Download Resonant Circuits - Ohio Wesleyan University

Parallel LC Resonant Circuit
• Consider the following parallel LC circuit: (Lab 3–1)
R
Vout
Vin
C
L
(Student Manual for The Art
of Electronics, Hayes and
Horowitz, 2nd Ed.)
Z LC
Vin
– Treating as a voltage divider, we have: Vout 
R  Z LC
– Calculate the (complex) impedance ZLC:
1
1
1
1
C
1 





 j C 

Z LC Z L Z C jL
j
L 

Z LC
jL


2
1
1


LC
 C
L
j
Parallel LC Resonant Circuit
• Thus we have:
Z LC
  jL  jL 
 


2
2
 1   LC  1   LC 
L 2
1   LC 
2
2

L
1   2 LC

jL 
jL 
L 

2
 R
 R 
 R 
2
2
1   LC 
1   LC 

1   2 LC
2
R  Z LC
Vout

Vin

L
1   LC 
2

L 
2
R2 
L 
2
1   LC 
2
L

 R 1   LC
2
2

2

2
2
Vout
1
1
– Note that for   0 
(resonant frequency):
Vin
LC
(Remember that  = 2pf )
Vout
– Otherwise
is small
Vin
Parallel LC Resonant Circuit
• Overall response (Vout / Vin vs. frequency):
Q = quality factor = f0 / Df3dB
= resonance frequency / width at –3 dB points
(Remember that at –3 dB point, Vout / Vin = 0.7 and
output power is reduced by ½ )
Q is a measure of the sharpness of the peak
For a parallel RLC circuit:
Q  0 RC
(The Art of Electronics, Horowitz and Hill, 2nd Ed.)
– This circuit is sometimes called a tank circuit
– Most often used to select one desired frequency from a
signal containing many different frequencies
• Used in radio tuning circuits
• Tuning knob is usually a variable capacitor in a parallel LC circuit
Oscillation in Parallel LC Resonant Circuit
(Introductory Electronics,
Simpson, 2nd Ed.)
Oscillation in Parallel LC Resonant Circuit
• For a pure LC circuit (no resistance), the current and
voltage are exactly sinusoidal, constant in amplitude,
1
and have angular frequency   0 
LC
– Can prove with Kirchhoff’s loop rule
– Analogous to mass oscillating on a spring with no friction
• For an RLC circuit (parallel or series), the current and
voltage will oscillate (“ring”) with an exponentially
decreasing amplitude
– Due to resistance in circuit
– Analogous to damped
oscillations of a mass
on a spring
(Lab 3–1)
(Introductory Electronics, Simpson, 2nd Ed.)
Series LC Resonant Circuit
• Consider the following series LC circuit: (HW #1.26)
Vout 
Z LC
R  Z LC
Vin
(The Art of Electronics, Horowitz and Hill, 2nd Ed.)
– Now ZLC = ZC + ZL = jL – j / C
• Overall response:
For series RLC circuit:
Q
f0
L
 0
Df 3 dB
R
(The Art of Electronics, Horowitz and Hill, 2nd Ed.)
(L and C in series)
Fourier Analysis
(Lab 3–1)
• In Lab 3–1, a parallel LC resonant circuit is used as a
Fourier Analyzer
– The circuit “picks out” the Fourier components of the input
(square) waveform
• Fourier analysis: Any function can be written as the
sum of sine and cosine functions of different
frequencies and amplitudes
– We can apply this technique to periodic voltage waveforms:
a0 
2pnt 
2pmt
V (t )    an cos
  bm sin
2 n 1
T
T
m 1
– Where T = minimum time voltage waveform repeats itself
and 1 / T = fundamental frequency = f0
– Could instead substitute   2p / T
Fourier Analysis
• The an and bm constants are determined from:
T /2
2
an 
V (t ) cos nt dt

T T / 2
T /2
2
bm 
V (t ) sin mt dt

T T / 2
• For a symmetric square wave voltage (assuming V(t)
is an odd function):
– an = 0
4
– bm 
T
– V (t ) 
n = 0, 1, 2, 3, …
T /2
 V (t ) sin mt dt
0
bm  0 m even
4V0

mp
m odd
4V0  sin t sin 3t sin 5t



 ...

