1 High-Frequency Analysis of Circuits Asst. Prof. MONTREE SIRIPRUCHYANUN, D. Eng. Dept. of Teacher Training in Electrical Engineering, Faculty of Technical Education King Mongkut’s Institute of Technology North Bangkok http://www.te.kmitnb.ac.th/msn mts@kmitnb.ac.th 1 Overview • Background – So far, our treatment of small-signal analysis of amplifiers has been for low frequencies where internal capacitances do not affect operation. However, we will see that internal capacitances do exist. We will see how these capacitances affect the frequency response of amplifiers and why we modeled real op amps to have a frequency response with a single dominant pole. To fully understand and model the frequency response of amplifiers, we utilize Bode plots again. We will look at two methods to facilitate the high-frequency circuit analysis. One technique called open-circuit time constant (OCT) method approximates the high-frequency response by calculating the time constant associated with each capacitor in a circuit. Miller’s theorem provides a way to deal with bridging capacitors (between input and output). – In this lecture, we will focus on understanding and analyzing the frequency response of MOSFET-based amplifiers. High-frequency analysis of BJT-based amplifiers are similar. 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 2 2 High-Frequency MOSFET Model • Let’s review the MOSFET high-frequency model complete simplified – when source is connected to the body, model is simplified (remove Csb) – in saturation, Cgb ≅ 0 – further simplify model by removing Cdb for hand calculations 3 224510 Advanced coomunication circuit design High-Frequency BJT Model Cμ rb B C gmvπ Cπ rπ rο vπ E • In BJTs, the PN junctions (EBJ and CBJ) also have capacitances associated with them – Cμ is the reverse-biased CB junction • Where mjc is between 0.2 and 0.5, V0 is between 0.5V and 1V – Cπ represents the capacitance of the forward-biased EBJ which exhibits both the junction cap and diffusion cap – At high frequencies, base resistance (rb) also plays an important role in device operation • Where Cje is the junction cap and τF is the base-transit time 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 4 3 Amplifier Transfer Function |A| (dB) |A| (dB) A0 ωH • • • • AM -3dB ω -3dB ωL ωH ω Voltage-gain frequency response of amplifiers seen so far take one of two forms – Direct-Coupled (DC) amplifiers exhibit low-pass characteristics Æ flat gain from DC to ωH – Capacitively-coupled amplifiers exhibit band-pass characteristics – attenuation at low frequency due to impedance from coupling capacitances increasing for low frequencies We will focus on the high-frequency portion of the response (ωH) – Gain drops due to effects of internal capacitances of the device Bandwidth is the frequency range over which gain is flat – BW = ωH or ωH-ωL ≈ ωH (ωH >> ωL) Gain-Bandwidth Product (GB) – Amplifier figure of merit – GB ≡ AMωH where AM is the midband gain – We will see later that it is possible to trade off gain for bandwidth 224510 Advanced coomunication circuit design 5 Gain Function A(s) • We can represent the frequency dependence of gain with the following expression: – Where FL(s) and FH(s) are the functions that account for the frequency dependence of gain on frequency at the lower and upper frequency ranges • We will use open-circuit time constant method to find FH(s) • Assume FL(s) = 1 for now – We can solve for AM by assuming that large coupling capacitors are short circuits and internal device capacitances are open circuits (what we have done so far for low-frequency small-signal analysis) 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 6 4 High-Frequency Response • We can express function FH(s) with the general form: • • Where ωP and ωZ represent the frequencies of high-frequency poles and zeros The zeros are usually at infinity or sufficiently high frequencies such that the numerator Æ 1 and assuming there is one dominant pole (other poles at much higher frequencies), we can approximate the function as… • – This simplifies the determination of the BW or ωH If a dominant pole does not exist, the upper -3dB frequency ωH can be found from a plot of |FH(jω)|. Alternatively, we can approximate with the following formula – Note: if ωP1 is a dominant pole, then reduces to ωH=ωP1 224510 Advanced coomunication circuit design 7 Open-Circuit Time Constant Method • It may be difficult to find the poles and zeros of the system (which is usually the case). We can find approximate values of ωH using the following method. – We can multiply out factors and represent FH(s) in an alternative form: – Where a and b are coefficients related to the zero and pole frequencies – We can show that and b1 can be obtained by considering the various capacitances in the highfrequency equivalent circuit one at a time while reducing all other capacitors to zero (or open circuits); and calculating (summing) the RC time constants due to the circuit associated with each capacitor. – This is called the open-circuit time constant (OCT) method 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 8 5 Calculating OCTs The approach: • For each capacitor: – set input signal to zero – replace all other capacitors with open circuits – find the effective resistance (Rio) seen by the capacitor Ci • Sum the individual time constants (RCs or also called the open-circuit time constants) • • This method for determining b1 is exact. The approximation comes from using this result to determine ωH. – This equation yields good results even if there is no single dominant pole but when all poles are real We will see an example of this method when we analyze the high-frequency response of different amplifier topologies 9 224510 Advanced coomunication circuit design Miller’s Theorem • Let’s review Miller’s Thoeorem Consider the circuit network below (left) with two nodes, 1 and 2. An admittance Y (Y=1/Z) is connected between the two nodes and these nodes are also connected to other nodes in the network. Miller’s theorem provides a way for replacing the “bridging” admittance Y with two admittances Y1 and Y2 between node 1 and gnd, and node 2 and gnd. 1 V1 – – I1 Y I2 2 1 I1 V2 V1 I2 2 Y1 The relationship between V2 and V1 is given by K=V2/V1 To find Y1 and Y2 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN Y2 V2 Caveat: The Miller equivalent circuit is valid only as long as the conditions that existed in the network when K was determined are not changed. 10 6 High-Frequency Response of CS Amp • Let’s now find the high-frequency response of a common-source amplifier – First, redraw using a high-frequency small-signal model for the nMOS • There are three ways to find the upper 3-dB frequency ωH – Use open-circuit time constant method – Use Miller’s theorem – Brute force calculations to find vout/vin Let’s investigate them all • 224510 Advanced coomunication circuit design 11 Using OCT on CS Amplifier • Find the RC time constants associated with Cgd and Cgs in the following circuit C gd Rs g mvgs vi • vgs C gs R L||r o vo Replace Cgd with an open circuit and find the resistance seen by Cgs Rs gmvgs vtst • RL||ro Itst vo Replace Cgs with an open-ckt and find the resistance seen by Cgd itst Rs vtst vgs gmvgs RL||ro vo 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 12 7 Using OCTs Cont’d • Summing the two time constants yields ωH – From the above equation, it is not difficult to imagine that Cgd has a more significant effect on reducing BW – The resulting frequency dependence of gain is… • Let’s compare this result with what we get using Miller’s theorem 13 224510 Advanced coomunication circuit design Using Miller’s Theorem on CS Amplifier • Redraw the high-frequency small-signal model using Miller’s theorem Rs Cgd(1+gmRL') vi vgs gmvgs RL' Cgs vo Cgd[1+1/(gmRL')] ~= Cgd CT – • • Assuming a dominant pole introduced by Cgd in parallel with Cgs – Miller multiplication of Cgd results in a large input capacitance Notice that this approximation for ωH is close to the approximation found using OCT assuming that RsCgd(1+gmRL’) dominates Let’s verify these two approaches by deriving the exact high-frequency transfer function of the CS amplifier 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 14 8 High-Frequency Response of CS Amplifier • Replace the input source and series resistance with a Norton equivalent sCgd(vgs-vo) Cgd vi/Rs gmvgs Rs vgs Cgs RL' vo Notice zero is negative. This has implications on stability (we will see later) • – The exact solution gives a zero (at a high frequency) and two poles – Notice that the s term (in the denominator) is the same as the solution using the OCT method Unfortunately, the denominator is too complicated to extract useful info as is. So, assuming the two poles are widely separated (greater than an order of magnitude), we can rewrite the expression for the denominator as… 224510 Advanced coomunication circuit design 15 HF Response of CS Amplifier • Rewrite the denominator as: – And from the solution on the previous slide we can write… • So the second pole is usually at a much higher frequency and we can assume a dominant pole (ωP1) • Using either Miller’s theorem or OCTs enables a way to quickly find approximations of the amplifier’s high-frequency response 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 16 9 Summary of 3 Approaches • To summarize, here are the dominant pole frequencies that we found using the three approaches – OCT method – Miller Theorem – Brute Force 17 224510 Advanced coomunication circuit design Frequency Response of CG Amplifier • One way to avoid the frequency limitations of Miller multiplication of Cgd is to utilize a CG amplifier configuration Cgd Rs vs • -(gm+gmb)vx Cgs vx Cdb -(gm+gmb)vx RD Csb CD=Cgd+Cdb Rs vs vx RD CS=Cgs+Csb Using OCT method, we find two time constants – At the input (source node) – At the output (drain node) • • The output drives additional load capacitance such that the output pole is dominant The frequency response of CG amplifiers is then combined with a CS stage to build a cascode circuit 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 18 10 Cascode Amplifier • Cascoding enables high bandwidth by suppressing Miller multiplication of Cgd. Let’s investigate how with the following high-frequency model of a cascode stage. RL vOUT Cgd2 -gm2+gmb2vx Cdb2+CL Cdb2+CL Vb Cgd1 Cgs2 Rs Rs vIN Cdb1+Csb2 Cgd1 gmvgs1 vin vx Cgs1 Cgs1 vout RL Cdb1+Csb2+Cgs2 – Use OCT method to find the time constants associated with each capacitor. The time constant associated with Cgd1 is… NOTICE: Miller multiplication is ~2 19 224510 Advanced coomunication circuit design Frequency Response of Source Followers • Start with a high-frequency small-signal model of the source follower circuit Rs Rs vOUT vIN vin Cgd vgs Cgs CL gmvgs vout CL • Directly solving for vout/vin yields: • – The zero is due to Cgs that directly couples the signal from the input to the output If poles are far apart, then the s term represents the dominant pole 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 20 11 More on Source Follower • Other important aspects of a source follower are its input and output impedances (since they are often used as buffers) Let’s calculate the input impedance using the high-freq small-signal models • Zin Cgs vgs gmvgs vout 1/gmb CL Now calculate the output impedance (ignoring gmb for simplicity) • Rs – – – At low frequency, Zout ≈ 1/gm At high frequency, Zout ≈ Rs Shape of the response depends on the relative size of Rs and 1/gm |Zout| Cgs vgs gmvgs vout |Zout| 1/gm Rs Rs 1/gm Zout ω Zout can look inductive or capacitive depending on Rs and 1/gm ω 21 224510 Advanced coomunication circuit design Differential Pair • We have seen that a symmetric differential amplifier can be analyzed with a differential half circuit. This still holds true for high-frequency small-signal analysis. RD +vd/2 -vd/2 Rs RD vout Rs Cgd gmvgs Rs vd/2 Cgs Cdb RD vout I – The response is identical to that of a common-source stage 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 22 12 High-Frequency CMRR • The CMRR of a differential pair degrades at high frequency due to a number of factors. The most important is the increase in CM gain with frequency due to capacitance on the tail node. Use the common-mode equivalent half circuit to understand how CM gain increases with frequency – Draw the small-signal equivalent model and see the effect of CTAIL on the vout/vin transfer function • RD vout,CM vin,CM 2ro vin,CM CTAIL I/2 gmvgs vgs Zero at ωZ = 1/roCTAIL (since ro is big, ωZ occurs at a low frequency) There are additional poles at higher frequencies due to CTAIL and other internal capacitances (that we have ignored) The zero causes the CM gain to increase with frequency until the higher frequency poles kick in Æ CMRR degrades due to the zero – – • vout,CM RD 2ro CTAIL 23 224510 Advanced coomunication circuit design HF CMRR plots • The impact of the zero in the CM gain on CMRR can be illustrated as shown – Remember CMRR = Ad/Acm |Acm| (dB) ωZ ω (log scale) |Ad| (dB) • There is a trade off between CMRR and voltage headroom – Wider current source devices enable lower vds – Wider current source device means larger CTAIL ωP ω (log scale) -20dB/dec CMRR (dB) -40dB/dec ωZ 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN ωP ω (log scale) 24 13 Example: Diff Pair with Active Load Q3 Q4 y z Q1 +vd /2 • diff half circuit Æ node x is virtual gnd Vo • Cy = Cgd1 + Cdb1 + Cgs3 + Cdb3 + Cgs4 + Cpar Cload Q2 • Cz = Cgd2 + Cdb2 + Cgd4 + Cdb4 + Cload x -vd /2 • Ro = ro4 // ro2 Q5 Vy = gm1 Vid 1 2 Io = (gm3 + sCy) A = gm1 Ro gm2 Vid 2 + gm3vy Vo = RoIo (1 + sCz/2gm) 1 (1 + sCzRo) (1 + sCz/gm) 25 224510 Advanced coomunication circuit design Example: 2 Stage CMOS Op Amp Q3 +vd /2 Q4 Q1 Cx Q6 Gm1 R1 Cc Q2 C1 Gm2 R2 C2 Cload -vd /2 Q5 Q7 • C1= Cgd2 + Cdb2+ Cgd4 + Cdb4+ Cgs6 • C2= Cgd7 + Cdb7 + Cdb6+ Cload • Cx= Cc + Cgd6 A = Gm1R1 Gm2R2 Using OCT: next pole is at C2/Gm2 • Gm2=gm6 • R1= ro2 // ro4 • R2= ro6 // ro7 • R1= ro2 // ro4 dominant pole is at (Gm2R2)CxR1 unity gain freq is at Gm1/Cx • Gm1=gm1 =gm2 How do you determine the phase margin? 224510 Advanced coomunication circuit design Asst. Prof. Dr. MONTREE SIRIPRUCHYANUN 26