Microelectronic Circuits, Kyung Hee Univ. Spring, 2016 1.6 … · 2018-09-10 · Microelectronic Circuits, Kyung Hee Univ. Spring, 2016 3 1.6.2 Amplifier Bandwidth •Amplifier bandwidth

Microelectronic Circuits, Kyung Hee Univ. Spring, 2016

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1.6 Frequency Response of Amplifiers

• Input signal = sum of sinusoidal signals

• Important its response to input sinusoids of different frequencies

• Amplifier frequency response


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1.6.1 Amplifier Frequency Response

• Whenever a sinewave signal is applied to a linear circuit, the resulting output is sinusoidal with the same frequency as the input

• Transfer function / amplifier transmission

• Magnitude of the amplifier gain @ 𝜔 : 𝑻(𝝎) =𝑉𝑜

𝑉𝑖

• Phase of the amplifier transmission: ∠𝑇 𝜔 = 𝜙

• At the test frequency, ω

• Change the frequency of the input sinusoid and measure the value of 𝑻 𝒂𝒏𝒅 ∠𝑇

• Magnitude or amplitude response

• Phase response

Figure 1.20 Measuring the frequency response of a linear amplifier: At the test frequency ω , the amplifier gain is characterized

by its magnitude (Vo /Vi) and phase ø .


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1.6.2 Amplifier Bandwidth

• Amplifier bandwidth• Band of frequencies over which the gain of the amplifier is almost constant

• 3dB

• If not, distort the frequency spectrum of the input signal with different components of the input signal being amplified by different amounts


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1.6.3 Evaluating the Frequency Response

• Analytically obtaining an expression for the frequency response

• To evaluate the frequency response of an amplifier,• Inductance L: 𝑗𝜔𝐿

• Capacitance C: 1

𝑗𝜔𝐶

• Frequency domain analysis (impedance and/or admittance)

• Amplifier transfer function 𝑇(𝜔) =𝑉𝑜(𝜔)

𝑉𝑖(𝜔)

• Using complex frequency variable, s,• Inductance L: 𝑠𝐿

• Capacitance C: 1

𝑠𝐶

• Performing standard circuit analysis

• 𝑇(𝑠) =𝑉𝑜(𝑠)

𝑉𝑖(𝑠)

• Replace s by 𝑗𝜔 to determine the transfer function for physical frequencies, 𝑇(𝑗𝜔) obtained from 𝑇(𝑠)


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Figure 1.22 Two examples of STC networks: (a) a low-pass network and (b) a high-pass network.

1.6.4 Single-Time-Constant Networks

• One reactive component(L or C) and one resistance

• Time constant, 𝜏 =𝐿

𝑅, 𝜏 = 𝐶𝑅

• Low-pass filter

• High-pass filter

• Bode plots, corner frequency, break frequency, pole frequency


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Figure 1.23 (a) Magnitude and (b) phase response of STC networks of the low-pass type.

Figure 1.24 (a) Magnitude and (b) phase response of STC networks of the high-pass type.



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1.6.5 Classification of Amplifiers

• Shape of magnitude response curve (or phase curve)

• 1.26(a) case:• Remain constant over a wide frequency range

• Internal capacitances in the device (effect on the high frequency)

• Coupling capacitors used to connect one amplifier stage to another

• 1.26(b) case: IC amplifiers, directly coupled or dc amplifiers

• 1.26(c) case: tuned amplifiers, bandpass amplifiers, bandpass filter

Figure 1.27 Use of a capacitor to couple amplifier stages.


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Figure 1.26 Frequency response for (a) a capacitively coupled amplifier, (b) a direct-coupled amplifier, and (c) a tuned or

bandpass amplifier.


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Homeworks

• Example 1.5


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2.5 Integrators and Differentiators

• Utilized resistors in the op-amp feedback and feed-in path

• Ideally independent of frequency

• Use of capacitors together with resistors in the feedback and feed-in paths of op-amp circuits

• Inverting closed-loop configuration with impedances 𝑍1(𝑠), 𝑍2(𝑠)

•𝑉𝑜(𝑠)

𝑉𝑖(𝑠)= −

𝑍2(𝑠)

𝑍1(𝑠)

• 𝑠 → 𝑗𝜔

Figure 2.22 The inverting configuration with general impedances in the feedback and the feed-in paths.


