Download Design Example –7.1.

7. FREQUENCY SELECTIVE RF CIRCUITS
Along the early days of radio, the tuned amplifier has been one of the most
important subject of the electronic engineering. It has been used as input RF amplifiers of
radio receivers tunable in a certain frequency range and as intermediate frequency (IF)
amplifiers tuned to a fixed frequency. From 30s to 50s the RF amplifiers using electron
tubes were one of the most important and interesting research area of the electronic
engineering and investigated in depth. TV and all other wireless systems were other
application areas of the tuned amplifiers.
After the emergence of transistor the knowledge already acquired has been
sufficient for transistorized tuned amplifiers, because the frequency range was still
limited to several hundreds of MHz. The active filters, one of the outcomes of the analog
ICs, replaced another class of the frequency selective circuits, the L-C filters. As a result
of these developments the importance of the inductors and circuits containing inductors
decreased and eventually these subjects disappeared from many electronic engineering
curricula and textbooks.
The rapid expansion of the wireless personal communication and data
communication along the recent decade, and the developments in the IC technology that
extended the operation frequencies into the GHz range, resulted in the “re-birth” of the
frequency selective circuits containing inductors. In parallel to the increase of the
frequencies up to the GHz range, necessary inductance values decreased to “nanohenry”
level that are now possible to realize as an integral part of ICs, as mentioned in Chapter
1. But the quality factor of these “on-chip inductors” is in the range of 5…15, that is
considerably small compared to the quality factor of classical discrete “wound”
inductors1. It must be kept in mind that the theory of tuned circuits developed in old
books is based on the hi-Q inductors. Therefore a re-consideration of the tuned circuits
for low-Q circuits is necessary.
In this chapter first a summary of the resonance circuits; basic definitions and
behaviors with special emphasis on the low quality factor circuits will be reminded. Than
the single tuned amplifiers, stagger tuning and amplifiers containing coupled resonance
circuits will be presented. The active and passive filters are considered out of the cover of
this book. But due to increasing importance of a class of active filters at high frequencies;
the gyrator based circuits will be given.
The quality factors of discrete wound inductors is in the range of 100…1000, depending on the structure
and material of the coil and the frequency.
1
7.1. Resonance Circuits
Resonance is one of the most important effects that occurs in many physical
systems comprising components capable to store energy as the potential energy and the
kinetic energy. These systems start to swing (oscilate) when excited; that means if a small
energy is injected into the system for example as a potential energy. This energy swings
in the system back and forth between fully potential to fully kinetic phases2. The
frequency of oscillations depends on the parameters of the system. The pendulum is the
most easy to understand oscillatory system. An oscillatory system looses its energy in
time, if there is (better to say; always there is) a reason to consume energy in the system
