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7.3. Cascaded Tuned Stages and the Staggered Tuning
The voltage gain of a tuned amplifier in the s domain was given as
s ( s  s0 )
(7.29)
C ( s  s p1 )( s  s p 2 )
From Fig. 7.14(a) it can be seen that for the vicinity of the resonance frequency
( s  s p 2 )  2s and ( s  s0 )  s0 . Therefore (7.29) can be simplified as
Av 
Cdg
Av  
Cdg
s0
2C ( s  s p1 )
(7.39)
and with Av (0 )   g m Reff and Qeff  0CReff ,
Av  Av (0 )
0
2Qeff
 p1
1
1
 Av ( p1 )
( s  s p1 )
2Qeff ( s  s p1 )
(7.40)
If n identical stages are connected in cascade, the total voltage gain becomes
n  p1
1
1
  Av ( p1 ) 
n
2Qeff ( s  s p1 )
2Qeff ( s  s p1 ) n
and the band-width of the amplifier shrinks to
AvT   Av (0 ) 
n
0
BT  B 21/ n  1
(7.41)
This is the appropriate solution if a high gain and narrow band amplifier is needed. But in
some cases a relatively broad bandwidth and a flat frequency characteristic in this band is
needed. It can be intuitively understood that to tune the stages of this multi-stage
amplifier to slightly different frequencies around the center frequency of the band can
lead to the solution. In this case the gain of this multi-stage amplifier can be written as
AvT  Av1 ( p1 )... Avn ( pn )
 p1
2Qeff 1
...
 pn
2Qeffn
1
( s  s p1 )....( s  s pn )
that has n poles.
The appropriate positions of the poles of the transfer function (the voltage gain in
our case) to obtain a desired frequency characteristic is investigated in the classical filter
theory in depth [7.4]. It is known that the number of poles and their relative positions
determine the band-width and the shape of the frequency characteristics. Among several
possibilities of the distribution of the poles in the s-domain, the Butterworth distribution
and the Chebyshev distribution have prime importance and extensive use in practice.
The Butterworth distribution provides a “maximally flat” frequency charactristic
in the band. It has been shown that to obtain a Butterworth type frequency characteristic,
poles must be on a semi-circle, whose center is at ω0 on the vertical axis of the s-plane
and they must be symmetrically positioned with respect to the horizontal diameter of the
circle, with equal distances. The diameter of the circle on the jω axis corresponds to the
band-width of the circuit in angular frequency, 2Δω [7.5] The appropriate positions of the
poles for n =2, 3 and 4 are shown in Fig. 7.181, where
j
j
j
( s  s p12 ) @ 0
12
s p12
 /2
s p14
s p13
s p13
2
0
0
11
s p11
 1,3
 1, 2
2
11
 1, 4
 2 ,3
(b)
(a)
2
12  / 4
s p12
 /3
s p11
13
2
0
s p12
11
s p11
13
13
(c)
Figure 7.18. The appropriate positions of the poles of a (a) 2-pole, (b) 3-pole and (c) 4-pole
circuit that has a maximally flat frequency characteristic.
11
 1,2  
2  
2Q1
0
2Q2
 1,4  
11
2Q1

12
for a 2-pole circuit,
2Q2
and  1,3  

14
11
2Q1

2  
0

2Q1
0
2Q2
 1,4  
,
0
2Q1
0
2Q2
2Q4
 Q1  Q2
 1,3  

0
2Q4
0
2Q1

for a 3-pole circuit,
12

13
2Q2
2Q3
0 , the sigmas can be written as
for a 2-pole circuit,
0
2Q3
,  2,3  
2Q3
and  2,3  
In case of the small relative band-widths, where 2
 1,2  
13
0
2Q2
 Q1  Q3

0
2Q3
for a 3-pole circuit,
 Q1  Q4 , Q2  Q3 for a 4-pole
circuit,
1
It must not be overlooked that these diagrams are the simplified versions of the pole-zero zero diagrams,
as shown in Fig. 7.12. The full pole-zero diagrams contain the complex conjugates of the poles shown in
Fig. 7.18.
--------------------------------------------------------------------Design Example 7.3.
A 3-stage staggered tuned amplifier having a Butterworth type frequency
characteristic will be designed. The center frequency of the frequency characteristic is
1GHz. The possible maximum effective Q value for the resonance circuits–without any Q
enhancement feature- is given as 20.
a) What is the realizable bandwidth?
b) Calculate the tuning frequencies of the stages.
c) Calculate the appropriate effective Q values of the resonance circuits.
Solution:
The pole-zero diagram of the voltage gain function of the amplifier is shown in
Fig. 1.18-b. From the geometry of the figure it can be easily seen that the real part of the
center pole, p12 must be equal to the half of the band width:
0
f
 0
2Q2
2 2f
Similarly, the real parts of p11 and p13 must be equal to Δω/2:
2  
0
 
