Alternating Current - ch. 31

ucla | PHYSICS 1C | 2023-02-06T11:06

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Table of Contents

• Definitions
• Big Ideas
  • Phasors and RMS
    • Phasors
    • RMS Values
  • AC Circuit Elements
    • Resistors
    • Inductors
    • Capacitors
    • Comparison
  • LRC Circuit (In Series)
    • Phase Angles
  • Power
  • Resonance
  • Transformers
  • Discussion 5 Review
  • Review
    • Imaginary
    • Polar Imaginary
    • RLC Circuit
    • Phase Angles
• Resources

Definitions

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Big Ideas

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Phasors and RMS

• an ac source supplies sinusoidally varying voltage or current
• the voltage, current with amplitude $V,I$ and angular frequency $\omega =2\pi f$ is

$v=V\cos\omega t$

• the US uses RMS values of 120 V at 60 Hz while others may use 240 V at 50 Hz

Phasors

• phasors are vectors that rotate counterclockwise with constant ang. speed $\omega$ which represents varying voltage and current
• a phasor sweeps an angle of $\theta (t)=\omega t$
• the projection of a phasor on the horizontal gives instantaneous values ($v,i$)

RMS Values

• root-mean-square values for voltage and current are used to represent “average”

$I_{\text{rms}}=\frac{I}{\sqrt{2}}{V}_{\text{rms}}=\frac{V}{\sqrt{2}}$

AC Circuit Elements

Resistors

Sps. $V_R=IR$ is the voltage amplitude across the resistor

$v_R=iR=(IR)\cos \omega t=V_R\cos \omega t$

Inductors

• bc inductors’ voltage are proportionate to the rate of change of current

$v_L=L\frac{di}{dt}=-I\omega L\sin\omega t=I\omega L\cos(\omega t+\phi)$

• thus, inductors are out of phase with current by phase angle $\varphi =90\text{degree}$ i.e. the phasor for the inductor is ahead of the voltage by the phase angle
• thus amplitude of voltage

$V_L=I\omega L$

• such that the inductive reactance $X_L=\omega L$ thus

$V_L=I{X}_L$

Capacitors

• because capacitors’ voltage is proportional to charge

$i=\frac{dq}{dt}=I\cos \omega t{v}_C=\frac{q}{C}=\frac{I}{\omega C}\sin \omega t$

• the phase angle is $\varphi =-90\text{degree}$

$v_C=\frac{I}{\omega C}\cos (\omega t+\varphi )=V_C\cos (\omega t-\frac{\pi }{2})$

• the voltage amplitude is given by the capacitive reactance

$X_C=\frac{1}{\omega C}⟹V_C=\frac{I}{\omega C}=I{X}_C$

Comparison

LRC Circuit (In Series)

• the current is $i=I\cos \omega t$
• amplitudes

$I,V,Q\propto e^{i\omega t}$

• Kirchhoff’s formula

$\mathcal E-i_1\space\V=L\frac{dI}{dt}+IR+\frac{Q}{C}\space$

• the parenthesized terms are complex impedances ($\tilde{Z}$) _(_added bc in series) such that the impedance is

$Z=\sqrt{R^2+{(\omega L-\frac{1}{\omega C})}^2}$

• the parenthesized terms are reactances $X_L,X_C$

$V_0=I_{0}Z{V}_{\text{rms}}=I_{\text{rms}}Z{\text{V}}_R=I_0{R}_0{\text{V}}_L=I_0{X}_L{\text{V}}_C=I_0{X}_C$

Phase Angles

• given the current $I$ is a real number, it scales the phasor parenthesized in the formula for the voltage → a phase angle

$V=I(R+i(\omega L-\frac{1}{\omega C}))$

${\varphi }_0=\arctan \left(\frac{\omega L-\frac{1}{\omega C}}{R}\right)$

Power

• we know

$P_{avg}=I^2{rms}R=V{rms}I_{rms}$

• power graphs

• in general

$p=vi=(I\cos \omega t)\cdot V\cos (\omega t+\varphi )$

$P_{\text{avg}}=\frac{1}{2}VI\cos \varphi =V_{\text{rms}}{I}_{\text{rms}}\cos \varphi$

• the power factor is $\cos \varphi$ s.t. $\varphi =0⟹P_{avg}=V_{rms}{I}_{rms}$ which occurs at pure resistance

Resonance

• the an freq of the source is varied → maximum current occurs at minimum impedance - resonance frequency

• the driving ang freq at which this occurs is the resonance angular frequency (${\omega }_0$)

$\omega ={\omega }_0⟹Z=R⟹X_L=X_C⟹{\omega }_0=\frac{1}{\sqrt{LC}}$

Transformers

• used to step-up/down voltages using a transformer

• a transformer has power supply to a primary coil around a core with high relative magnetic permeability $K_m$ (iron) and a secondary coil is wrapped around that delivers power to a resistor → via an emf in the core

${\mathcal{E}}_k=-N_k\frac{d{\mathrm{\Phi }}_B}{dt}$

• the flux per turn is the same in both coils so the ratio of emfs is proportional to the turns:

$\frac{{\mathcal{E}}_2}{{\mathcal{E}}_1}=\frac{N_2}{N_1}⟹\frac{V_2}{V_1}=\frac{N_2}{N_1}=\frac{I_1}{I_2}$

• $N_2>N_1⟹V_1>V_2$step-up transformer and vice versa
• the resistance in the secondary coil allows us to find

$V_1{I}_1=V_2{I}_2⟹\frac{V_1}{I_1}=\frac{R}{(N_2/N_1{)}^2}$

Discussion 5 Review

Review

Imaginary

• complex conjugate operator

$\ast :i\to -i$

• of form

$z=a+ib⟹(z_1+z_2)=(a_1+a_2)+i(b_1+b_2)$

• magnitude

$

z

=\sqrt{z\cdot z^*}=\sqrt{(a+ib)(a-ib)}=\sqrt{a^2+b^2}$

Polar Imaginary

$z=

z

\cos\phi\space+\space i

z

\sin\phi=

z

(\cos\phi\space+\space i\sin\phi)\space$

• time dependent phasor

$z(t)=

z

(\cos\omega t\space+\space \sin\omega t)$

RLC Circuit

• amplitudes

$I,V,Q\propto e^{i\omega t}$

• Kirchhoff’s formula

$V=L\frac{dI}{dt}+IR+\frac{Q}{C}\space$

• the parenthesized terms are complex impedances ($\tilde{Z}$) _(_added bc in series) such that the impedance is

$

\tilde Z

=\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}$

• the parenthesized terms are reactances $X_L,X_C$

$V_0=I_0

\tilde Z

\V_R=I_0R_0\V_L=I_0X_L\V_C=I_0X_C$

Phase Angles

• given the current $I$ is a real number, it scales the phasor parenthesized in the formula for the voltage → a phase angle

$V=I(R+i(\omega L-\frac{1}{\omega C}))$

$\varphi =\arctan \left(\frac{\omega L-\frac{1}{\omega C}}{R}\right)$

Resources

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