(for example the air friction for a pendulum), the amplitude of the oscillation eventually
decreases down to zero.
In electric circuits the capacitor is the component that is capable to store energy in
potential form. The energy stored in a capacitor is Ek  (1/ 2)CV 2 if the voltage is V. An
inductor stores the energy in kinetic form. If a current I is flowing through an inductor,
the stored energy stored in the inductor is Ep  (1/ 2) LI 2 . All resistors in a circuit
consumes energy. The consumed (dissipated) energy in a time interval t is Ed  I 2 Rt in
terms of the current and Ed  (V 2 / R)t in terms of voltage.
The components of an electrical resonance circuit is an inductor and a capacitor.
The series and/or parallel parasitic resistances of them; for example the resistance of the
inductor material and the parallel dielectric losses of the capacitor are the energy
consuming components of the system. To form a resonance circuit there are two
possibilities; to connect the inductor and the capacitor in series or in parallel. It will be
shown that the resonance effect of a parallel resonance circuit is pronounced when it is
driven by a current source. In contrast, a series resonance circuit exhibits the resonance
effect when it is driven by a voltage source.
7.1.1. The Parallel Resonance Circuit
A parallel resonance circuit is shown in Fig. 7.1. The inductor is modeled with its
self inductance (L) and the series resistance (rL) that represents all losses related to the
inductor (for an on-chip inductor the resistance of the strip, the losses of the magnetically
induced currents and the substrate resistance, shown in Fig. 1.32.). C is the value of the
capacitance. For on-chip capacitors, since the insulator is silicon dioxide, the parallel
dielectric losses are negligibly small, but there is a considerable series resistance as
shown in Fig. 1.25 that is the main cause of the low Q value of an on-chip capacitor3. Rp
represent the internal resistance of the signal (current) source, and other parallel losses if
2
For an excellent reading on resonance see [7.1]
Note that for discrete capacitors of the classical resonance circuits the series resistance of the capacitor is
neglected during the derivation of the expressions. Therefore these expressions do not sufficiently represent
the behavior of the on-chip resonance circuits.
3
there is any. The impedance seen by the current source (Z’) is the parallel equivalent of
Rp and Z (Fig. 7.1.). First we’ll calculate Z and then includu the effect of Rp.
The impedance Z can be calculated as
C
Rp
i
Z’
rC
L
rL
Z
Figure 7.1. The parallel resonance circuit with the series resistances of the inductor and the
capacitor.
Z
rL  s( L  CrC rL )  s 2 LCrC
1  sC (rC  rL )  s 2 LC
and in ω domain,
(rL   2 LCrC )  j ( L  CrC rL )
Z ( ) 
(1   2 LC )  jC (rC  rL )
(7.1)
The real and imaginary parts of (7.1) are
Re{Z } 
(rL   2 LCrC )(1   2 LC )   2C (rL  rC )( L  CrL rC )
(1   2 LC )   2C 2 (rL  rC )2
(7.2)
Im{Z } 
 ( L  CrL rC )(1   2 LC )  C (rL   2 LCrC )(rL  rC )
(1   2 LC )   2C 2 (rL  rC )2
(7.3)
From these expressions some information related to the extreme cases can be
extracted:
- For ω → 0 the imaginary part of the impedance is zero and the real part is equal
to rL, as can be intuitively seen from Fig. 7.1.
- For ω → ∞ the imaginary part of the impedance is again zero and the real part
is equal to rC .
- The imaginary part of the impedance is zero (the impedance is resistive) for
2
(Re)

