REPASO EXAMEN 2INEL 4202 - M. Toledo

TEMAS

• Análisis de amplificadores para obtener Af, Rif y Rof

• Estabilidad

• Osciladores senoidales

Análisis de amplificadores para obtener Af, Rif y Rof

HINTS FEEDBACK AMPS• connect to vout (with node): voltage-sampling, else current-sampling

• connect to input (with node): current-mixing, else voltage-mixing

• From table: use correct feedback type, source, load and formulas to find β, R11, R22, gain, Rif and Rof

• If requested gain is different from table, find Af using feedback method and convert

• R seen by source + thevenin resistance: remove Rth from Rif

• R seen by load: remove RL from Rof

• Input and output resistances can be increased or decreased by choosing the feedback type/network

HINTS FEEDBACK AMPS• Feedback reduces gain by 1+Aβ, increases bandwidth and reduces

sensitivity by the same amount

• For a single-pole amplifier, bandwith = (1+Aβ) x (frequency of the pole of non-feedback amplifier)

• For the feedback amplifier, bandwidth is also called high-frequency corner, and equals (1+Aβ)fp where fp is the pole frequency for the non-feedback amplifier

• Non-feedback gain may also be called open-loop gain

• Feedback gain may also be called closed-loop gain, and approximates 1/β if |Aβ| >> 1

Stability

Basics

• Basic feedback equation:

Af(s) =a(s)

1 + β(s)a(s)

Thus, feedback moves the poles of the amplifier’s transfer func-tion.

• Poles of Af are roots of 1+βa. Thus, feedback moves the polesof the amplifier’s transfer function.

• The idea is to determine information about the stability of Af

from the loop gain T(s) = β(s)a(s).

Nyquist Theorem

Let ω180◦ be the frequency at which the loop gain’s phase angle is−180◦. If

| T(jω180◦) |=| β(jω180◦)A(jω180◦) |> 1

then the amplifier is unstable. Otherwise, it is stable.

Nyquist theorem allows us to answer questions about the stabilityof Af by analyzing the loop gain βA.

1

Phase and Gain Margin

• Gain margin: decibels below zero of | T(jω180◦) |.

• Phase margin: degrees above −180◦ at the frequency ω0dB atwhich | T(jω0dB) |= 1, or 0 db.

φm = 180 + ∠T(jω0dB)

Note that ∠T(jω0dB) is usually negative.

• The amplifier is unstable if the gain and phase margins are neg-ative. If the margins are positive or zero the amplifier is stableor marginally stable, respectively.

2

Amplifier gain, as a function of frequency and with n poles:

A(jf) =A0

�ni=1

�j f

fi+ 1

�

Amplifier transfer function (gain)

L=Aβ

• Solve the following for f0dB (frequency at which |A| = 0dB) and/or f180�

(frequency at which phase �A = 180�)

180� =n�

i=1

arctan

�f180�

fi

�

1 =A0

�ni=1

��f0dB

fi

�2+ 1

or A0,dB =1

2

n�

i=1

log

��f0dB

fi

�2

+ 1

�

• Calculate phase margin �m and/or gain margin Gm:

�m = 180 �n�

i=1

arctan

�f0dB

fi

�

Gm =1

2

n�

i=1

log

��f180�

fi

�2

+ 1

�� A0,dB

Analysis

To design, use the same formulas to select poles, feedback-network β, etc.

L L

L

Osciladores senoidales

Finding oscillation frequency ω0, gain• Given a circuit with feedback, you can calculate the loop gain T=Aβ(s) by

• finding A, β and multiplying, or

• opening the loop at a convenient point, and

• take care of including loading effects, and

• may apply a voltage test source Vt, find the return’s output voltage Vr and T= Vr / Vt

• Hint for phase condition

• if numerator of T=Aβ(jω0) is imaginary, denominator must be imaginary (real part must vanish)

• if numerator of T=Aβ(jω0) is real, denominator must be real (imaginary part must vanish)

• Magnitude condition: Select components so that |Aβ(jω0)| ≥1

Microelectronics: Analysis and Design© April 13, 2004 Sundaram Natarajan

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Ad (s ) 104

(1 53.05×10 6 s ) (1 2.653×10 9 s).

10.7: CONCLUSIONS

An important problem faced by the designer of feedback amplifiers is its stability, which is determined

by the loop-gain of the feedback amplifiers. Therefore, the stability analysis using the Nyquist criterion was

also discussed in this chapter. Normally, the designer is not only interested in determining the absolute stability

but also the stability margins. We illustrated the process of designing amplifiers for a given phase margin with

specific examples. The frequency compensation, which helps to stabilize the amplifiers with sufficient phase

margins, was also examined, and the frequency compensation of the most popular op-amp 741-type was

addressed. The frequency compensation of a MOSFET configuration using a RHP zero was also discussed.