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2.5.2 The Inverting Integrator

• 𝑖1 𝑡 = 𝑣𝐼 𝑡 𝑅

• Charge to accumulate on C = 0𝑡𝑖1 𝑡 𝑑𝑡

• Capacitor voltage: 1

𝐶 0𝑡𝑖1 𝑡 𝑑𝑡, initial voltage on C(t=0): 𝑉𝐶

• 𝑣𝐶 𝑡 = 𝑉𝐶 +1

𝐶 0𝑡𝑖1 𝑡 𝑑𝑡

• 𝑣𝑂 𝑡 = −𝑣𝐶 𝑡 = −1

𝑅𝐶 0𝑡𝑣𝐼 𝑡 𝑑𝑡 − 𝑉𝐶

• 𝑅𝐶 : integrator time constant

• Proportional to the time

integral of the input with

𝑉𝐶 being the initial

condition of integration

Figure 2.24 (a) The Miller or inverting integrator. (b) Frequency response of the integrator.


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Inverting Integrator (Miller Integrator)

• 𝑍1 𝑠 = R, 𝑍2 𝑠 = 1 𝑠𝐶, 𝑉𝑜(𝑠)

𝑉𝑖(𝑠)= −

1

𝑠𝐶𝑅

• 𝑠 → 𝑗𝜔, 𝑉𝑜(𝑗𝜔)

𝑉𝑖(𝑗𝜔)= −

1

𝑗𝜔𝐶𝑅

•𝑉𝑜

𝑉𝑖=

1

𝜔𝐶𝑅, 𝜙 = +90°, -20 dB/decade

• 𝜔𝑖𝑛𝑡 =1

𝐶𝑅: integrator frequency

• Low-pass filter with a corner frequency of zero

• 𝜔 = 0, • Magnitude of the integrator transfer function = infinite = open loop

• Tiny dc component produce an infinite output

Figure 2.24 (a) The Miller or inverting integrator. (b) Frequency response of the integrator.


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• Alleviate by connecting a resistor 𝑅𝐹 across the integrator capacitor C

• Gain @ DC of −𝑅𝐹 𝑅 (dc feedback path)

•𝑉𝑜(𝑠)

𝑉𝑖(𝑠)= −

𝑅𝐹 𝑅

1+𝑠𝐶𝑅𝐹(Example 2.4)

• Lower 𝑅𝐹, higher the corner frequency ( 1 𝐶𝑅𝐹)

• Become more nonideal integrator

• Trade-off between dc performance and

signal performance

Figure 2.25 The Miller integrator with a large resistance RF connected in parallel with C

in order to provide negative feedback and hence finite gain at dc.


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2.5.3 Op-Amp Differentiator

• Current through C: 𝑖 𝑡 = 𝐶(𝑑𝑣𝐼 𝑑𝑡)

• 𝑣𝑂 𝑡 = −𝐶𝑅𝑑𝑣𝐼(𝑡)

𝑑𝑡

• 𝑍1 𝑠 = 1 𝑠𝐶, 𝑍2 𝑠 = 𝑅, 𝑉𝑜(𝑠)

𝑉𝑖(𝑠)= −𝑠𝐶𝑅

• 𝑠 → 𝑗𝜔, 𝑉𝑜(𝑗𝜔)

𝑉𝑖(𝑗𝜔)= −𝑗𝜔𝐶𝑅

•𝑉𝑜

𝑉𝑖= 𝜔𝐶𝑅,𝜙 = −90°, +20 dB/decade

• 𝐶𝑅 : differentiator time constant

• STC highpass filter with a corner frequency

at infinity (Noise magnifier)

• Stability problem: small valued resistor

in series with the capacitorFigure 2.27 (a) A differentiator. (b) Frequency response of a

differentiator with a time-constant CR.


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2.6 DC Imperfections

• Assume the op amp to be ideal except of op amp with a finite gain A on the closed-loop gain of the inverting and noninverting configuration

• Familiar with the characteristics of practical op amps

• Effects of such characteristics on the performance of op amp circuits

• Op amp: direct-coupled devices with large gains at dc

• Input offset voltage (𝑉𝑂𝑆): opposite polarity of external source which balances out the input offset of the op amp

• As a result of the unavoidable mismatches

• 𝑉𝑂𝑆 depends on temperature

• Finite CMRR

• Finite input resistance

• Nonzero output resistanceFigure 2.28 Circuit model for an op amp with input offset voltage VOS.


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• Effect of the op-amp 𝑉𝑂𝑆 on their performance

• 𝑉𝑂 = 𝑉𝑂𝑆 1 +𝑅2

𝑅1

• Provide two additional terminals

to trim to zero the output dc

voltage due to 𝑉𝑂𝑆

• Introduce an imbalance

• Remain the problem of variation of 𝑉𝑂𝑆

with temperature

Figure 2.29 Evaluating the output dc offset voltage due to VOS in

a closed-loop amplifier.

