# I. Frequency Response of Voltage Amplifiers

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1 I. Frequency Response of Voltage Amplifiers A. Common-Emitter Amplifier: V i SUP i OUT R S V BIAS R L v OUT V Operating Point analysis: 0, R s 0, r o --->, r oc --->, R L ---> Find V BIAS such that I C I SUP with the BJT in the forward active region Frequency Response - Set V BIAS 0. Substitute bipolar small signal model (with capacitors) including R S, R L, r o, r oc, and perform impedance analysis Full Analysis Miller Approximation Open Circuit Time Constants EECS 6.02 Spring 998 Lecture 2

2 B. Small-Signal Model R S C µ rπ V π C π g m V π r o r oc R L (a) II. Full Analysis of CE Voltage Amplifier A. See Pages for Analysis r π C µ g R j ω m out R r S π g m jω R C R C g R R C ω 2 R R C C out µ in µ m out in π out in µ π where R in R S r π and R out r o r oc R L Zero at g m /C µ - Ignore since higher than f T EECS 6.02 Spring 998 Lecture 2

3 B. Examine Denominator of System Transfer Function jωτ jωτ jω τ τ ω 2 τ τ 2 2 is DC gain and τ and τ 2 are the two time constants db 20 decade db 40 decade /τ /τ 2 ω log scale τ τ R C R C g R R C 2 out µ in µ m out in π τ τ R R C C 2 out in µ π Could solve for τ and τ 2 but algebraically complex Assume that τ >> τ 2. This implies that τ τ 2 τ This is a conservative estimate since τ is actually smaller τ R C g R C R C in π m out µ out µ ω dB τ R C g R C R C in π m out µ out µ EECS 6.02 Spring 998 Lecture 2

4 III. The Miller Approximation A. Input Impedance Caused by C µ I t C µ V t g m V t R'out r o r oc R L The effect of C µ on the input can be found by finding the input impedance Z i which is found from V t /I t g V R I R m t out t out The Miller Approxiamtion is that the current through C µ is small compared to that in the transconductance generator I «t g m V t g V R m t out We can relate V t and by V V t out I t jωc µ The effect of C µ at the input is Miller multiplied by (-A vcµ ) V t Z I eff jωc g R t µ m out j ω C A µ vcµ EECS 6.02 Spring 998 Lecture 2

5 B. Generalized Miller Effect An impedance connected across an amplifier with voltage gain can be replaced by an impedance to ground... divided by (- ) Z V i V o V i Z eff V o Z eff Z/( ) Common-emitter (and common-source): large and negative Capacitance at the input is magnified EECS 6.02 Spring 998 Lecture 2

6 C. Frequency for CE Volatage Amplifier Using the Miller Approximation The Miller capacitance is lumped together with C π, which results in a single-pole low-pass RC filter at the input R S rπ Vπ Cπ C M g m Vπ R' out C M C µ ( g m R' out ) r π g R out V m r R s π S jω C C R r π M S π At DC the small signal voltage gain is v out r π g R out v m r R s π S The frequency at which the magnitude of the voltage gain is reduced by / 2 ω 3dB R r C C S π π M R r S π C g R π C m out µ EECS 6.02 Spring 998 Lecture 2

7 III. Open Circuit Time Constant Analysis A. Introduction No Zeros One dominant pole (/τ << /τ 2, /τ 3,..., /τ n ) N capacitors ( jωτ ) ( jωτ 2 ) ( ) ( jωτ n ) (the example shows a voltage gain... it could by I out / or / I s ) Multiplying out the denominator b jω b 2 ( jω) 2 b n ( jω) n where b τ τ 2 τ 3... τ n b It can be shown (Gray & Meyer, 3rd ed., pp text) coefficient b can be found exactly by: N N R Ti C i τ,where N is the number of capacitors Cio i i τ Cio is the open-circuit time constant for capacitor C i C i is the i th capacitor and R Ti is the Thevenin resistance across the i th capacitor terminals (with all capacitors open-circuited) EECS 6.02 Spring 998 Lecture 2

8 B. Estimating the Dominate Pole The dominant pole of the system can be estimated by: b τ τ 2 τ 3... τ n τ exact approximate N b R Ti C i τ i , ω R Ti C i is the open-circuit time constant for capacitor C i C Power of the Technique Estimates the contribution of each capacitor to the dominant pole frequency separately. Enables the designer to understand what part of a complicated circuit is responsible for limiting the bandwidth of the amplifier. The approximate magnitude Bode plot is db approximation is valid up to ω < /τ /τ ω EECS 6.02 Spring 998 Lecture 2

9 IV. Common Emitter Amplifier Analysis Using OCT A. Small Signal Model R S C µ rπ Vπ Cπ g m Vπ r o r oc R L From the Full Analysis r π C µ g R j ω m out R r S π g m jω R C R C g R R C ω 2 R R C C out µ in µ m out in π out in µ π where R in R S r π and R out r o r oc R L b R C R C g R R C out µ in µ m out in π ω dB b R C g R C R C in π m out µ out µ Next Lecture we will show calculate b EECS 6.02 Spring 998 Lecture 2

10 V. Compare the Three Methods for analyzing the CE amp A. Full Analysis - assumes τ τ 2 τ ω dB τ R C g R C R C in π m out µ out µ B. Miller Approximation - does not take into account R out ω dB R in C g R π C m out µ C. Open Circuit Time Constant - Assumes dominant pole same as the full analysis ω db b R C g R C R C in π m out µ out µ EECS 6.02 Spring 998 Lecture 2

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