p  1
3
5

Fourier Analysis
• Thus for a square wave of fundamental frequency 0:
(Student Manual for The Art
of Electronics, Hayes and
Horowitz, 2nd Ed.)
– When we apply an input square wave voltage of frequency
0 to the parallel LC circuit, we are in essence applying
frequencies 0, 30, 50, etc. simultaneously with relative
amplitudes 1, 1/3, 1/5, etc. (respectively)
– The LC circuit is a “detector” of its resonance frequency f0,
including contributions from the harmonics of the input
fundamental frequency
• “Mini-resonance” peaks will occur in the output voltage at driving
frequencies of f0 / 3, f0 / 5, etc.
Diodes
• Diodes are semiconductor devices that are made when p–
type and n–type semiconductors are joined together to form a
p–n junction
– With no external voltage applied, there is some electron
flow from the n side to the p side (and similar for holes),
but equilibrium is established and there is no net current
(Introductory Electronics, Simpson, 2nd Ed.)
Diodes
• With a reverse bias external voltage applied, there is only a
small net flow of electrons from the p side to the n side, and
hence a small positive current from the n to the p side
(Introductory Electronics, Simpson, 2nd Ed.)
Diodes
• With a forward bias external voltage applied, electrons are
“pushed” in the direction they would tend to move anyway,
and hence there is a large positive current from the p side to
the n side
(Introductory Electronics, Simpson, 2nd Ed.)
Diodes
• Thus diodes pass current in one direction, but not
(Student Manual for The Art
the other
of Electronics, Hayes and
Horowitz, 2nd Ed.)
When diodes are forwardbiased and conduct current,
there is an associated
voltage drop of about 0.6 V
across the diode (for Si
diodes) – “diode drop”
• The diode’s arrow on a circuit diagram points in the
direction of current flow
Current can flow
X
Current can’t flow
Diodes in Voltage Divider Circuits
• Consider diodes as part of the following voltagedivider circuits:
(1)
Vin
Vout
• This diode circuit is called a rectifier (specifically, a
half-wave rectifier) (Lab 3–2)
Diodes in Voltage Divider Circuits
(2)
Vin
Vout
• This circuit is called a diode clamp circuit because
the output voltage is “clamped” at about –0.6 V
(Lab 3–6)
Diodes in Voltage Divider Circuits
(3)
Vin
Vout
• This is another clamp circuit: the output voltage is
clamped at about +5.6 V and –0.6 V
(Lab 3–6)
Diode Applications
• Rectification: conversion of AC to DC voltage
– We already saw how this could be done with a half-wave
rectifier
– A much better way is with a full-wave bridge rectifier:
(Lab 3–3)
(The Art of Electronics, Horowitz and Hill, 2nd Ed.)
– Two diodes are always in series with the input (so there will
always be 2 forward diode drops)
– Gap at 0 V occurs because of diodes’ forward voltage drop
Diode Applications
• Although more efficient than the half-wave rectifier,
the bridge rectifier still produces a lot of “ripple”
(periodic variations in the output voltage)
– The ripple can be reduced by attaching a low-pass filter:
(Lab 3–4)
(The Art of Electronics, Horowitz and Hill, 2nd Ed.)
– The resistor R is actually unnecessary and is always
omitted since the diodes prevent flow of current back out
of the capacitors
– C is chosen to ensure that RloadC >> 1 / fripple so the time
constant for discharge >> time between recharging
Diode Applications
• We have almost finished building our own DC power
supply!
• For further power supply design details, see Class 3
Worked Example in the Lab Manual (p. 71–74)
(Student Manual for The Art of Electronics, Hayes and Horowitz, 2nd Ed.)
Diode Applications
• Signal rectifier (Lab 3–5)
– Eliminates an unwanted polarity of a waveform
– Example: Remove sharp negative spikes from the output
of a differentiator
– An RC differentiator is used to generate the spikes, and a
diode is used to rectify the spikes:
(The Art of Electronics, Horowitz and Hill, 2nd Ed.)
Diode Applications
• Voltage limiter (Lab 3–7)
– In the circuit below, the output voltage is limited to the
range –0.6 V  Vout  +0.6 V
– This is just another example of a diode clamp circuit
– Useful as an input protection circuit for a high-gain
amplifier (otherwise amplifier may “saturate”)
(The Art of Electronics,
Horowitz and Hill, 2nd Ed.)
Example Problem: Chap. 1 AE 7
Sketch the output for the circuit
shown at right. (Solution details
will be discussed in class.)