sin ⁡ ( ω t ) I(t)=I_(max)sin(omega t)I(t)=I_{\max } \sin (\omega t)I(t)=Imaxsin⁡(ωt)

AC Circuits: Resistors

  • Simple circuit example:
  • Function generator + resistor
The voltage across the resistor is then simply:
V R = I R V R = I R V_(R)=IRV_{R}=I RVR=IR
V R ( t ) = I ( t ) R = I max R sin ⁡ ( ω t ) ≡ V max sin ⁡ ( ω t ) V R ( t ) = I ( t ) R = I max R sin ⁡ ( ω t ) ≡ V max sin ⁡ ( ω t ) {:[V_(R)(t)=I(t)R],[=I_(max)R sin(omega t)],[-=V_(max)sin(omega t)]:}\begin{aligned} V_{R}(t) & =I(t) R \\ & =I_{\max } R \sin (\omega t) \\ & \equiv V_{\max } \sin (\omega t) \end{aligned}VR(t)=I(t)R=ImaxRsin⁡(ωt)≡Vmaxsin⁡(ωt)

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:
Q = C V C ⇓ ∫ I ( t ) d t = C V C Q = C V C ⇓ ∫ I ( t ) d t = C V C {:[Q=CV_(C)],[⇓],[int I(t)dt=CV_(C)]:}\begin{gathered} Q=C V_{C} \\ \Downarrow \\ \int I(t) d t=C V_{C} \end{gathered}Q=CVC⇓∫I(t)dt=CVC
  • Therefore, when the current is sinusoidal the voltage is given by:
V C ( t ) = 1 C ∫ I ( t ) d t = 1 C ∫ I max sin ⁡ ( ω t ) d t = − I max ω C cos ⁡ ( ω t ) V C ( t ) = 1 C ∫ I ( t ) d t = 1 C ∫ I max sin ⁡ ( ω t ) d t = − I max ω C cos ⁡ ( ω t ) {:[V_(C)(t)=(1)/(C)int I(t)dt],[=(1)/(C)intI_(max)sin(omega t)dt],[=-(I_(max))/(omega C)cos(omega t)]:}\begin{aligned} V_{C}(t) & =\frac{1}{C} \int I(t) d t \\ & =\frac{1}{C} \int I_{\max } \sin (\omega t) d t \\ & =-\frac{I_{\max }}{\omega C} \cos (\omega t) \end{aligned}VC(t)=1C∫I(t)dt=1C∫Imaxsin⁡(ωt)dt=−ImaxωCcos⁡(ωt)

AC Circuits: Capacitors

  • The voltage is sinusoidal:
V C ( t ) = − I max ω C cos ⁡ ( ω t ) = − V max cos ⁡ ( ω t ) = V max sin ⁡ ( ω t − π 2 ) ) V C ( t ) = − I max ω C cos ⁡ ( ω t ) = − V max cos ⁡ ( ω t ) = V max sin ⁡ ω t − π 2 {:[V_(C)(t)=-(I_(max))/(omega C)cos(omega t)],[=-V_(max)cos(omega t)],[{:=V_(max)sin(omega t-(pi)/(2)))]:}\begin{aligned} V_{C}(t) & =-\frac{I_{\max }}{\omega C} \cos (\omega t) \\ & =-V_{\max } \cos (\omega t) \\ & \left.=V_{\max } \sin \left(\omega t-\frac{\pi}{2}\right)\right) \end{aligned}VC(t)=−ImaxωCcos⁡(ωt)=−Vmaxcos⁡(ωt)=Vmaxsin⁡(ωt−π2))
  • The extra π / 2 π / 2 pi//2\pi / 2π/2 in the expression is the phase of the voltage.
  • Voltage across the capacitor lags behind the current by:
ϕ = − π / 2 ϕ = − π / 2 phi=-pi//2\phi=-\pi / 2ϕ=−π/2

AC Circuits: Inductors

  • The voltage is still sinusoidal:
V L ( t ) = L d I d t = ω L I max cos ⁡ ( ω t ) = V max sin ⁡ ( ω t + π 2 ) V L ( t ) = L d I d t = ω L I max cos ⁡ ( ω t ) = V max sin ⁡ ω t + π 2 {:[V_(L)(t)=L(dI)/(dt)],[=omega LI_(max)cos(omega t)],[=V_(max)sin(omega t+(pi)/(2))]:}\begin{aligned} V_{L}(t) & =L \frac{d I}{d t} \\ & =\omega L I_{\max } \cos (\omega t) \\ & =V_{\max } \sin \left(\omega t+\frac{\pi}{2}\right) \end{aligned}VL(t)=LdIdt=ωLImaxcos⁡(ωt)=Vmaxsin⁡(ωt+π2)
  • Inductor voltage is also phase shifted w.r.t. current.
  • Voltage across the inductor anticipates the current through it by:
ϕ = π / 2 ϕ = π / 2 phi=pi//2\phi=\pi / 2ϕ=π/2

Voltage Maxima: A Closer Look

  • Given our expression for V R V R V_(R)V_{R}VR, the maximum value of the voltage across the resistor is just given by Ohm's Law:
V R ( t ) = I ( t ) R = I max R sin ⁡ ( ω t ) ◻ V R max = I max R ≡ V max sin ⁡ ( ω t ) V R ( t ) = I ( t ) R = I max R sin ⁡ ( ω t ) ◻ V R max = I max R ≡ V max sin ⁡ ( ω t ) {:[V_(R)(t)=I(t)R],[=I_(max)R sin(omega t)quad◻quadV_(R)^(max)=I_(max)R],[-=V_(max)sin(omega t)]:}\begin{aligned} V_{R}(t) & =I(t) R \\ & =I_{\max } R \sin (\omega t) \quad \square \quad V_{R}^{\max }=I_{\max } R \\ & \equiv V_{\max } \sin (\omega t) \end{aligned}VR(t)=I(t)R=ImaxRsin⁡(ωt)◻VRmax=ImaxR≡Vmaxsin⁡(ωt)
  • The maximum voltage across the capacitor is a function of ω ω omega\omegaω :
Capacitive reactance
Given an oscillating input current the capacitor voltage is higher for small frequencies and lower for high frequencies

Voltage Maxima: A Closer Look

  • The maximum voltage across the capacitor is a function of ω ω omega\omegaω :
V C ( t ) = 1 C ∫ I ( t ) d t = 1 C ∫ I max sin ⁡ ( ω t ) d t = − I max ω C cos ⁡ ( ω t ) V C ( t ) = 1 C ∫ I ( t ) d t = 1 C ∫ I max sin ⁡ ( ω t ) d t = − I max ω C cos ⁡ ( ω t ) {:[V_(C)(t)=(1)/(C)int I(t)dt],[=(1)/(C)intI_(max)sin(omega t)dt],[=-(I_(max))/(omega C)cos(omega t)]:}\begin{aligned} V_{C}(t) & =\frac{1}{C} \int I(t) d t \\ & =\frac{1}{C} \int I_{\max } \sin (\omega t) d t \\ & =-\frac{I_{\max }}{\omega C} \cos (\omega t) \end{aligned}VC(t)=1C∫I(t)dt=1C∫Imaxsin⁡(ωt)dt=−ImaxωCcos⁡(ωt)
V C max = i i − 1 I max ≡ I max X C V C max = i i − 1 I max  ≡ I max  X C V_(C)^(max)=i_(i)^(-1)I_("max ")-=I_("max ")X_(C)V_{C}^{\max }=i_{i}^{-1} I_{\text {max }} \equiv I_{\text {max }} X_{C}VCmax=ii−1Imax ≡Imax XC

Capacitive reactance

Given an oscillating input current the capacitor voltage is higher for
X C = 1 ω C X C = 1 ω C X_(C)=(1)/(omega C)X_{C}=\frac{1}{\omega C}XC=1ωC small frequencies and lower for high frequencies

