CE/CS Amplifier Response at High Frequencies

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1 .. CE/CS Amplifier Response at High Frequencies INEL Manuel Toledo August 20, 2012 INEL Manuel Toledo CE/CS High Frequency Analysis 1/ 24

2 Outline.1 High Frequency Models.2 Simplified Method.3 Common-emitter.4 Miller Theorem.5 Unity Gain Frequency INEL Manuel Toledo CE/CS High Frequency Analysis 2/ 24

3 High Frequency Models G C gd r d D S C gs g m v gs S B C µ C r π v π C π g m v π r O E E small-signal incremental model PARASITIC CAPS LIMIT GAIN AT HIGH FREQS. INEL Manuel Toledo CE/CS High Frequency Analysis 3/ 24

4 CE High Frequency Model R TH B C µ C v S R B r π vπ C π g m v π R LL =R C R L R B =R 1 R 2 E INEL Manuel Toledo CE/CS High Frequency Analysis 4/ 24

5 Open-circuit time constant method.1 Replace all coupling and bypass caps by shorts.2 Select one parasitic cap; call it C H1.3 Replace all other parasitic caps by open circuits.4 Find resistance seen by C H1 ; call it R H1.5 High frequency pole associated with C H1 is 1 ω H1 = C H1 R H1.6 Repeat above steps for each parasitic cap.7 Find equivalent high frequency cutoff ω H = 1 n i=1 1 ω Hi INEL Manuel Toledo CE/CS High Frequency Analysis 5/ 24

6 Single-stage amplifier V CC R 1 R C C C2 R TH C C1 v S R 2 R E C E R L INEL Manuel Toledo CE/CS High Frequency Analysis 6/ 24

7 Single-stage amplifier R TH B C µ C v S R B r π vπ C π g m v π R LL =R C R L R B =R 1 R 2 E INEL Manuel Toledo CE/CS High Frequency Analysis 7/ 24

8 Resistance Seen by C π : Seen by C µ : R π = r π R B R TH i test R B R TH r π v π v test R µ g m v π R LL =R C R L INEL Manuel Toledo CE/CS High Frequency Analysis 8/ 24

9 R µ Resistance seen by C µ i test R B R TH r π v π v test R µ g m v π R LL =R C R L v π = i test (R B R TH r π ) INEL Manuel Toledo CE/CS High Frequency Analysis 9/ 24

10 R µ Applying KVL on the external loop yields v test = v π + (i test + g m v π )R LL = i test (R B R TH r π ) +(1 + g m (R B R TH r π ))i test R LL R µ = v test i test = R B R TH r π + R LL +g m (R B R TH r π )R LL INEL Manuel Toledo CE/CS High Frequency Analysis 10/ 24

11 Miller Theorem Y=sC i IN i OUT v in A m v out Assume that A m = v OUT v IN Use is negative and is independent of Y = sc. v OUT = A m v IN v IN = v OUT /A m INEL Manuel Toledo CE/CS High Frequency Analysis 11/ 24

12 Miller Theorem Input: i IN = Y (v IN v OUT ) = sc(1 A m )v IN = sc IN v IN i.e. from the input C looks like a bigger capacitor C(1 A m ). Output: i OUT = Y (v OUT v IN ) = sc(1 1 A m )v OUT = sc OUT v OUT i.e. from the output C looks like a capacitor C(1 1 A m ) C. INEL Manuel Toledo CE/CS High Frequency Analysis 12/ 24

13 Miller Theorem v IN C IN A m v OUT C OUT INEL Manuel Toledo CE/CS High Frequency Analysis 13/ 24

14 Miller Theorem To apply Miller s Theorem, make sure that A m is negative A m is real, i.e. load is resistive INEL Manuel Toledo CE/CS High Frequency Analysis 14/ 24

15 Unity-gain frequency: f t f t : frequency at which the transistor s β = 1. i c C µ i b i b Z b r π vπ C π g m v π i c INEL Manuel Toledo CE/CS High Frequency Analysis 15/ 24

16 Unity-gain frequency: f t i c = g m v π v π sc µ Z b = r π = = v π = i b Z b 1 1 sc π 1 1 sc µ r π + sc π + sc µ r π 1 + sr π (C π + C µ ) INEL Manuel Toledo CE/CS High Frequency Analysis 16/ 24

17 Unity-gain frequency: f t Midband β β 0 = g m r π β has a pole at β(s) = i c = ω β = i b g m r π sr π C µ 1 + sr π (C π + C µ ) β sr π (C π + C µ ) 1 r π (C π + C µ ) INEL Manuel Toledo CE/CS High Frequency Analysis 17/ 24

18 Unity-gain frequency: f t f t : f at which β(s) = 1 β 2 0 = 1 + ω2 t ω 2 β ω t = ω β β β 0ω β Data sheet often specifies f t and C µ ; C π can then be found from above equations. INEL Manuel Toledo CE/CS High Frequency Analysis 18/ 24

19 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 Manuel Toledo CE/CS High Frequency Analysis 19/ 24

20 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: A v (s) = 10 4 s s s 2π INEL Manuel Toledo CE/CS High Frequency Analysis 20/ 24

21 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 A v (s) = v OUT v S as a function of complex frequency s, valid for midand high-frequencies. 10k C 1 =10-11 F 1k v S 5k - v v 1 1k + v OUT - INEL Manuel Toledo CE/CS High Frequency Analysis 21/ 24

22 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 A v (s) = v OUT v S as a function of complex frequency s, valid for midand high-frequencies. 10k C 1 =10-11 F 1k v S 5k - v v 1 1k + v OUT - ANSWER: (i) 297krps; (ii) 297krps (iii) A v (s) = s 297krps +1 INEL Manuel Toledo CE/CS High Frequency Analysis 22/ 24

23 Prob A CS amplifier is specified to have g m = 5mA/V, r o = 40kΩ, C gs = 2pF, C gd = 0.1pF, C L = 1pF, R sig = 20kΩ, and R L = 40kΩ. (a) Find the low-frequency gain A M and use open-circuit time constants to estimate the 3-dB frequency f H. Hence determine the gain-bandwidth product. (b) If a 500Ω resistance is connected in the source lead, find the new values of A M, f H, and the gain-bandwidth product. Assume g mb = 1mA/V. INEL Manuel Toledo CE/CS High Frequency Analysis 23/ 24

24 Prob INEL Manuel Toledo CE/CS High Frequency Analysis 24/ 24

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