CE/CS Amplifier Response at High Frequencies
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High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency . . CE/CS Amplifier Response at High Frequencies INEL 4202 - Manuel Toledo August 20, 2012 INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 1/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Outline 1. High Frequency Models 2. Simplified Method 3. Common-emitter 4. Miller Theorem 5. Unity Gain Frequency INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 2/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency High Frequency Models C gd G D rd C gs S g m vgs S Cµ B C rπ vπ rO Cπ g m vπ E E small-signal incremental model PARASITIC CAPS LIMIT GAIN AT HIGH FREQS. INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 3/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency CE High Frequency Model RTH vS Cµ B RB rπ vπ RB=R1||R2 INEL 4202 - Manuel Toledo Cπ C RLL=RC||RL g m vπ E CE/CS High Frequency Analysis 4/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Open-circuit time constant method . 2. 3. 4. 5. 1 Replace all coupling and bypass caps by shorts Select one parasitic cap; call it CH1 Replace all other parasitic caps by open circuits Find resistance seen by CH1 ; call it RH1 High frequency pole associated with CH1 is ωH1 = 1 CH1 RH1 . Repeat above steps for each parasitic cap 7. Find equivalent high frequency cutoff 6 ωH = ∑n 1 1 i=1 ωHi INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 5/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Single-stage amplifier VCC R1 RC CC2 RTH CC1 vS R2 INEL 4202 - Manuel Toledo RE CE CE/CS High Frequency Analysis RL 6/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Single-stage amplifier RTH vS Cµ B RB rπ vπ RB=R1||R2 INEL 4202 - Manuel Toledo Cπ C RLL=RC||RL g m vπ E CE/CS High Frequency Analysis 7/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Resistance Seen by Cπ : Rπ = rπ || RB || RTH Seen by Cµ : itest vtest RB||RTH||rπ vπ Rµ INEL 4202 - Manuel Toledo g m vπ RLL=RC||RL CE/CS High Frequency Analysis 8/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Rµ Resistance seen by Cµ itest vtest RB||RTH||rπ vπ Rµ g m vπ RLL=RC||RL vπ = itest (RB || RTH || rπ ) INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 9/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Rµ Applying KVL on the external loop yields vtest = vπ + (itest + gm vπ )RLL = itest (RB || RTH || rπ ) Rµ +(1 + gm (RB || RTH || rπ ))itest RLL vtest = itest = RB || RTH || rπ + RLL +gm (RB || RTH || rπ )RLL INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 10/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Miller Theorem Y=sC iIN vin Assume that Am = Use vOUT vIN iOUT Am vout is negative and is independent of Y = sC . vOUT = Am vIN vIN = vOUT /Am INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 11/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Miller Theorem Input: iIN = Y (vIN − vOUT ) = sC (1 − Am )vIN = sCIN vIN i.e. from the input C looks like a bigger capacitor C (1 − Am ). Output: iOUT = Y (vOUT − vIN ) 1 = sC (1 − )vOUT = sCOUT vOUT Am i.e. from the output C looks like a capacitor C (1 − INEL 4202 - Manuel Toledo 1 Am ) CE/CS High Frequency Analysis ≈ C. 12/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Miller Theorem vIN CIN vOUT Am INEL 4202 - Manuel Toledo COUT CE/CS High Frequency Analysis 13/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Miller Theorem To apply Miller’s Theorem, make sure that Am is negative Am is real, i.e. load is resistive INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 14/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Unity-gain frequency: ft ft : frequency at which the transistor’s β = 1. Cµ ic ib ib Zb rπ INEL 4202 - Manuel Toledo ic vπ Cπ g m vπ CE/CS High Frequency Analysis 15/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Unity-gain frequency: ft ic = gm vπ − vπ sCµ vπ = ib × Zb 1 1 || sCπ sCµ 1 1 rπ + sCπ + sCµ rπ 1 + srπ (Cπ + Cµ ) Zb = rπ || = = INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 16/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Unity-gain frequency: ft β(s) = = ≈ ic ib gm rπ − srπ Cµ 1 + srπ (Cπ + Cµ ) β0 1 + srπ (Cπ + Cµ ) Midband β ≡ β0 = gm rπ β has a pole at ωβ = 1 rπ (Cπ + Cµ ) INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 17/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Unity-gain frequency: ft ft : f at which | β(s) |= 1 β02 = 1 + √ ωt = ωβ ωt2 ωβ2 β02 − 1 ≈ β0 ωβ Data sheet often specifies ft and Cµ ; Cπ can then be found from above equations. INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 18/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Example A common-source amplifier is constructed with a 10µF bypass capacitor in parallel with a 1kΩ resistor, both connected to the FET’s source terminal. The equivalent resistance “seen” by the bypass capacitor is 100Ω. At high frequencies there is a single pole located at 1MHz. If the amplifier’s midband gain is 80dB, find an expression for the amplifer’s gain as a function of the complex frequency s, valid for low-, mid- and high-frequencies. INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 19/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency A common-source amplifier is constructed with a 10µF bypass capacitor in parallel with a 1kΩ resistor, both connected to the FET’s source terminal. The equivalent resistance “seen” by the bypass capacitor is 100Ω. At high frequencies there is a single pole located at 1MHz. If the amplifier’s midband gain is 80dB, find an expression for the amplifer’s gain as a function of the complex frequency s, valid for low-, mid- and high-frequencies. ANSWER: Av (s) = −104 s + 102 1 3 s s + 10 ⧸2π × 106 + 1 INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 20/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Example For the circuit shown below, find (i) the pole frequency applying Miller’s theorem; (ii) the pole frequency using the open-circuit time constant method; and (iii) an expression for the voltage gain Av (s) = vOUT vS as a function of complex frequency s, valid for midand high-frequencies. C1=10-11F 10k vS 5k v1 + INEL 4202 - Manuel Toledo 1k 100v1 1k + vOUT - CE/CS High Frequency Analysis 21/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Example For the circuit shown below, find (i) the pole frequency applying Miller’s theorem; (ii) the pole frequency using the open-circuit time constant method; and (iii) an expression for the voltage gain Av (s) = vOUT vS as a function of complex frequency s, valid for midand high-frequencies. C1=10-11F 10k vS 5k v1 + 1k 100v1 1k + vOUT - ANSWER: (i) 297krps; (ii) 297krps (iii) Av (s) = −16.7 s⧸ 1 297krps +1 INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 22/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Prob. 6.110 A CS amplifier is specified to have gm = 5mA/V , ro = 40kΩ, Cgs = 2pF , Cgd = 0.1pF , CL = 1pF , Rsig = 20kΩ, and RL = 40kΩ. (a) Find the low-frequency gain AM and use open-circuit time constants to estimate the 3-dB frequency fH . Hence determine the gain-bandwidth product. (b) If a 500Ω resistance is connected in the source lead, find the new values of |AM |, fH , and the gain-bandwidth product. Assume gmb = 1mA/V . INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 23/ 24 High Frequency Models Simplified Method Common-emitter Miller Theorem Unity Gain Frequency Prob. 6.110 INEL 4202 - Manuel Toledo CE/CS High Frequency Analysis 24/ 24