( L  CrL2 )
LC ( L  CrC2 )
(7.4)
In Fig. 7.2-a the root locus of Z for rC rL is shown, that correspond to the
conventional resonance circuits where the series resistance of the capacitor is negligible.
As seen from the figure the impedance is resistive and equal to rL. The impedance is
inductive up to a frequency for which the impedance is again resistive. The frequency
corresponding to this case was found as (7.4) and can be simplified as
1 rL2
2
(7.5)
(Re) 

LC L2
for rC rL . Note that this frequency is not same as the ω0 natural frequency 4 that is
defined as
1
02 
(7.6)
LC
In addition it can be shown that the frequency for which the magnitude of the impedance
is maximum, is another important frequency and is equal to
rL2
1
2 C
(7.7)
1

2
r

L
(max)
LC
L L2
for rC rL . The differences between these three frequencies that are characteristic for a
resonance circuit are small but may have important effects on the behavior of the circuits
containing resonance circuits. To assume them equal, may hide certain delicate properties
of the circuit, as will be exemplified below.
2 
For example for an L-C oscillator, according to the Barkhausen criterion the
circuit oscillates at a frequency for which the loop gain satisfies the gain condition and
the total phase shift on the loop is equal to zero. It is obvious that the ideal case is the
coincidence of the frequency for which the magnitude of the impedance (and
consequently the gain) is maximum and the frequency for which the phase shift is zero
(the impedance is resistive); with the notation we used above, ω(max)=ω(Re). The
conventional way to satisfy (at least to approach to) this condition is to use a low loss
(high Q) resonance circuit as the resonator of the oscillator.
For on-chip resonance circuits there is a possibility to equate ω(max) and ω(Re). It
can be seen from (7.4) that for rC  rL , ω(Re) becomes equal to ω0 and the beginning of
the root locus (ω=0) coincides with its end (ω=∞) at Re[Z]= rL =rC. The root-locus
corresponding to this special case is shown in Fig. 7.2-b. It can be seen that due to the symmetry
of the plot not only ω0 and ω(Re), but also ω(max) coincides. This may be a valuable hint for
the phase noise minimization of L-C oscillators.
4
“Natural frequency” is used for the resonance frequency of the losless L-C combination
Im(Z)
Im(Z)
ω
Z(ω)
Z(ω)
ω (Re)
ω= ∞
Z(0) = rL
ω
ω0
Zmax
ω0= ω (Re) = ω (max)
ω= 0
Re(Z)
Re(Z)
ω=∞
ω (max)
rL= rC
(a)
(b)
Figure 7.2. The root-locus of the impedance of a parallel resonance circuit; (a) for
rC rL , (b) for rC  rL .
-----------------------------------------Example 7.1.
To check these results let us simulate a resonance circuit for two different cases:
a) The inductance is L=2nH and series resistance of the inductance is rL=10 0hm.
C has the appropriate value to tune the circuit to f0 = 5GHz. The series resistance of the
capacitor is negligibly small.
b) The inductance and the resonance frequencies are same as (a). But the series
resistances of L and C are same and equal to 5 ohm.
Draw the variations of the magnitude and phase of the impedance as a function
and compare.
The value of the capacitance can be calculated as 506.7 fF from (7.6). The pSpice
results of the magnitude and phase are shown in Fig. 7.3-a. As can be seen from these
plots the frequencies corresponding to the maximum of the magnitude of the impedance
and the frequency where the impedance is resistive (the phase angle is zero) are not same.
Or from even more important point of view, the phase angle corresponding to the
maximum value of the impedance is not zero, but Φ = -8.05 degree.
The simulation results for case (b) are shown in Fig.7.3-b. As can be seen from
these plots the frequencies corresponding to the maximum of the impedance and to the
zero phase shift are same, or in other words the phase shift corresponding to the maximun
value of the impedance is zero, as expected.
|Z| (Ω)
Φ (degree)
|Z| (Ω)
Φ (degree)
400
100
400
100
300
50
300
50
200
0
200
0
100
-50
100
-50
0
-100
3.0
4.0
5.0
6.0
7.0
0
-100
3.0
4.0
5.0
Frequency (GHz)
Frequency (GHz)
(a)
(b)
6.0
7.0
Figure 7.3. The phase and magnitude curves of a parallel resonance circuit: (a) with a lossy
inductor and losless capacitor, (b) with a lossy inductor and lossy capacitor, if the series
resistances of L and C are equal.
-----------------------------------------------7.1.1.2. The Quality Factor of a Resonance Circuit
Before going on further, let us remind the basic definition of the quality factor (Q)
of any oscillatory system, and apply this definition to calculate the quality factor of the
circuit shown in Fig.7.4.
The quality factor of an oscillatory system is defined as
Q  2
The total energy of the system
The energy lost in one period
Since the total energy in an oscillatory system swings back and forth between fully
potential energy and fully kinetic energy, to find the total energy of the system it is
convenient to calculate the value of the potential or kinetic energy at one of these extreme
conditions. For the circuit shown in Fig.7.4. the circuit is excited with a sinusoidal
voltage, v  V sin 0t , whose frequency is equal to the natural frequency ω0 of the
resonance circuit. The maximum value of the potential energy stored in the capacitor
corresponds to the case when the voltage on this capacitor is maximum, that is the peak
(7.8)
value of v. Then the maximum of the potential energy, that is equal to the total energy of
the system is
v  V sin t
C
Rp
rC
L
rL
Figure 7.4. The parallel resonance circuit in its most general form.
1
E  CV 2
(7.9)
2
The energy lost in one period (T) in the parallel resistor Rp is
1V2
1 V2
Ep  T 