Figure 2.30 The output dc offset voltage of an op amp can be trimmed to

zero by connecting a potentiometer to the two offset-nulling terminals. The

wiper of the potentiometer is connected to the negative supply of the op

amp.


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Capacitive Coupling

• 𝑉𝑂𝑆 1 +𝑅2

𝑅1without the coupling capacitor

• Gain to be zero at dc

• dc output voltage with coupling capacitor, 𝑉𝑂 = 𝑉𝑂𝑆

• STC high-pass circuit with a corner frequency of 𝜔0 = 1 𝐶𝑅1

• Gain from a magnitude of (1 +𝑅2

𝑅1) at high frequency and 3 dB down

at 𝜔0

Figure 2.31 (a) A capacitively coupled inverting amplifier. (b) The equivalent circuit for determining its dc output offset voltage VO.


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2.6.2 Input Bias and Offset Currents

• Two input terminals have to be supplied with dc currents (𝐼𝐵1, 𝐼𝐵2)

• Independent of the fact that a real op amp has finite input resistance

• Input bias current, 𝐼𝐵 =𝐼𝐵1+𝐼𝐵2

2

• Input offset current, 𝐼𝑂𝑆 = 𝐼𝐵1 − 𝐼𝐵2

• 𝑉𝑂 = 𝐼𝐵1𝑅2 ≅ 𝐼𝐵𝑅2

• Upper limit on the value of 𝑅2

Figure 2.32 The op-amp input bias currents represented by two current

sources IB1 and IB2.

Figure 2.33 Analysis of the closed-loop amplifier, taking into account

the input bias currents.


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• Introducing 𝑅3 in series with the noninverting input lead

• 𝑉𝑂 = −𝐼𝐵2𝑅3 + 𝑅2 𝐼𝐵1 − 𝐼𝐵2𝑅3 𝑅1

• Consider 𝐼𝐵1 = 𝐼𝐵2 = 𝐼𝐵, 𝑉𝑂 = 𝐼𝐵 𝑅2 − 𝑅3 1 + 𝑅2 𝑅1

• 𝑉𝑂 = 0, when 𝑅3 =𝑅2

1+ 𝑅2 𝑅1=

𝑅1𝑅2

𝑅1+𝑅2: parallel equivalent of 𝑅1, 𝑅2

• Evaluate the effect of a finite offset current 𝐼𝑂𝑆• 𝐼𝐵1 = 𝐼𝐵 + 𝐼𝑂𝑆/2, 𝐼𝐵2 = 𝐼𝐵 − 𝐼𝑂𝑆/2

• 𝑉𝑂 = 𝐼𝑂𝑆𝑅2

• Minimize the effect of input bias current• Place in the positive lead a resistance equal

to the equivalent dc resistance seen by the

inverting terminal

Figure 2.34 Reducing the effect of the input bias currents by introducing a resistor R3.


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Figure 2.36 Illustrating the need for a continuous dc path for each of the

op-amp input terminals. Specifically, note that the amplifier will not work

without resistor R3.

• AC-coupled, select 𝑅3 = 𝑅2

• AC-coupled amplifiers• Provide a continuous dc path between each of the input terminals of the op

amp and ground

• Not work without 𝑅3• Lower considerably the input resistance

Figure 2.35 In an ac-coupled amplifier the dc resistance seen by the

inverting terminal is R2; hence R3 is chosen equal to R2.


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2.6.3 𝑉𝑂𝑆 and 𝐼𝑂𝑆 in Inverting Integrator

• Small dc voltages or currents increases linearly with time until saturate

• When 𝑉𝐶 = 0, 𝑣𝑂 = 𝑉𝑂𝑆 +𝑉𝑂𝑆

𝐶𝑅𝑡

• By adding a resistance R in the op amp positive-input lead

• 𝐼𝐵, 𝐼𝑂𝑆 flow through 𝐶 and cause 𝑣𝑂 to ramp linearly until saturate

• Resistor 𝑅𝐹 across the 𝐶, provide a dc path: 𝑣𝑂 = 𝑉𝑂𝑆(1 + 𝑅𝐹 𝑅) +𝐼𝑂𝑆𝑅𝐹

Figure 2.37 Determining the effect of the op-amp input offset voltage VOS on the

Miller integrator circuit. Note that since the output rises with time, the op amp

eventually saturates.

Figure 2.38 Effect of the op-amp input bias and offset currents on

the performance of the Miller integrator circuit.


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Homeworks

• Example 2.4

• Example 2.5

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Microelectronic Circuits, Kyung Hee Univ. Spring, 2016 1.6 … · 2018-09-10 · Microelectronic Circuits, Kyung Hee Univ. Spring, 2016 3 1.6.2 Amplifier Bandwidth •Amplifier bandwidth