Voltage Maxima: A Closer Look

  • But, the maximum voltage across the inductor is also a function of the driving frequency:
V L ( t ) = L d I d t = ω L I max cos ⁡ ( ω t ) = V max sin ⁡ ( ω t + π 2 ) V L ( t ) = L d I d t = ω L I max cos ⁡ ( ω t ) = V max sin ⁡ ω t + π 2 {:[V_(L)(t)=L(dI)/(dt)],[=omega LI_(max)cos(omega t)],[=V_(max)sin(omega t+(pi)/(2))]:}\begin{aligned} V_{L}(t) & =L \frac{d I}{d t} \\ & =\omega L I_{\max } \cos (\omega t) \\ & =V_{\max } \sin \left(\omega t+\frac{\pi}{2}\right) \end{aligned}VL(t)=LdIdt=ωLImaxcos⁡(ωt)=Vmaxsin⁡(ωt+π2)
The inductor voltage is instead higher for large frequencies and lower for small ones
Inductive reactance
X L = ω L X L = ω L X_(L)=omega LX_{L}=\omega LXL=ωL

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 ) X C (X_(C))\left(X_{C}\right)(XC)
Hard to charge = high reactance ( X C ) X C (X_(C))\left(X_{C}\right)(XC)

Physical Explanation: Capacitors

  • Rapidly varying signals (high frequency) quickly charge/discharge capacitor before it fills with charge → → rarr\rightarrow→ low impedance.
  • Slowly varying signals (low frequency) charge the capacitor to its limit, slowing down the rate: that is, decreasing the current!
  • Now that we have introduced the language of reactances, you can think about the capacitor somehow as a resistor with ω ω omega\omegaω-dependent resistance

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 → → rarr\rightarrow→ induction!
  • Lenz's Law: a coil will oppose changes in magnetic flux. Self-induced EMF is:

Physical Explanation: Inductors

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

RLC Circuits

  • Let's see what happens when we combine all these three components in a series:
  • From Kirchhoff's first law (loops):
V ( t ) = V R + V L + V C = I R + L d I d t + Q C V ( t ) = V R + V L + V C = I R + L d I d t + Q C V(t)=V_(R)+V_(L)+V_(C)=IR+L(dI)/(dt)+(Q)/(C)V(t)=V_{R}+V_{L}+V_{C}=I R+L \frac{d I}{d t}+\frac{Q}{C}V(t)=VR+VL+VC=IR+LdIdt+QC

RLC Circuits: Phase Shift

  • After passing through the three components the voltage will have some phase shift
  • Let's then impose V ( t ) V ( t ) V(t)V(t)V(t) to look like:
V ( t ) = V max sin ⁡ ( ω t + ϕ ) = V max [ sin ⁡ ( ω t ) cos ⁡ ϕ + cos ⁡ ( ω t ) sin ⁡ ϕ ] V ( t ) = V max sin ⁡ ( ω t + ϕ ) = V max [ sin ⁡ ( ω t ) cos ⁡ ϕ + cos ⁡ ( ω t ) sin ⁡ ϕ ] {:[V(t)=V_(max)sin(omega t+phi)],[=V_(max)[sin(omega t)cos phi+cos(omega t)sin phi]]:}\begin{aligned} V(t) & =V_{\max } \sin (\omega t+\phi) \\ & =V_{\max }[\sin (\omega t) \cos \phi+\cos (\omega t) \sin \phi] \end{aligned}V(t)=Vmaxsin⁡(ωt+ϕ)=Vmax[sin⁡(ωt)cos⁡ϕ+cos⁡(ωt)sin⁡ϕ]
  • Comparing with the equation from the previous slide it must necessarily be:
{ V max cos ⁡ ϕ = I max R V max sin ⁡ ϕ = I max ( X L − X C ) V max cos ⁡ ϕ = I max R V max sin ⁡ ϕ = I max X L − X C {[V_(max)cos phi=I_(max)R],[V_(max)sin phi=I_(max)(X_(L)-X_(C))]:}\left\{\begin{aligned} V_{\max } \cos \phi & =I_{\max } R \\ V_{\max } \sin \phi & =I_{\max }\left(X_{L}-X_{C}\right) \end{aligned}\right.{Vmaxcos⁡ϕ=ImaxRVmaxsin⁡ϕ=Imax(XL−XC)
  • And hence the phase shift is:
tan ⁡ ϕ = X L − X C R = ω L − 1 / ω C R tan ⁡ ϕ = X L − X C R = ω L − 1 / ω C R tan phi=(X_(L)-X_(C))/(R)=(omega L-1//omega C)/(R)\tan \phi=\frac{X_{L}-X_{C}}{R}=\frac{\omega L-1 / \omega C}{R}tan⁡ϕ=XL−XCR=ωL−1/ωCR
  • The phase shift will depend both on the characteristics of the circuit ( R , C , L R , C , L R,C,LR, C, LR,C,L ) and on the frequency of the input signal!

