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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 ) jC (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 j0 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 j0t j0t v(t ) I e (e e ) C 2 j0 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 7.96 104 S 9 L0Q 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(3.98 104 ) (W / L) 9.37 nCox (VGS VT ) 374(4.54 107 )1 For this aspect ratio and (VGS -VT) =1V, the drain current is 1 W 1 nCox (VGS VT ) 374 9.3(4.54 107 ) 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 4n (VGS VT ) 0 4 374 1 3.45 104 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 (32 104 )(3.5 104 )(4.54 107 ) 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 / L0 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 L0 - 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 )( jC0 ) i (0 ) Reff ( jC0 ) 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 )= = jL0 jL0 (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 L0 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 jL0 ) v(0 ) r j.v (0 ) L Qeff reff and similarly, It means that; vC (0 ) j.v(0 ) Qeff rL jL0 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.