AC-generator

AC-Circuits

Alternating current

AC-circuits

AC-generator

Induced EMF

=N\ \left| \tfrac{d}{dt}(AB\cos\phi)\right|

\phi=\omega t

\mathcal E=N\ \left|\tfrac{d}{dt}(\Phi_m)\right|

for

\text{The induced EMF }

\text{each with area $A$,}

\text{rotating at a constant rate $\omega=d\phi/dt$, }

\text{in a uniform magnetic field $B$ :}

\text{in a coil of $N$ loops,}

\mathcal E=N AB\omega \sin\omega t

AC-circuits

AC-generator

Experimental results

\omega=d\phi/dt

\phi=\omega t

For constant

B_{\small \perp}=B\cos\phi

=B\cos\omega t

\mathcal E=N\ \tfrac{d}{dt}(AB\cos\omega t)

=N A B\ \omega\ \sin\omega t

AC-circuits

AC-generator

.

The output of an AC source flips polarity periodically.

The output of an DC source has a fixed polarity.

AC-circuits

AC-generator

AC-source

V =V_0\sin\omega t

(a) The output of an AC generator:

(b) Symbol used to represent an AC voltage source in a circuit diagram.

Parameters of an AC-source:

Peak EMF (Volt)

V_0

Angular Frequency (rad/s)

\omega

Frequency (1/s = Hz)

f

Period (seconds)

T

f=\tfrac{\omega}{2\pi}

T=\tfrac{1}{f}

AC-circuits

AC-generator

.

(a) The dc voltage and current are constant in time (once the current is established.)

(b) The voltage and current versus time are quite different for ac power. In this example, which shows 60-Hz ac power and time t in milliseconds, voltage and current are sinusoidal and are in phase for a simple resistance circuit.

AC-circuits

AC-generator

AC Sources

simple AC-circuits

AC-Circuits

Alternating current

AC-circuits

Section 15.2

AC-circuits

AC-source + Resistor

v_R(t)=v_s(t)=V_0 \sin\omega t

i_R(t)=\tfrac{v_R(t)}{R}=\tfrac{V_0}{R} \sin\omega t=I_0 \sin\omega t

AC-circuits

AC-source + Resistor

AC-circuits

AC-source + Capacitor

v_C(t)=V_0 \sin\omega t

i_C(t)=\tfrac{dQ}{dt}

=\tfrac{d}{dt} (C\ v_c(t))

={CV_0 \omega} \cos\omega t

={I_0} \cos\omega t

= C\tfrac{d}{dt} ( v_c(t))

X_C=\frac{V_0}{I_0}= \frac{1}{\omega C}

Current Leads Voltage by

\pi/2

AC-circuits

AC-source + Capacitor

AC-circuits

AC-source + Inductor

v_L(t)=V_0 \sin\omega t

=L\frac{di_L(t)}{dt}

i_L(t)=\int_0^t\tfrac{V_0}{L} \sin \omega t\ dt

=-\frac{V_0}{ \omega L} \cos\omega t

=-{I_0} \cos\omega t

X_L=\frac{V_0}{I_0}= {\omega L}

Current Lags Voltage by

\pi/2

AC-circuits

AC-source + Inductor

AC-circuits

Summary

RLC-circuits

AC-Circuits

Alternating current

AC-circuits

RLC-circuits

Section 15.3

AC-circuits

RLC-circuits

RLC in series

Consider a circuit where R, L, and C are in series with an AC-source. What is the electric current response?

How do we reconcile the different phase relationships?

AC-circuits

RLC-circuits

RLC in series

Consider a circuit where R, L, and C are in series with an AC-source. What is the electric current response?

Since the elements are in series, the same current flows through each element at all points in time.

The relative phase between the current and the emf is not obvious when all three elements are present. Consequently, we represent the current by the general expression:

i(t)=I_0 \sin(\omega t-\phi)

where the peak current and the phase-shift are unknowns.