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**Filters and Tuned Amplifiers**

C H A P T E R 11 Filters and Tuned Amplifiers

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Introduction The electronic filter is an important building block of communications and instrumentation systems. The oldest technology for realizing filters makes use of inductors and capacitors, and the resulting circuits are called passive LC filters. The passive LC filters work well at high frequencies; however, in low-frequency application (dc to 100 KHz) the required inductors are large and physically bulky. Additionally, it is incompatible with modern VLSI technology. The inductor-less filter is of interest and we shall study active-RC filters and switched-capacitor filters.

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Filter Transmission The filters we are going to study are linear circuits that can represented by a general 2-port network. The transfer function T(s) is the ratio of output voltage Vo(s) to the input voltage Vi(s). The filter transmission is found by evaluating T(s) and can be expressed in terms of its magnitude and phase as Figure 11.1

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Filter Types The filters can perform a frequency-selection function: passing signals whose frequency lies within a specified range, and stopping signal whose frequency falls outside this range. Figure 11.2 Ideal transmission characteristics of the four major filter types: (a) low-pass (LP), (b) high-pass (HP), (c) bandpass (BP), and (d) bandstop (BS).

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Filter Specification The transmission of a low-pass filter is specified by 4 parameters: The passband edge wp. The stopband edge ws. The maximum allowed variation in passband transmission Amax. The minimum required stopband attenuation Amin. Lower Amax, higher Amin, ws/wp closer to 1 are desirable in an ideal filter. Figure 11.3 Specification of the transmission characteristics of a low-pass filter. The magnitude response of a filter that just meets specifications is also shown.

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**Filter Design Procesdure**

The following is the filter design procedure: Decide the filter specification. Find the transfer function whose magnitude meets the specification. The process of the above (2nd) step is known as filter approximation. Transmission specification for a bandpass filter. Figure 11.4 Transmission specifications for a bandpass filter. The magnitude response of a filter that just meets specifications is also shown. Note that this particular filter has a monotonically decreasing transmission in the passband on both sides of the peak frequency.

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**11.2 The Filter Tranfer Function**

The filter transfer function T(s) can be written as the ratio of 2 polynomials as: The degree of the denominator, N, is the filter order. For stability consideration, M ≦ N. The numerator/denominator coefficients ai and bi are real numbers. The polynomials in the numerators/denominators can be factored and T(s) can be written in the form: Zero Pole/Nature Modes Poles and zeros can be either a real or a complex number. Complex zeros and poles must occur in conjugate pair.

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**Tranfer Function of a Low-Pass Filter (1)**

Consider the filter whose transfer function is shown below. The filter has transmission zeros at s= ± jwl1 and s= ± jwl2. The transmission decreases toward -∞ as w approaches ∞. The filter must have one or more zeros at s=∞. The number of transmission zeros at s=∞ is N-M. assume the filter is of fifth order (N=5) Figure 11.5 Pole–zero pattern for the lowpass filter whose transmission is sketched in Fig This is a fifth-order filter (N = 5).

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**Tranfer Functionof a Bandpass Filter**

Consider the filter whose transfer function is shown below. The filter has transmission zeros at s= ± jwl1 and s= ± jwl2. The transmission decreases toward -∞ as w approaches 0 and ∞. The filter must have one or more zeros at s=∞ and s=0. Assume only 1 zero at s=∞ and s=0. The filter must be of sixth order (N=6). Figure 11.6 Pole–zero pattern for the bandpass filter whose transmission function is shown in Fig This is a sixth-order filter (N = 6).

