The inductor and capacitor elements have much more application
when they are connected to an alternating current (ac) source.
One important application is the resonance circuit (LCR circuit
on P.912).
The concept of resonance and natural frequency are very general,
resonance also occur in mechanical systems (such as a mass-spring,
pendulum, vibrations,...)
The LCR circuit has a natural frequency (or called resonance
frequency). When the applied ac source has the same frequency
as the natural frequency, the current through the circuit is
maximum. (Demo)
We will first describe the characteristic of an ac source
We study the current-voltage relationship for R, C, and L.
Remember we don't need any new equations. All you need to know
is:
Vr = IR
Vc=Q/C
VL=LdI/dt
We will analyse the circuit in two ways:
(1) Using differential equations
(2) Using complex impedance (similar to phasors)
AC source
A sinusoidal alternating source has an EMF given by the equation
above 28-2.
Omega = 2*pi*f ; f=frequency (e.g. f=60 for 60 cycles/sec)
The period T =1/f
Shfiting the origin of the cosine function: Discuss in class.
R, C, L-circuit
R-circuit.
Setup: Fig. 28-1, P.899
Key : Vr = EMF and Vr=IR ; solve for I
I is in phase with Vr (i.e. both I anf V are coswt)
There is power consumed by the resisitor. The average is
1/2 the maximum value.
C-circuit
Setup:Fig. 28-6, P.904
Key: Vc = EMF and Vc=Q/C; solve for Q(t) and find I =dQ/dt
I and Vc are out of phase (Vc =coswt, I = -sinwt; see Fig.
28-7, P.905)
The average power output is zero (because I and V are 90
degree out of phase. The capacitor consumes zero power. The capacitor
is charging for half of the cycle and discharging for the other
half.
1/wC is the capacitive reactance; it has the unit of resistance
(Ohms)
L-circuit
Setup: Fig. 28-4, P. 902.
Key: VL=EMF and VL=LdI/dt; integrate to get I(t).
I and VL are out of phase (VL = coswt; I=sinwt; see Fig.
28-5, P.903)
Again, the average power consumed is zero.
wL is the inductive reactance; it has the unit of resistance
(Ohms)
** Only resistor consumes power.
LCR-circuit (resonance circuit)
Setup: Fig. 28-15, P. 912
Key: EMF = Vr + Vc + VL
=> differential equation (DE) for Q(t).
For starter: Assume R=0. Then Q(t)=Q(max) coswt will satisfy
the DE.
Substitue Q(t) into DE ans solve for Q(max)
What happens to Q(max) when w = square root of (1/LC) = natural
frequency (in rad/sec) ?
From Q(t), calculate I(t)
Can you write I(max) = EMF (max) /something ? What is that
something?
Calculate dI/dt and hence VL
Compare Vc and VL, are they in phase or out of phase?
At low frequency (w < natural frequency), which one is
bigger, Vc or VL?
At high frequency , which one is bigger?
Can Vc or VL greater than EMF ?
Now: Do not assume R=0. Try again Q(t)=Q(max) coswt. Does
it work?
You can try Q(t)=Q(max) cos (wt - delta). Substitue into
DE and solve for Q(max) and delta. (Math may get messy. Try it.
You should get Eq. 28-48 for delta)
The math is not as messy if we use "complex" function
and introduce complex impedance, see next lecture. Click here
for a brief review of complex number: