Analogue ElectronicsPaolo Colantonio

A.A. 2015-16

Università degli Studi di Roma Tor VergataDipartimento di Ingegneria Elettronica

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Linear permanent (LP) systemThe output signal Su(t) is a replica of the input signal Si(t), eventually amplified and phase shifted

The transfer function H() represents the in‐out relationship in frequency domain

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Frequency response

• The frequency response of a network is identified by the variation of its transfer function (i.e., voltage, current or power) vs. frequency, both in terms of amplitude and phase shift.Example

Transfer function

The frequency response is characterized by the Bode diagrams

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High Pass RC filter

Defining the resonating frequency

f << fC f = fC f >> fCIm(AV)

Re(AV)

Im(AV)

Re(AV)

Im(AV)

Re(AV)

1

12

1

1

1

1

tan

11 1

→0

→1

12

0,707 3→90°

→0°45°

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High Pass RC filter

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Low Pass RC filterDefining the resonating frequency

f << fC f = fC f >> fCIm(AV)

Re(AV)

Im(AV)

Re(AV)

Im(AV)

Re(AV)

1

11

11

21

1

1

1tan

111

→0

→1 1

20,707 3

→90°

→0° 45°

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Low Pass RC filter

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RL Filters: Low Pass

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RL Filters: High Pass

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A Comparison of RC and RL Networks

• Circuits using RC and RL techniques have similar characteristics

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Bode Diagrams• Straight‐line approximations

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Bode Diagrams• Creating more detailed Bode diagrams

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Cascade of networks: example of HP‐LP

If fC1<<fc2 Band pass filter

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Band Pass and Band Stop (Notch filter)

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Parallel RLC

Defining the resonating frequency

And the parameter Q (quality factor)

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Parallel RLC

The frequency behavior of the transfer function AI is reported in the followingamplitude and phase plots

The resonating phenomenon is highlighted in the amplitude plot and the frequencybehavior is strictly depending on the value of Q, that is also referred as resonatingcoefficient.

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Series RLC

The voltage across each component and the currentcan be easily determined in frequency domain

Defining the resonating frequency

And the parameter Q (quality factor)

The same consideration as for theparallel RLC case can be applied.

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Summary of resonances in RLC circuits

Series RLC

• The impedance is given by

• if the magnitude of the reactance of theinductor and capacitor are equal, theimaginary part is zero, and the impedance issimply R. This occurs when

• This situation is referred to as resonance• The frequency at which is occurs is the 

resonant frequency• In the series resonant circuit, the 

impedance is at a minimum at resonance while the current is at a maximum

• The series RLC circuit is an acceptor circuit 

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Summary of resonances in RLC circuits

Parallel RLC

• The admittance is given by

• if the magnitude of the reactance of theinductor and capacitor are equal, theimaginary part is zero, and the admittance issimply 1/R. This occurs when

• This situation is referred to as resonance• The frequency at which is occurs is the 

resonant frequency• In the parallel resonant circuit, the 

impedance is at a maximum at resonance while the current is at a minimum

• The parallel RLC circuit is a rejector circuit 

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Filters

• The RC networks considered earlier are first‐order or single‐pole filters.• These have a maximum roll‐off of 6 dB/octave• they also produce a maximum of 90° phase shift

• Combining multiple stages can produce filters with a greater ultimate roll‐off rates (12 dB, 18 dB, etc.) but such filters have a very soft ‘knee’

• An ideal filter would have constant gain and zero phase shift for frequencies within its pass band, and zero gain for frequencies outside this range (its stop band)

• Real filters do not have these idealized characteristics

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FiltersThe use of combination of inductors and capacitors can produce very high performance filters.

Common forms include:• Butterworth

• optimised for a flat response• Chebyshev

• optimised for a sharp ‘knee’• Bessel

• optimised for its phase response

• The use of inductors is inconvenient since they are expensive, bulky and suffer from greater losses than other passive components.

• Actually, active filters are realized by using operational amplifiers

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Notes

• All circuits have losses (i.e., intrinsically resistors)

• All circuits have stray capacitance and stray inductance

• These unintended elements can dramatically affect circuit operation and have to be carefully and properly taken into account

Cs adds an unintended low‐pass filter

Ls adds an unintended low‐pass filter

Cs produces an unintended resonant circuit and can produce instability