RLC Circuits: Phase Shift

  • What about the maximum amplitude for the voltage?
  • Let take again: { V max cos ⁡ ϕ = I max R V max cos ⁡ ϕ = I max R {V_(max)cos phi=I_(max)R:}\left\{V_{\max } \cos \phi=I_{\max } R\right.{Vmaxcos⁡ϕ=ImaxR
{ V max sin ⁡ ϕ = I max ( X L − X C ) V max sin ⁡ ϕ = I max X L − X C {V_(max)sin phi=I_(max)(X_(L)-X_(C)):}\left\{V_{\max } \sin \phi=I_{\max }\left(X_{L}-X_{C}\right)\right.{Vmaxsin⁡ϕ=Imax(XL−XC)
  • Let's now square both equations and add them together:
V max = I max R 2 + ( X L − X C ) 2 = I max Z V max = I max R 2 + X L − X C 2 = I max Z {:[V_(max)=I_(max)sqrt(R^(2)+(X_(L)-X_(C))^(2))],[=I_(max)Z]:}\begin{aligned} V_{\max } & =I_{\max } \sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}} \\ & =I_{\max } Z \end{aligned}Vmax=ImaxR2+(XL−XC)2=ImaxZ
  • The quantity Z Z ZZZ 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

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
High Resonant Low
  • We therefore expect to have a particular frequency ( ω 0 ) ω 0 (omega_(0))\left(\omega_{0}\right)(ω0) in the middle range that goes through the system almost untouched
I max = V max Z = V max R 2 + ( X L − X C ) 2 I max = V max Z = V max R 2 + X L − X C 2 I_(max)=(V_(max))/(Z)=(V_(max))/(sqrt(R^(2)+(X_(L)-X_(C))^(2)))I_{\max }=\frac{V_{\max }}{Z}=\frac{V_{\max }}{\sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}}}Imax=VmaxZ=VmaxR2+(XL−XC)2
  • For a given input voltage, the current in the circuit is maximum when Z Z ZZZ is minimum i.e. when X L = X C X L = X C X_(L)=X_(C)X_{L}=X_{C}XL=XC.
  • The resonant frequency is given by:
ω 0 L = 1 ω 0 C ⟹ ω 0 = 1 L C ω 0 L = 1 ω 0 C ⟹ ω 0 = 1 L C omega_(0)L=(1)/(omega_(0)C)Longrightarrowomega_(0)=(1)/(sqrt(LC))\omega_{0} L=\frac{1}{\omega_{0} C} \Longrightarrow \omega_{0}=\frac{1}{\sqrt{L C}}ω0L=1ω0C⟹ω0=1LC

Resonance

Terminology

FWHM: Full Width at Half Maximum Is the full width of the resonance peak at the point where its height is halfway between zero and the maximum.
Recall that the resistor voltage V R V R V_(R)V_{R}VR is directly proportional to the magnitude of current.
The Experiment

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 φ varphi\varphiφ, between the driving signal and the three components ( R , L R , L R,LR, LR,L and C C CCC ) of the circuit
  • Compare with expected value