(7.10)
2 Rp 2 f0 Rp
To calculate the energy lost in rL , the peak value of the sinusoidal current flowing
through the inductor branch must be calculated:
V
IL 
2
rL  02 L2
Then the energy lost in one period in rL is
r
1 2
V 2 L2 2
2 f0
rL  0 L
Similarly the energy consumed in rC in one period can be found as
EL 
(7.11)
rC
1 2
V
(7.12)
1
2 f0
2
rC  2 2
0 C
The total energy lost in one period is the sum of (7.10), (7.11) and (7.12). Hence from
(7.8) and (7.9) the effective quality factor containing all of the losses of the circuit can be
obtained as
1
1
1
1
(7.13)



Qeff QL QC Qp
where
r 2  02 L2
L
QL  L
 0
(7.14)
0 LrL
rL
that correspond to the losses on rL, in other words is the quality factor of the inductor at
ω0.
EC 
02C 2 rC2  1
1
QC 

0CrC
0CrC
(7.15)
that correspond to the losses on rC, or it is the quality factor of the capacitor at ω0.
Qp  0CRp 
Rp
(7.16)
0 L
that correspond to the losses in Rp.
Note that the approximate values of QL and QC are only valid for small values of
the corresponding resistances and cause considerable error for Q values smaller than 5.
---------------------------------------Problem 7.1.
Calculate the error of the approximate form of (7.14) for QL=50, QL=10, QL=5
and QL=3.
---------------------------------------It is a common practice to represent the total losses of a resonance circuit with a
lumped parallel resistor (effective resistance) Reff, as shown in Fig. 7.5.
If we want to express the losses of the inductor branch with a parallel resistor RLp,
from (7.14) and (7.16) we can obtain
02 L2
L
(7.17)
RLp  rL 
 rL (1  QL2 ) 
rL
rLC
Similarly the parallel resistor representing the losses of the capacitor branch is
RCp  rC 
1
L
 rC (1  QC2 ) 
2
 C rC
rC C
2
0
(7.18)
Then the effective resistance can be expressed as the parallel equivalent of RLp, RCp and
Rp:
Reff  ( RLp // RCp // R p )
(7.19)
or in terms of parallel conductances;
Geff  GLp  GCp  G p
(7.19-a)
Then the effective quality factor containing all of the losses becomes
Qeff 
C
1
 0
0 LGeff Geff
or Qeff 
Reff
0 L
 0CReff
(7.20)
This equivalence considerably decreases the complexity of the expressions and
extensively used in the literature but hides some properties of the resonance circuit. For
example; although the natural frequency ( f0), the frequency where the impedance is real
(f(Re)) and the frequency corresponding to the maximum magnitude of the impedance
(f(max)) are not same for the original circuit shown in Fig.7.5-a, they are all same for
Fig.7.5-b and the skew of the frequency characteristics exemplified in Example 7.1. is
now hidden.
C
Rp
rC
L
C
Reff
L
rL
(a)
(b)
Figure 7.5. (a) Parallel resonance circuit with its lossy components. (b) The
effective parallel resistance representing all losses of the circuit at f0.
The Quality Factor from a Different Point of view:
As previously mentioned, if an oscillatory system is excited, it starts to oscillate
but due to the losses the amplitude of oscillation gradually decreases. Since the rate of
this decrease of the amplitude and the quality factor of the system are related to the losses
of the system, there must be an inter-relation between them.
Assume a parallel resonance circuit simplified as shown in Fig.7.5-b. The
impedance of this circuit in the s-domain can be arranged as
1
s
1
s
(7.21)
Z

C s2  s 1  1
C ( s  s p1 )(s  s p 2 )
Reff C LC
The poles of the impedance function are
2
s p1, p 2
1

