Parallel LC Resonant Circuit• Consider the following parallel LC circuit:

– Treating as a voltage divider, we have:

– Calculate the (complex) impedance ZLC:

R

C

Vin Vout

L(Student Manual for The Art of Electronics, Hayes and Horowitz, 2nd Ed.)

inout VZR

ZV

LC

LC

LCj

j

C

LjZZZ CLLC

11111

LC

Lj

CL

jZLC 211

(Lab 3–1)

Parallel LC Resonant Circuit• Thus we have:

– Note that for (resonant frequency):

– Otherwise is small

LC

L

LC

L

LC

Lj

LC

LjZLC 222

2

22 1111

22

22

22111 LC

LR

LC

LjR

LC

LjRZR LC

2222

22

222in

out

1

11 LCRL

L

LC

LRLC

L

V

V

LC

10 1

in

out V

V

in

out

V

V (Remember that = 2f)

Parallel LC Resonant Circuit• Overall response (Vout / Vin vs. frequency):

– 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

(The Art of Electronics, Horowitz and Hill, 2nd Ed.)

Q = quality factor = f0 / f3dB = 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: RCQ 0

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, and have angular frequency– 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

LC

10

(Introductory Electronics, Simpson, 2nd Ed.)

(Lab 3–1)

Series LC Resonant Circuit• Consider the following series LC circuit:

– Now ZLC = ZC + ZL = jL – j / C (L and C in series)

• Overall response:

inout VZR

ZV

LC

LC

(The Art of Electronics, Horowitz and Hill, 2nd Ed.)

(The Art of Electronics, Horowitz and Hill, 2nd Ed.)

R

L

f

fQ 0

dB3

0

For series RLC circuit:

(HW #1.26)

Fourier Analysis• 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:

– Where T = minimum time voltage waveform repeats itself and 1 / T = fundamental frequency = f0

– Could instead substitute T

1 1

0 2sin

2cos

2)(

n mmn T

mtb

T

nta

atV

(Lab 3–1)

Fourier Analysis• The an and bm constants are determined from:

• For a symmetric square wave voltage (assuming V(t) is an odd function):– an = 0 n = 0, 1, 2, 3, …

–

–

2/

2/

cos)(2 T

T

n dttntVT

a

2/

2/

sin)(2 T

T

m dttmtVT

b

2/

0

sin)(4 T

m dttmtVT

b odd

4

even0

0 mm

V

mbm

...

5

5sin

3

3sin

1

sin4)( 0 tttV

tV

Fourier Analysis• Thus for a square wave of fundamental frequency 0:

– When we apply an input square wave voltage of frequency 0to the parallel LC circuit, we are in essence applying frequencies , 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.

(Student Manual for The Art of Electronics, Hayes and Horowitz, 2nd Ed.)

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

the other

• The diode’s arrow on a circuit diagram points in the direction of current flow

When diodes are forward-biased and conduct current, there is an associated voltage drop of about 0.6 V across the diode (for Si diodes) – “diode drop”

(Student Manual for The Art of Electronics, Hayes and Horowitz, 2nd Ed.)

Current can flow

Current can’t flowX

Diodes in Voltage Divider Circuits• Consider diodes as part of the following voltage-

divider circuits:

(1)

• This diode circuit is called a rectifier (specifically, a half-wave rectifier)

Vin

Vout

(Lab 3–2)

Diodes in Voltage Divider Circuits(2)

• This circuit is called a diode clamp circuit because the output voltage is “clamped” at about –0.6 V

Vin

Vout

(Lab 3–6)

Diodes in Voltage Divider Circuits(3)

• This is another clamp circuit: the output voltage is clamped at about +5.6 V and –0.6 V

Vin

Vout

(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:

– 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

(The Art of Electronics, Horowitz and Hill, 2nd Ed.)

(Lab 3–3)

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:

– 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

(The Art of Electronics, Horowitz and Hill, 2nd Ed.)

(Lab 3–4)

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

– 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.)

(Lab 3–5)

Diode Applications• Voltage limiter

– 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.)

(Lab 3–7)

Example Problem: Chap. 1 AE 7

Sketch the output for the circuit shown at right. (Solution details will be discussed in class.)