Another closely related subject matter is the topic of sinusoidal oscillators, and their performance

characteristics depend on the loop-gain of a feedback circuit. Thus, in Sections 10.5 and 10.6, some important

sinusoidal oscillator circuits were discussed.

Many versatile electronic circuits can be designed for both linear and nonlinear applications using op-

amps and other modern monolithic IC devices. Several functional building blocks are considered in the next

chapter.

PROBLEMS

SECTION 10.1

10.1. An op-amp has a gain of

What maximum constant value of F (<1) can one achieve so that the pole-Q factor of the closed-loop

poles is less than or equal to 0.5?

10.2. A feedback amplifier, after suppressing the biasing details, is shown in Fig. P10.2. The BJTs have ȕ

= 200. The dc bias currents are: IC1 = 1 mA, IC2 = 2 mA, and IC3 = 5 mA. RC1 = RC2 = 0.621.8 kȍ ,

kȍ, RF = 12 kȍ, and RE = 0.5 kȍ. Find the loop-gain by finding A and F.

10.3. In Problem 10.2, assume that we add the junction capacitances of the transistors and find the loop-gain

to find the stability margins. Assume that Cµ1 = Cµ2 = Cµ3 = 2 pF, Cʌ1 = Cʌ2 = 72 pF, and Cʌ352 pF,

= 130 pF. Find the loop-gain as a function of frequency. (Hint: Use the unilateral models.)

10.4. In the circuit of Fig. P10.4, assume that Rs = RL = 10 kȍ, Rf = 1 kȍ, and r = The op-amp has0.1 kȍ .

Rid = 100 kȍ, Ro = 1 kȍ, and it has a finite gain of

Microelectronics: Analysis and Design© April 13, 2004 Sundaram Natarajan

786

Ad (s ) 105

(1 0.01s ) (1 10 6s) (1 10 7s).

A (s ) 100(1 10 4s ) (1 10 6s )

,

Fig. P10.2.

+-

RC1 RC21 k:

VsRF

+Vo

Q1

Q2

RE

Q3

Fig. P10.4.

+

-

Is Rs

Io

r

Ad

Rf

RL

Fig. P10.6.

K

1 nF10.45 k: 12.1 k:

+Vi +Vo

1 nF

Fig. P10.5.

+-Vs

10 k:Rif

0.5 mARof

+12 V

+VoQ1

10 k:

1 k:

Q2

330 :

Find the loop-gain.

10.5. In the circuit of Fig. P10.5, the BJTs have ȕ = 100, Cjeo = 20 pF, Cjco= 4 pF, and IJF = Find the500 ps .

loop-gain in the high frequency range.

10.6. Assume that the amplifier in the circuit of Fig. P10.6 is an ideal VCVS of gain K. Find the loop-gain.

Plot the movement of the characteristic roots as K varies from 0 to 3. Can you suggest the form of the

impulse response of the circuit, if (a) K = 1, (b) K = 2, and (c) K = 3.

10.7. If

Microelectronics: Analysis and Design© April 13, 2004 Sundaram Natarajan

785

Ad (s ) 104

(1 53.05×10 6 s ) (1 2.653×10 9 s).

10.7: CONCLUSIONS

An important problem faced by the designer of feedback amplifiers is its stability, which is determined

by the loop-gain of the feedback amplifiers. Therefore, the stability analysis using the Nyquist criterion was

also discussed in this chapter. Normally, the designer is not only interested in determining the absolute stability

but also the stability margins. We illustrated the process of designing amplifiers for a given phase margin with

specific examples. The frequency compensation, which helps to stabilize the amplifiers with sufficient phase

margins, was also examined, and the frequency compensation of the most popular op-amp 741-type was

addressed. The frequency compensation of a MOSFET configuration using a RHP zero was also discussed.

Another closely related subject matter is the topic of sinusoidal oscillators, and their performance

characteristics depend on the loop-gain of a feedback circuit. Thus, in Sections 10.5 and 10.6, some important

sinusoidal oscillator circuits were discussed.

Many versatile electronic circuits can be designed for both linear and nonlinear applications using op-

amps and other modern monolithic IC devices. Several functional building blocks are considered in the next

chapter.

PROBLEMS

SECTION 10.1

10.1. An op-amp has a gain of

What maximum constant value of F (<1) can one achieve so that the pole-Q factor of the closed-loop

poles is less than or equal to 0.5?