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**Tranfer Function of a Low-Pass Filter (2)**

Consider the filter whose transfer function is shown below. No finite values of w at which the transmission is zero. Thus all the transmission zeros are at s=∞. Such a filter is known as “all pole filter”. assume the filter is of fifth order (N=5) Figure 11.7 (a) Transmission characteristics of a fifth-order low-pass filter having all transmission zeros at infinity. (b) Pole–zero pattern for the filter in (a)

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**Figure 11.8 The magnitude response of a Butterworth filter.**

Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

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**Butterworth Filters (1)**

2 functions that are frequently used in approximating the transmission characteristics of low-pass filters are Butterworth and Chebyshev filters. Butterworth filter exhibits a monotonically decreasing transmission with all the zeros at w =∞, making it an all-pole filter. The magnitude of an N-th order Butterworth filter with a passband edge wp is given by where ε determines the maximum variation in passband transmission.

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**Microelectronic Circuits, International Sixth Edition**

Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

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**Butterworth Filters (2)**

It can be shown that the first 2N-1 derivatives of |T| relative to w are zero at w=0. This property make the Butterworth response very flat near w=0 and gives the response the name maximally flat response. As the order increases, the response approaches the ideal brick-wall type of transmission. Figure 16.9

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**Butterworth Filters (3)**

Its poles can be determined from the graphical construction. N=2 N=3 N=4 Figure 11.10

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**Butterworth Filters (4)**

The transfer function of the Butterworth filter is The procedure to find a Butterworth transfer function that meets the specification (Amin, Amax, ws(the edge of the stopband), wp): Determine ε from Amax. Determine the required filter order as the lowest integer value of N that results in A(ws) ≧ Amin. Determine the N poles. Determine T(s). The detailed procedure can be found in Ex. 12.1

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**Butterworth Filters (5)**

Poles of the ninth-order Butterworth filter. Figure Poles of the ninth-order Butterworth filter of Example 11.1.

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**Normailized Denominator Polynomials in Factored Form**

The Butterworth Response n Normailized Denominator Polynomials in Factored Form 1 (1+s) 2 ( s+s2) 3 (1+s)(1+s+s2) 4 ( s+s2)( s2) 5 (1+s)( s+s2)( s2) 6 ( s+s2)( s+s2)( s+s2) 7 (1+s)( s+s2)( s+s2)( s+s2) 8 ( s+s2)( s+s2)( s+s2)( s+s2) 9 (1+s)( s+s2)(1+s+s2)( s+s2)( s+s2) 10 ( s+s2)( s+s2)( s+s2)( s+s2)( s+s2)

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**3rd-order Butterworth Low Pass Filter**

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Chebyshev Filters (1) The Chebyshev filter exhibits an equiripple response in the passband and a monotonically decreasing transmission in the stopband. It is an all-pole filter. While the odd-order filter has |T(0)| =1, the even-order filter exhibits its maximum magnitude deviation at w=0. N=4 N=5 Figure Sketches of the transmission characteristics of representative (a) even-order and (b) odd-order Chebyshev filters.

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Chebyshev Filters (2) The poles of a Chebyshev filter can be determined by The transfer function of the Chebyshev filter can be written as For the same order and the same Amax, the Chebyshev filter provides greater stopband attenuation than the Butterworth filter. To meet identical specification, one requires a lower order for the Chebyshev than for the Butterworth. Ex is a useful practice.

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**11.4 First-order and second-order filters Cascade Filter Design**

First and second order filters can be cascaded to realize a high-order filter. It is one of the most popular methods for the design of active filters. Cascading does not change the transfer functions of individual blocks if the output impedance level is low.

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First Orer Filters (1) The general first-order transfer function is given by Figure First-order filters.

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**Figure 11.14 First-order all-pass filter.**

phase shifter Figure First-order all-pass filter. Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

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Second Orer Filters (1) The general second-order (bi-quadratic) transfer function is given by The natural modes (poles) are w0 is the radial distance of the poles from the origin and is known as the pole frequency. Q is called the pole quality factor. Figure 11.15

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**Figure 11.16 Second-order filtering functions.**

Second Orer Filters (2) The second-order filter function. Figure Second-order filtering functions.