2 Reff C
 1 
1

 
 2 Reff C  LC
or
s p1, p 2  
where
 
1
Reff C
j 02   2  
,
0 
1
LC
j0
and
(7.22)
0  02   2
(7.22-a)
If the circuit is excited with a current pulse, the voltage in the s-domain between
the terminals of the circuit can be written as
s
I 1
V  
 s  C ( s  s p1 )(s  s p 2 )
and the voltage in the t-domain,
1
1
s t
s t
v(t )  I
(e p1  e p 2 )
C (s p1  s p 2 )
that can be arranged as
1 1  t j0t  j0t
v(t )  I
e (e  e
)
C 2 j0
1 1 t
v(t )  I
e sin 0t
C 0
and with   (0 / 2Qeff ) and 0  0
v(t )  V(ini ) e
 (0 / 2Qeff ) t
sin 0t
(7.23)
The variation of the voltage of an excited parallel resonance circuit is shown in
Fig.7.6. From (7.23) two important properties of this damped oscillation can be extracted:
v(t)
Vini
t
Figure 7.6. The damped oscillation of an excited parallel resonance circuit.
- At the and of n  Qeff swings, the amplitude decreases to Vini e  .
- The number of swings5 corresponding to v  (1/ e)Vini is equal to (Qeff /  )
5
In 60s Rohde und Schwarz was introduced a Q-meter (QDM) based on these interesting relations.
The “Q enhancement”
To increase the low quality factor of an on-chip resonance circuit there is a
possibility; to connect a “negative conductance” in parallel to the resonance circuit. We
know that the Q factor of the resonance circuit at its resonance frequency (ω0) can be
expressed as Qeff=1/L ω0Geff . If we connect a negative conductance (-Gn ) parallel to Geff,
the enhanced effective parallel conductance becomes G’eff = (Geff – Gn). This decrease of
the parallel conductance consequently increases the Q factor to Q’eff =1/L ω0 (Geff – Gn)6.
Although this negative conductance, that is introduced by an appropriate
electronic circuit, adds an additional noise and nonlinearity, can be effectively used to
increase the quality factor of a resonance circuit [7.2], [7.3]. One of the simplest negative
resistance circuits is a capacitive loaded source follower. In Section 5.2 we have seen that
the input resistance of a source follower can be negative under appropriate conditions,
and can be used to enhance the quality factor of a resonance circuit.
Design Example –7.1.
The problem that we are going to solve is to increase the Q=10 of an on-chip
parallel resonance circuit to 20. The resonance frequency and the inductance
(consequently the capacitance) are given as f0=2 GHz, L=10 nH and C=0.633 pF. The
effective parallel conductance of the resonance circuit can be calculated as
1
1
Geff 

 7.96  104 S
9
L0Q 10 10  2  (2 109 )
that correspond to Reff =1256.6 ohm. It is obvious that to increase the Q factor to 20, the
effective conductance must be decreased to G’eff = 3.98×10-4 S and the necessary negative
conductance Gn= -3.98×10-4 S.
From (5.17) we see that the negative input conductance of a source follower is
maximum for g mC GCgs and for   p , and becomes equal to
g i ( ) 
CCgs
gm
(5.17-b)
(C  Cgs ) 2
and has its maximum value for C=Cgs. Under this condition the pole frequency given in
(5.15) becomes
g G
g
 (VGS  VT )
p  m
 m 
(5.17-c)
2Cgs
2Cgs
L2
for g m G , that can be satisfied by using a current source instead of R, as shown in Fig
7.7. From last two expressions we obtain
6
Note that for Gn = Geff the quality factor becomes infinite. According to (7.23) once the circuit is
excited, the magnitude of the oscillation remains constant at its initial value. It means that the circuit
operates as a sinusoidal oscillator. It can be shown that in all types of the L-C oscillators there exist a
negative conductance parallel to the L-C circuit introduced by a positive-feedback circuit or by a device,
like a tunnel diode, that inherently exhibits a negative conductance.
gm
(5.17-d)
4
which is the first design hint. In addition, (5.17-c) can be used to calculate the channel
length, L.
g i ( )  
Inspecting Fig. 5.11 we see that, to obtain a high negative conductance with a
small sensitivity, ω0 must be chosen where the slope of the curve is small, or in other
words on (or close to) the asymptote. It can be seen that for ω>3ωp the slope of the curve
(or the sensitivity of the negative conductance against the frequency) is sufficiently small.
It means that for ω0>3ωp, the value of the negative conductance becomes approximately
equal to gi () .
The aspect ratio of the transistor can be calculated in terms of Gn and the gate
bias voltage. From (5.17-d) and (1.26)
gm  4 Gn  nCox (W / L)(VGS  VT )
4 Gn
(W / L) 
nCox (VGS  VT )
that indicates a trade-off between gate overdrive and the aspect ratio. (VGS-VT)=1V is a
reasonable value for the AMS 035 micron technology, for which the maximum supply
voltage is given as 3.3V, and the aspect ratio can be calculated as
4 Gn
4(3.98 104 )
(W / L) 