01


Q2 

f

 Q1  01  01
2Q1
2
 f


f

 3   03  
 Q3  03  03
2Q3
2
 f
1  
ω01 and ω03 can be calculated from the geometry:
01  0   cos( / 6)
03  0   cos( / 6)
Since ω03 is the highest among the three tuning frequencies, the quality factor
corresponding to this resonance circuit is the highest and must be equal to the possible
maximum Q value, that is 20:
Q3 
03 0   cos( / 6) 0


 cos( / 6)



f0
 0.866  20
f
that yields Δf = 52.26 MHz (Δω = 328.36 rad/s). Now the tuning frequencies and the
quality factors can be calculated as

f 01  f 0  f cos( / 6)  1000  52.26  0.866  954.74 [MHz]
Q1 
f 01 954.74

 18.27
f
52.26
f 02  f 0  1000 [MHz], Q2 
f0
1000

 9.57
f 2  52.26
f 03  f 0  f cos( / 6)  1000  52.26  0.866  1045.26 [MHz]
f 03 1045.26

 20
f
52.26
Note that since the quality factors are not high and the relative band width (2Δf / f0) is not
small, Q1 and Q3 are not equal.
Q3 
The frequency characteristic of the circuit calculated with 01  03  20 and
02  10 is given in Fig. 7.19. Note the slight irregularity of the curve in the flat region
since the quality factors used in the calculation are not exactly equal to the calculated
values.
(dB)
0
-10
-20
-30
-40
800
900
1000
1100
1200
Frequency (MHz)
Figure 9.19. The calculated frequency characteristic of the circuit, calculated with
MATLAB.
---------------------------------------------------------
Problem 7.5.
The center frequency and the bandwidth of a 4-stage, staggered tuned amplifier
are 2GHz and 80 MHz, respectively. Calculate the tuning frequencies and the quality
factors to obtain a Butterworth type frequency characteristic.
The second important type of pole distribution provides a Chebyshev type
frequency characteristic. The side-walls of a Chebyshev type (or equi-ripple)
characteristic is steeper than that of a same order Butterwoth type characteristic, but
exhibits a typical ripple on the top of the curve, as shown in Fig. 7.20-a. The number of
ripples depend on the order of the circuit.
The poles of a Chebyshev type circuit are positioned on an ellipse, whose longer
axis is on the jω axis and the length of the longer axis corresponds to the band width of
the circuit. It has been shown that the appropriate positions of the poles for a certain
amount of ripple can be obtained from the positions of a Butterworth type circuit that has
the same band width. As shown from Fig. 7.20-b, the tuning frequencies of the resonance
circuits are same, but the real parts of the poles of the Chebyshev type circuit are smaller.
For a nth order Chebyshev type circuit with r (dB) ripple, the magnitude of σ of the
Chebyshev pole can be calculated in terms of the corresponding Butterworth pole from
j
(s p3 )B
3
A (dB)
( s p 3 )C
r
0
-3
(s p 2 )B
( s p 2 )C
( s p1 )C
2  0 2
1
( s p1 ) B
f
0
2f
(a)
( 1 )C
(b)
Figure 7.20. (a) The frequency characteristic of a 3rd order C type circuit. (b) The positioning of
the poles of a Butterworth type circuit and a Chebyshev type circuit that have the same band width.
( i )C  tanh   ( i ) B
where
(7.42)
1
n
  sinh 1
1

  log 1
,
r (dB)
1
10
(7.42-a)
For convenience, the values of (tanh α) for n = 2, 3 and 4 and for several ripple values are
given below [7.5]:
Table 7.1
r (dB)
n=2
n=3
n=4
0.05
0.1
0.2
0.3
0.4
0.5
0.898
0.859
0.806
0.767
0.736
0.709
0.750
0.696
0.631
0.588
0.556
0.524
0.623
0.567
0.505
0.467
0.439
0.416
----------------------------------------Design Example 7.4.
A 3-stage staggered tuned amplifier for 2 GHz, having a voltage gain of 40 dB
and a Chebyshev type frequency characteristic with 0.5 dB ripple and 380 MHz band
width will be designed. The design will be made for a technology similar to the 0.35
micron AMS technology, but with an additional thick metal layer, 10 nH inductors
having an efficient quality factor of 10 at 2 GHz are available. The Q values can be
further increased with Q-enhancement circuit similar to the circuit given in Design
Example-7.1
From Fig. 7.20-b we see that the bandwidth of the circuit is B  2  2  2  B ,
where  2 B is the negative real center pole of a Butterworth type circuit having the
same center frequency and band-width. The center (real) Chebyshev pole can be
calculated from (7.40) and Table 7.1:
( 2 )C  0.524 ( 2 ) B  0.524  
At the other hand,
( 2 )C 
Now Q2 can be calculated as Q2 
0
2Q2
0
f
1
1
 0
 10.04
2 0.524 2f 0.524
It means that the center pole can be realized without any Q- enhancement.
From Fig. 7.18-b and Fig. 7.20-b the tuning frequency and the quality factor
corresponding to (sp1)C can be calculated as