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**Second Orer Filters (3) The second-order filter function.**

Figure (continued)

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**Second Orer Filters (4) The second-order filter function.**

Figure (continued)

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**The Second Orer LRC Resonator**

The nature mode of the parallel resonance circuit can be determined by applying an excitation that does not change the natural structure of the circuit. Current Excitation Voltage Excitation Figure 11.17

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**Microelectronic Circuits, International Sixth Edition**

Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

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**Realization of Transmission Zeros**

Having realized a given pair of complex conjugate natural mode, find out where to inject the voltage signal Vi so that the transfer function VO/Vi is the desired filter type. The transmission zeros are the values of s at which the Z2(s) is zero (provided Z1(s) is not simultaneously zero), and the values of s at which the Z1(s) is infinite (provided Z2(s) is not simultaneously infinite),

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**Realization of the Low -Pass and High-Pass Function**

High-Pass Filter Low-Pass Filter For low-pass filter, it has 2 zeros. One is at the value of s for which the series impedance becomes infinite and shunt impedance becomes zero.

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**Realization of the All -Pass**

The all-pass transfer function can be written as The disadvantage of the circuit is the lack of a common ground terminal between the input and the output. Figure Realization of the second-order all-pass transfer function using a voltage divider and an LCR resonator.

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**The Antoniou Inductance-Simulation Circuit**

It is a amp-RC circuits that could simulate the operation of an inductor. Prof. Antoniou has proved that this circuit is the best (tolerant of the finite gain and bandwidth of a real OP) among other approaches. Figure (a) The Antoniou inductance-simulation circuit. (b) Analysis of the circuit assuming ideal op amps. The order of the analysis steps is indicated by the circled numbers.

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**The Op Amp-RC Resonator**

An op amp-RC resonator obtained by replacing the inductor L with a simulated inductance. Vr is not a convenient node to use as the filter output terminal because connecting a load there would change the filter characteristics. This problem can be solved by utilizing a buffer amplifier. Figure (a) An LCR resonator. (b) An op amp–RC resonator obtained by replacing the inductor L in the LCR resonator of (a) with a simulated inductance realized by the Antoniou circuit of Fig (a). (c) Implementation of the buffer amplifier K.

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**Realization of the Various Filter Types (1)**

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**Realization of the Various Filter Types (2)**

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**Realization of the Various Filter Types (3)**

The all-pass circuit with a flat gain of unity can be written as AP = 1- (BP with a center-frequency gain of 2) A simple procedure to obtain the complement of a given linear circuit. Disconnect all the nodes connected to ground and then connect them to Vi. Disconnect all the nodes connected to Vi and then connect them to ground. That is, interchange the input and ground. 2 circuits whose transfer functions are related in this fashion are said to be complementary.

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**11.7 Second-Order Active Filters Based on the two-Integrator-Loop**

It is another family of op amp-RC circuits that realize second-order filter function. First consider a high-pass transfer function. The negative sign in the integrator blocks means the use of inverting Miller integrator to implement each integrator. Figure Derivation of a block diagram realization of the two-integrator-loop biquad.

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**Derivation of the Two-Integrator-Loop Biquad**

Rearrange the equation, the high-pass function can be written as High-Pass Function It is known as universal active filter Band-Pass Function Low-Pass Function Figure Derivation of a block diagram realization of the two-integrator-loop biquad.

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Figure (a) The KHN biquad circuit, obtained as a direct implementation of the block diagram of Fig (c). The three basic filtering functions, HP, BP, and LP, are simultaneously realized. (b) To obtain notch and all-pass functions, the three outputs are summed with appropriate weights using this op-amp summer. Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

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**Circuit Implementation-KHN Biquad**

The circuit known as KHN biquad can implement such function.