 9.37
nCox (VGS  VT ) 374(4.54 107 )1
For this aspect ratio and (VGS -VT) =1V, the drain current is
1
W
1
nCox (VGS  VT )  374  9.3(4.54 107 )  0.79 mA
2
L
2
Now L can be calculated from (5.17-c). With ω0=4ωp as a value that satisfies the ω0>3ωp
ID 
L
and
4n (VGS  VT )
0

4  374 1
 3.45 104 cm  3.5  m
9
2  (2 10 )
W  9.37  3.45  32  m
The diagram of the circuit together with the source current source is given in
Fig.7.7. The VDD supply is 3.3 V. The gate of M1 is directly biased from VDD . To satisfy
(VGS1 -VT)=1V (or VGS1)=1.5V, the d.c. voltage of the source of M1 must be 1.8V, which
is the drain-source voltage of the current source transistor, M2. M2 must be biased such
that its current is also 0.79 mA and is in the saturation region. For VGS2 =1V, since
ID1=ID2, the aspect ratio of M2 can be calculated as
 W   W  (VGS1  VT )(1  n .VDS1 )
   
 L 2  L 1 (VGS 2  VT )(1  n .VDS 2 )
 9.37
(1.5  0.5)2 (1  0.073 1.5)
 37
(1  0.5)2 (1  0.073 1.8)
and with L2=0.35 m, W2=13 m.
The final value to be calculated is C which is equal to Cgs1 :
Cgs1  W1L1Cox  (32 104 )(3.5 104 )(4.54 107 )  508.48 fF
The designed Q enhancement circuit is simulated with pSpice. It has been
necessary to bring the current of M1 to 0.79 mA, the gate voltage of the current source
M2 adjusted to 1.22V. The frequency characteristics of the original resonance circuit and
the Q-enhanced circuit are given in Fig.7.7-b. As seen from this graphics there are certain
discrepancies between the target and the simulation results. One of them is the decrease
of the resonance frequency. This is an expected shift due to the input capacitance of M1
and can be compensated with the decrease of C. The resonance impedance –that is
proportional with the quality factor- was not increased twice but only %48, that
correspond an enhanced quality factor 14.8. One of the reasons is the losses associated
with the body resistances of the devices that were not included into account in analytical
expressions. The second reason is the value of gm, that is usually smaller than the
calculated value as discussed in Chapter-1. To compensate it the channel width can be
increased in order to reach the target value.
Another valuable possibility of this simple Q-enhancement circuit shown in
Fig.7.7-a is to control the transconductance of M1 (and hence the negative conductance)
with the gate bias voltage of the current source, M2,. In Fig. 7.7-c the frequency
characteristics of the circuit for different values of VG2 are shown. Note that the value of
C has been decreased to compensate the effect of the input capacitance of M1 and bring
the resonance frequency to the target value.
---------------------------------Problem 7.2.
Make transient simulations of the original circuit and the Q-enhanced circuit in
Design example 7.1 for sinusoidal input currents with an amplitude of 1mA and 5 mA.
Compare the results and interpret.
│Z│ (ohm)
2.0k
B
(f0, L, C, Q)
+VDD
1.5k
A
M1
ii
vi
1.0k
ID1
C
0.5k
M2
+VG2
0
(a)
(b)
1.50
1.75
2.00
2.25
2.50
Frequency (GHz)
│Z│ (ohm)
3.0k
2.0k
1.0k
0
1.50
1.75
2.00
2.25
Frequency (GHz)
(c)
Figure 7.7. (a) The complete circuit diagram of the Q enhancement circuit. (b) The frequency
characteristic of the original resonance circuit (A) and the Q-enhanced circuit (B). (c) Q
adjustment of the resonance circuit. (curves correspond to VG2 = 0.6V to 1.4V with 0.2V intervals).
7.1.1.3. Bandwidth of a Parallel Resonance Circuit
We know that a parallel resonance circuit has a frequency selective behavior. Its
impedance is maximum at its resonance frequency and decreases at below and above of
this frequency. We already know that due to the series resistances of the inductance
and/or capacitance branches there may exist a skew as seen in Fig. 7.3-a. We also know
that this asymmetry is associated with a shift of the zero crossing frequency of the phase,
from the top of the magnitude curve. Keeping in mind these imperfections, we’ll prefer
the conventional approach to obtain universal expressions for the bandwidth, assuming
that the quality factor of the circuit is sufficiently high, or the series resistances of the
inductance and capacitance branches are sufficiently balance each other, as seen in
7.1.1.1. Since the equivalent circuit seen in Fig.7.5.b can represent both of these cases,
we’ll base our derivations on this circuit.
The impedance of the circuit was obtained as (7.21), that can be written in the
frequency domain as
1
Z  Reff
R
1  j eff ( 2 LC  1)
L
2
With LC  1/ 0 and Q  R / L0 this expression can be arranged as
1
1