Experimental Setup

"Unknown inductance" coil

Oscilloscope
Function generator
Resistors in series

Experimental Setup

  • Recommendations:
  • Set the function generator peak-to-peak voltage to 20 V , the maximum allowed.
  • There is a 0 − 2 V / 0 − 20 V 0 − 2 V / 0 − 20 V 0-2V//0-20V0-2 \mathrm{~V} / 0-20 \mathrm{~V}0−2 V/0−20 V selector button in addition to the voltage knob.
  • Make sure the oscilloscope is set to trigger on channel 1 , the function generator signal. You can do this by pressing the TRIGGER button and checking in the window menu that CH 1 is selected.
  • Use the MEASURE tools to observe peak-peak amplitudes, signal periods, and signal frequencies. Let the scope do the work for you!
  • Make sure that both peaks are in the viewable range of the scope!

Resonance Measurements

  • Set the oscilloscope to look at the potential across the two ends of the resistor
  • First localize ω 0 ω 0 omega_(0)\omega_{0}ω0 by looking at when the amplitude of the signal gets amplified
  • Then, for about 20 frequencies above and below ω 0 ω 0 omega_(0)\omega_{0}ω0, record peakpeak voltage across resistor
  • Normalize your values such that the maximum is V p p = 1 V p p = 1 Vpp=1V p p=1Vpp=1
  • Repeat for three values of the resistance ( 1.2 K Ω 1.2 K Ω 1.2KOmega1.2 \mathrm{~K} \Omega1.2 KΩ, 3.3 K Ω , 4.5 K Ω 3.3 K Ω , 4.5 K Ω 3.3KOmega,4.5KOmega3.3 \mathrm{~K} \Omega, 4.5 \mathrm{~K} \Omega3.3 KΩ,4.5 KΩ )
  • Plot V p p V p p V_(pp)V_{p p}Vpp vs. ω ω omega\omegaω as shown in the figure
  • Tip: Make sure you are going to sufficiently high frequencies in order to identify FWHM!

Resonance Measurements

  • Use the plots to determine the value of the resonant frequencies (with errors!)
  • Also use the plots to measure the FWHM of the three curves.
  • Compare the results from the three measurements
  • When you're finished with this part, replace the inductor with the large copper coil
  • Repeat the previous measurements and from the value of the average resonant frequency compute the inductance, L L LLL, of the wire

Phase Shift Measurements

  • Replace the copper ring with the known inductor again and set R = 3.3 K Ω R = 3.3 K Ω R=3.3KOmegaR=3.3 \mathrm{~K} \OmegaR=3.3 KΩ
  • Locate the resonant frequency and for 5 values at, above and below it measure the phase shift across the resistor
  • Now increase the frequency well above the resonant one. This makes the inductor much more important than the capacitor. Measure ϕ ϕ phi\phiϕ
  • Now make the capacitor more important by going way below the resonant frequency. Measure ϕ ϕ phi\phiϕ again
ϕ = 2 π t R − t d T d ϕ = 2 π t R − t d T d phi=2pi(t_(R)-t_(d))/(T_(d))\phi=2 \pi \frac{t_{R}-t_{d}}{T_{d}}ϕ=2πtR−tdTd
Compare the results obtained with the expected ones

Tips

  • Don't get confused-the frequency reported by the oscilloscope (in the MEASURE mode) is f f fff. It is related to what we called frequency so far by ω = 2 π f ω = 2 π f omega=2pi f\omega=2 \pi fω=2πf.
  • Once you manage to locate the maximum of the V p p V p p V_(pp)V_{p p}Vpp curve and you have taken 20 points above and below it, try to take more points right around your maximum. This will reduce your uncertainty on ω 0 ω 0 omega_(0)\omega_{0}ω0.
  • When plotting V p p V p p V_(pp)V_{p p}Vpp vs ω ω omega\omegaω remember to take enough data to measure the FWHM. When R R RRR is large, the peak is very broad (especially towards higher frequencies). Keep taking data until you pass half height.