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**Circuit Implementation- Two-Thomas Biquad**

An alternative two-integrator-loop biquad circuit in which all the three op amps are used in a single-ended mode can be developed as follows. Coefficients of the summer have the same sign. High-pass function is no longer available. Figure (a) Derivation of an alternative two-integrator-loop biquad in which all op amps are used in a single-ended fashion. (b) The resulting circuit, known as the Tow–Thomas biquad.

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**Figure 11. 26 The Tow–Thomas biquad with feed forward**

Figure The Tow–Thomas biquad with feed forward. The transfer function of Eq. (11.68) is realized by feeding the input signal through appropriate components to the inputs of the three op amps. This circuit can realize all special second-order functions. The design equations are given in Table 11.2. Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

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**11.8 Single-Amplifier Biquadratic Active Filters**

The previous op amp-RC biquadratic circuits are versatile and are easy to design. However, they are not economic in their use of op amps. It is a problem in applications where the power-supply current is to be conserved. Therefore the circuit with only one op amp is desirable. These minimal realizations suffer 2 problems than multiple op-amp biquads. A dependence of limited gain and bandwidth of the op amp. Sensitive to the unavoidable tolerances in the values of R and C. The single-amplifier-biquads (SABs) are therefore limited to the less stringent filter specifications. The synthesis of SABs follows a 2-step process. Synthesis of a feedback loop that realizes a pair of complex-conjugate poles characterized by w0 and Q. Injecting the input signal in a way that realizes the desired zeros.

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**Synthesis of Feedback Loop**

The synthesis of the feedback loop. The closed-loop characterization equation is The poles of the closed-loop system are identical to the zeros of the RC network. Figure (a) Feedback loop obtained by placing a two-port RC network n in the feedback path of an op amp. (b) Definition of the open-circuit transfer function t(s) of the RC network.

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**Networks with Complex Zeros**

2 RC networks (bridged-T networks) that can have complex transmission zeros are shown below. Figure Two RC networks (called bridged-T networks) that can have complex transmission zeros. The transfer functions given are from b to a, with a open-circuited.

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**Feedback Loop Using Bridged-T Network**

The pole polynomial of the active-filter circuit will be equal to the numerator polynomial of the bridged-T network; Figure An active-filter feedback loop generated using the bridged-T network of Fig (a).

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**Injecting the Input Signal**

Any circuit node connected to ground can instead be connected to the input voltage source without causing the poles to change. Depending on the components through which the input signal is injected, different transmission zeros are obtained. Figure (a) The feedback loop of Fig with the input signal injected through part of resistance R4. This circuit realizes the bandpass function. (b) Analysis of the circuit in (a) to determine its voltage transfer function T(s) with the order of the analysis steps indicated by the circled numbers.

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**Generation of Equivalent Feedback Loops (1)**

The complementary transformation of feedback loop. Figure Interchanging input and ground results in the complement of the transfer function.

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**Generation of Equivalent Feedback Loops (2)**

A 2-step process to generate an equivalent feedback loop. Nodes of feedback network and op-amp input that are connected to ground should be disconnected and connected to op-amp output. The 2 input terminals of the op amp should be interchanged. Characteristic Equation Characteristic Equation Figure Application of the complementary transformation to the feedback loop in (a) results in the equivalent loop (same poles) shown in (b).

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**Generation of Equivalent Feedback Loops (3)**

An example of applying the complementary transformation to get the equivalent feedback loop. A second-order high-pass function is realized. Figure (a) Feedback loop obtained by applying the complementary transformation to the loop in Fig (b) Injecting the input signal through C1 realizes the high-pass function. This is one of the Sallenand-Key family of circuits.

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**Generation of Equivalent Feedback Loops (4)**

Another example of applying the complementary transformation to get the equivalent feedback loop. Injecting signal to C4 results in a bandpass function. Injecting signal to R1 results in a low-pass function. Figure (a) Feedback loop obtained by placing the bridged-T network of Fig (b) in the negative feedback path of an op amp. (b) Equivalent feedback loop generated by applying the complementary transformation to the loop in (a). (c) A low-pass filter obtained by injecting Vi through R1 into the loop in (b).