R
eff
1  j Q
  2 
1  jQ 0  2  1
  0 
  2 / 0
Z  Reff
where
(7.24)
(7.24-a)
and  is the difference from the resonance frequency. From (7.24) the magnitude and
phase of the impedance can be written as
Z  Z (0 )
1
1  (  Q) 2
   arctan(  Q)
(7.24-a)
It can be seen from these expressions that;
- At resonance, the magnitude of the impedance is maximum and equal to Reff,
and from (7.20)
Z (0 )  Reff  Qeff L0
- It means that if the effective quality factor is known, higher the inductance
value gives higher the resonance impedance. This is an important fact that is useful to
obtain higher voltage gain from a tuned amplifier, as will be shown later on.
- For βQ = ±1 the magnitude of the impedance decreases to
correspond to the –3dB frequencies of the magnitude curve:
f
f
f ( 3dB )  f 0  0 and the bandwidth B  0
2Q
Q
2.Z (0 ) , that
(7.25)
- The phase angle of the impedance is zero at the resonance frequency (the
impedance is resistive), as expected.
- Below the resonance frequency the phase angle is positive (impedance is
inductive) and above f0 the impedance is capacitive. The phase angles corresponding to
the band ends (-3dB frequencies) are  / 4 . The normalized magnitude and phase
characteristics of Z are given in Fig. 7.8.
Figure 7.8 The normalized magnitude and phase characteristics of a parallel resonance circuit
7.1.1.4. Currents of L and C Branches of a Parallel Resonance Circuit
Assume that the input current of the parallel resonance circuit shown in Fig. 7.9
under resonance is i(0 ) . The voltage between the terminals of the circuit is
v(0 )  i (0 ) Reff . The current of the capacity branch is
iC(ω0)
iR(ω0)
i(0 )  I sin(0t )
Reff
iL(ω0)
C
L
Figure 7.9. The branch currents of a parallel resonance circuit under resonance.
iC (0 )  v(0 )( jC0 )  i (0 ) Reff ( jC0 )
iC (0 )  j [i(0 )  Qeff ]
(7.26)
It means that:
- The current flowing trough the capacitance branch leads 90º the input current.
- The magnitude of the current flowing trough the capacitance branch is Qeff times
bigger than the input current.
Similarly the current of the inductance branch can be written as
v(0 ) i(0 )Reff
iL (0 )=
=
jL0
jL0
(7.27)
iL (0 )=  j [i(0 )  Qeff ]
It means that:
- The current flowing trough the inductance branch lags 90º behind the input
current.
- The magnitude of the current flowing trough the inductance branch is Qeff times
bigger than the input current. This effect can lead to a long term effect called as the
“electro migration” that means the weakening of a conductor due to the momentum
transfer between the electrons and the metal atoms where the current density exceeds a
certain value (approximately 2 mA/m2 for aluminum7). Note that a similar problem is
valid for the interconnects of the capacitor of the resonance circuit.
Problem 7.3.
Calculate the branch currents of the circuit shown in Fig. 7.5-a that corresponds
to a more realistic case.
7.1.2. The Series Resonance circuit
A series resonance circuit can be formed by connecting a capacitor and an
inductor in series (Fig. 7.10-a). The series resistances representing the losses of the
inductor and the capacitor are shown with rL and rC , and rS is the internal resistance of
the voltage source. Circuit can be simplified as shown in Fig. 7.10-b, where reff is the
sum of rL, rC and rS. The impedance of the circuit is
Z  (reff  sL 
7
1
)
sC
(7.28)
This value is given for d.c. It is known that the electromigration is less effective at high frequencies. But
the current crowding due to the skin effect and the proximity effects must be kept under considerations.
and the admittance
1
Y
reff  sL 
Y
1
sC