10.2. A feedback amplifier, after suppressing the biasing details, is shown in Fig. P10.2. The BJTs have ȕ

= 200. The dc bias currents are: IC1 = 1 mA, IC2 = 2 mA, and IC3 = 5 mA. RC1 = RC2 = 0.621.8 kȍ ,

kȍ, RF = 12 kȍ, and RE = 0.5 kȍ. Find the loop-gain by finding A and F.

10.3. In Problem 10.2, assume that we add the junction capacitances of the transistors and find the loop-gain

to find the stability margins. Assume that Cµ1 = Cµ2 = Cµ3 = 2 pF, Cʌ1 = Cʌ2 = 72 pF, and Cʌ352 pF,

= 130 pF. Find the loop-gain as a function of frequency. (Hint: Use the unilateral models.)

10.4. In the circuit of Fig. P10.4, assume that Rs = RL = 10 kȍ, Rf = 1 kȍ, and r = The op-amp has0.1 kȍ .

Rid = 100 kȍ, Ro = 1 kȍ, and it has a finite gain of

Microelectronics: Analysis and Design© April 13, 2004 Sundaram Natarajan

786

Ad (s ) 105

(1 0.01s ) (1 10 6s) (1 10 7s).

A (s ) 100(1 10 4s ) (1 10 6s )

,

Fig. P10.2.

+-

RC1 RC21 k:

VsRF

+Vo

Q1

Q2

RE

Q3

Fig. P10.4.

+

-

Is Rs

Io

r

Ad

Rf

RL

Fig. P10.6.

K

1 nF10.45 k: 12.1 k:

+Vi +Vo

1 nF

Fig. P10.5.

+-Vs

10 k:Rif

0.5 mARof

+12 V

+VoQ1

10 k:

1 k:

Q2

330 :

Find the loop-gain.

10.5. In the circuit of Fig. P10.5, the BJTs have ȕ = 100, Cjeo = 20 pF, Cjco= 4 pF, and IJF = Find the500 ps .

loop-gain in the high frequency range.

10.6. Assume that the amplifier in the circuit of Fig. P10.6 is an ideal VCVS of gain K. Find the loop-gain.

Plot the movement of the characteristic roots as K varies from 0 to 3. Can you suggest the form of the

impulse response of the circuit, if (a) K = 1, (b) K = 2, and (c) K = 3.

10.7. If

-+ + +

-

(Note: their F is our β)

Microelectronics: Analysis and Design© April 13, 2004 Sundaram Natarajan

787

A (s ) 105

(1 10 2s ) (1 10 4s )2.

A (s ) 105

(1 10 4s )3.

Ad (s ) 105

(1 10 3s ) (1 10 6s ) (1 10 8s ).

Ad (s ) 300×103

(1 4×10 3s ) (1 72.3×10 9s ).

for what constant value of F will the closed-loop poles have (a) the same value and (b) the magnitudes

of the imaginary and real parts are equal?

10.8. An op-amp has a gain of

If the feedback is constant, at what frequency will the phase of the loop-gain be -180 ? For what value

of F will the magnitude of the loop-gain be unity at this frequency? This is the critical F at which the

oscillations will commence.

10.9. Repeat Problem 10.8, if A is

10.10. For the loop-gain obtained in Problem 10.3, find the values of the gain-crossover frequency, phase-

crossover frequency, and the phase and gain margins.

10.11. Repeat the Problem 10.10 for the circuit described in Problem 10.3.

10.12. Recall the noninverting amplifier designed in Example 8.4 using circuit of Fig. 8.3.6 in which R1 = 2

kȍ and R2 = 18 kȍ. The op-amp has Rid = 100 kȍ and Ro = 100 ȍ. The op-amp has a frequency

dependent gain of

Find the stability margins of the closed-loop amplifier.

10.13. Repeat Problem 10.12 with R1 = , and R2 = 0 (unity-gain buffer).

10.14. The circuit shown in Fig. P10.14 is an equivalent circuit of an inverting amplifier. CS1 is the lumped

effect of the stray and input capacitances, which is 20 pF. Assume that R1 = 10 kȍ and R2 = 100 kȍ.

The op-amp has Rid = 2 Mȍ, Ro = 100 ȍ, and

CL is a load capacitor having a value of 500 pF. Note that CL and CS1 introduce two additional poles

to the loop-gain. Find the loop-gain and determine the stability margins.

10.15. In the circuit of Fig. P10.14, if we connect a 3-pF capacitor across the resistor R2, find the new loop-

gain and the stability margins.

D10.16.In the design referred to in Problem 10.12, assume that the op-amp has Rid and Ro 0. However,

--+ +