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**Microelectronic Circuits, International Sixth Edition**

Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

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**11.10 Switched-Capacitor Filters (1)**

Previous active-RC filters need large-valued capacitors and the requirement of accurate RC time constants that makes production in monolithic IC form difficult. The switched-capacitor filter design would lend itself more naturally to IC implementation.

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**Switched-Capacitor Filters (2)**

Basic principle of switched-capacitor filter techniques. Inverting integrator Accuracy is obtainable with 0.1 pF. The integrator is stray-sensitive. Figure 11.35

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Figure A pair of complementary stray-insensitive, switched-capacitor integrators. (a) Noninverting switched-capacitor integrator. (b) Inverting switched-capacitor integrator. Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

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**Switched-Capacitor Filters (3)**

Using switched-capacitor technique, implement the circuit shown below so that the product of RC is 1 ms. The product of RC may be written as RC2 = Tc(C2/C1). RC is determined by T, which is an accurate period derived from a crystal oscillator and the ratio of C2 and C1 which may be within 1% on a chip. Set C1 is larger than the associated parasitics, say 1pF and C2= 100 pF with T=10-5 s. The value of switched-capacitor resistor is 10 MΩ. Implementing this resistor in a CMOS process requires large area.

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**Switched-Capacitor Filters (3)**

A 2-integrator-loop active filter is composed of one inverting and one noninverting integrator. Therefore we need a pair of complementary switched-capacitor integrators. Non-inverting integrator Inverting integrator These integrators are stray-insensitive.

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**Switched-Capacitor Filters (4)**

A 2-integrator-loop active filter biquad and its switched-capacitor counterpart. Figure (a) A two-integrator-loop, active-RC biquad (b) its switched-capacitor counterpart.

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**Figure 11.38 Frequency response of a tuned amplifier.**

11.11 Tuned Amplifiers Figure Frequency response of a tuned amplifier. Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

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sedr42021_1239a.jpg Figure The basic principle of tuned amplifiers is illustrated using a MOSFET with a tuned-circuit load. Bias details are not shown.

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**Figure 11.40 Inductor equivalent circuits.**

Inductor Losses sedr42021_1240ab.jpg Figure Inductor equivalent circuits.

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sedr42021_1241.jpg 可得到大L, 小C, Wo 不變 Figure A tapped inductor is used as an impedance transformer to allow using a higher inductance, L’, and a smaller capacitance, C’.

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Figure (a) The output of a tuned amplifier is coupled to the input of another amplifier via a tapped coil. (b) An equivalent circuit. Note that the use of a tapped coil increases the effective input impedance of the second amplifier stage.

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**雙tuner, selective更好, BW ↓**

sedr42021_1243.jpg 雙tuner, selective更好, BW ↓ Figure A BJT amplifier with tuned circuits at the input and the output.

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**Cascode, CE+CB CC+CB sedr42021_1244a.jpg**

Figure Two tuned-amplifier configurations that do not suffer from the Miller effect: (a) cascode and (b) common-collector common-base cascade. (Note that bias details of the cascode circuit are not shown.)

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**N tuner cascade, BW ↓ sedr42021_1245.jpg**

Figure Frequency response of a synchronously tuned amplifier.

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**max flat in pass band, BW ↑**

sedr42021_1246.jpg Figure Stagger-tuning the individual resonant circuits can result in an overall response with a passband flatter than that obtained with synchronous tuning (Fig ).

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Figure 11.47 Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

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**by transforming a second-order low-pass, maximally flat response.**

Figure Obtaining the poles and the frequency response of a fourth-order stagger-tuned, narrow-band bandpass amplifier by transforming a second-order low-pass, maximally flat response. Microelectronic Circuits, International Sixth Edition Sedra/Smith Copyright © 2011 by Oxford University Press, Inc.

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