1
L
s2  s
s
reff
L

1
LC
1
s
L (s  s p1 )( s  s p 2 )
(7.29)
rL
reff
rS
i
+
L
v
+
i
L
v
rC
C
C
(a)
(b)
Figure 7.10. (a) The series resonance circuit, (b) its simplified form.
The poles of this admittance function is same as the poles of the impedance of a parallel
resonance circuit:
s p1, p 2    j 02   2
where
 02 
1
,
LC
 
reff
2L

0
2Qeff
(7.30)
Qeff 
,
0 L0

2
reff
(7.31)
Now the impedance of the circuit can be written as
Z ( )  Z (0 )(1   Qeff )
(7.32)
where,   2 / 0 and  is the difference from the resonance frequency, similar to
the parallel resonance circuit. The magnitude and phase of the impedance are
Z ( )  Z (0 ) 1  (  Qeff ) 2
( )  arctan(  Qeff )
(7.33)
The normalized variations of the magnitude and the phase of the impedance are
given in Fig. 7.11. From expressions (7.33 ) and Fig. 7.11 it can be seen that;
- The magnitude of the impedance is minimum and equal to reff at ω0.
- For βQ = ±1 the magnitude of the impedance increases to
be expressed as that the +3dB frequencies of the magnitude are
f ( 3dB )  f 0 
f0
2Q
and the bandwidth B 
f0
Q
2.Z (0 ) . This can
(7.25)
- The phase angle of the impedance is zero at the resonance frequency (the
impedance is resistive), as expected.
- Below the resonance frequency the phase angle is negative (impedance is
capacitive) and above f0 the impedance is inductive. The phase angles corresponding to
the band ends (+3dB frequencies) are  / 4 .
Figure 7.11. The normalized magnitude and phase characteristics of a series resonance circuit
7.1.2.1. Voltages in a Series Resonance Circuit
Assume a series resonance circuit as shown in Fig.7.12. The current at resonance
is i (0 )  v / reff ,where veff  rS  rL  rC .Then the voltage between the terminals of the
inductor can be written as
vL (0 )  i (0 )(rL  jL0 )  v(0 )
r
 j.v (0 )  L  Qeff
 reff

and similarly,
It means that;
vC (0 )   j.v(0 )  Qeff
rL  jL0
reff

  j.v(0 )  Qeff

(7.26)
(7.27)
- The voltage between the terminals of the inductor is approximately Qeff times
bigger than the input voltage and leads 90º the input voltage.
- The voltage on the capacitor is approximately Qeff times bigger than the input
voltage and lags 90º behind the input voltage. This is an important property that needs
care to prevent the breakdown of the capacitor dielectric, that is especially important if
the capacitor is a MOS capacitor or MOS varactor.
rL
rS
v(0 )  V sin(0t )
+
i( 0 )
vL( 0 )
L
rC
vC( 0 )
C
Figure 7.12. The voltages in a series resonance circuit at resonance.