Lecture 27: Frequency response. Context

Transcription

1 Lecture 27: Frequency response Prof J. S. Smith Context Today, we will continue the discussion of single transistor amplifiers by looking at common source amplifiers with source degeneration (also common Emitter amplifiers with emitter degeneration. We will then start discussing the frequency response of single stage amplifier, the frequency response of CE amps, and the Miller approximation. 1

2 Lecture Outline Emitter Degeneration Frequency response of the CE and CS current amplifiers Unity-gain frequency ω T Frequency response of the CE as voltage amp The Miller approximation Bias sensitivity When a transistor biasing circuit is designed, it is important to realize that the characteristics of the transistor can vary widely, and that passive components vary significantly also. Biasing circuits, must therefore be designed to produce a usable bias without counting on specific values for these components. One example is a BJT base bias in a CE amp. A slight change in the base-emitter voltage makes a very large difference in the quiescent point. The insertion of a resistor will improve sensitivity. 2

3 Typical Discrete Biasing R 1 R 2 V CC R C R E In this scheme, the base bias voltage is given by a voltage divider: V R1 B V CC R + R 1 2 The emitter will approximately follow the base voltage, so the emitter current is: I V R V / R 1 E CC BE, on E R1+ R2 Gain for Discrete Design R r + ( β +1) in R E R ( β +1) in R E R 1 R C ~ vr / R s C E v s R 2 v / R s ~ v s R E E R ( β + 1) R R R in2 E 1 2 Can be made large to couple All of source to input (even with R S ) 3

4 Insensitivity to transistor parameters Most of the circuit parameters are independent of variation of the transistor parameters, and depend only on resistance ratios. That is often a design goal, but in integrated circuits we will not want to use so many resistors. Emitter Degeneration The addition of the resistor at the emitter will have several additional effects: The transconductance will be reduced The output resistance will be increased The input resistance will be increased Each of these are a result of the negative feedback from the presence of the emitter resistor. 4

5 Source Degeneration Source degeneration for a FET is not as common as emitter degeneration is for a BJT, because the gain of a FET is already lower than for an BJT, and its input impedance is already. It is widely used, however, to raise the output impedance of CS amplifiers Source degeneration in a CS amp The input impedance of the amplifier is still Assume that the body is still connected to ground 5

6 Small signal model CS amp with source degeneration: i v in + R D v 0 v s R S Circuit gain calculation From KCL at the source, vs vs + gm( vi vs) + gmb(0 vs) RS r0 At the Drain: vs i0 + gm( vi vs) + gmb(0 vs) r The circuit s transconductance: 0 G i0 v 1+ ( g + g ) R Because of the body effect, the output is still dependant on the transistor characteristics m i m g m mb S R + r 0 S 6

7 Output impedance To calculate the output impedance, we put in a test current and calculate the voltage: i vin + it v t v s R S Output impedance Since all of the test current must go through R S, if we take i 1 to be the current through r o, we have: v [1 t vs + i r0 vs + r0 + gm + g From which we get the output resistance: ) R 1 mb S vt R RS + r0[1 + ( gm + g i ) R 0 mb S t ] ] 7

8 Frequency response So far, we have modeled the small signal response of the stages for low frequencies (but not so low that we couldn t neglect the DC blocking coupling capacitors!) Now we will put in the parasitic capacitances, and analyze the changes in the transfer functions of the circuits at higher frequencies. Parasitic Capacitances If we look at the small signal model of the FET that we developed a few weeks ago, we have capacitances between the gate and the source, and the gate and the drain 8

9 CS with parasitics When we take into account a finite source impedance in a common source amplifier, the capacitances will reduce the voltage swing at the gate at high frequencies. Parasitic Capacitances The transfer function will be a low pass filter, with a pole at the frequency determined by the source resistance and the capacitance. r s v s 9

10 Miller approximation However, the capacitance from the gate to the drain has a large effect, because the voltage on the drain is amplified, with a negative gain coefficient. This means that the contribution of the capacitance from the gate to the drain is comparable to the same capacitance to ground, multiplied by the gain of the transistor. This is called the Miller effect. We can approximate the effect, by simply putting in a capacitance to ground, multiplied by the low frequency gain. This is called the Miller approximation High frequency zero At very high frequencies, the gain flattens out again, because the capacitor couples from the gate to the drain directly, as a passive circuit r s v s 10

11 Magnitude Bode Plot pole Low frequency gain β o Unity current gain 0 db zero ω p ω T ω z CE Amplifier with Current Input Find intrinsic current gain by driving with infinite source impedance and Zero load impedance 11

12 Short-Circuit Current Gain Pure input current (R S 0 Ω) Small-signal short circuit (could be a DC voltage source) Substitute equivalent circuit model of transistor and do the math Small-Signal Model: A i Note that r o, C cs play no role (shorted out) 12

13 Phasor Analysis: Find A i KCL at the output node: I ( 0 V ) Z µ out gmv + / KCL at the input node: Solve for V : ( V ) Z µ I in V / Z + 0 / V Z Z I 1 (1/ + 1/ µ ) in Phasor Analysis for A i (cont.) I out ( g m jωcµ ) V Substituting for V ( gm jωcµ ) Ai ( jω) (1/ Z ) + jωc Substituting for Z r (1/jωC ) β (1 / ) 0 jωcµ gm Ai ( jω) 1 + jωr ( C + C ) µ µ Z r 1 + j rc ω 13

14 Short-Circuit Current Gain Transfer Function Transfer function has one pole and one zero: βo(1 jω[ C Ai ( jω) 1+ jω[ r ( C A ( jω) i µ / g + C m µ ]) )] βo(1 jω / ω z ) (1 + jω / ω ) Note: Zero Frequency much larger than pole: gm β 1 1 ωz > β βωp C r C r ( C + C ) µ µ µ p Magnitude Bode Plot β o pole βo 1 Ai ( jω) jω / ω jω/ ω Unity current gain p T 0 db zero ω p ω T ω z 14

15 Transition Frequency ω T ω T β ω o p C g m + C µ Dependence on DC collector current: C C C + je diff diff mτ F τ F / C g qi kt Base Region Transit Time! Limiting case: f T ωt 1 2 2τ F Common Source Amplifer: A i (jω) DC Bias is problematic: what sets V GS? 15

16 CS Short-Circuit Current Gain Transfer function: A ( jω) i g m ( 1 jωc / g ) jω( C gs gd + C gd ) m MOS Unity Gain Frequency Since the zero occurs at a higher frequency than pole, assume it has negligible effect: A i g g m m 1 ω T jω( C + C ) ( C + C ) gs gd gs gd W µ Cox ( VGS V ) g T m L 3 µ ( VGS VT) ωt 2 C 2 gs WLC 2 L ox 3 Performance improves like L^2 for long channel devices! For short channel devices the dependence is like ~ L^1 VGS VT µ 3 µ ( VGS VT ) Eeff v T ~ L µ ω τ 2 L 2 L L L L Time to cross channel 16

17 Miller Impedance Consider the current flowing through an impedance Z hooked up to a black-box where the voltage gain from terminal to the other is fixed (as you can see, it depends on Z) A v 2 v I v1 Z v1 v2 I v v v Av 1 Av v1 Z Z Z v 1 Miller Impedance Notice that the current flowing into Z from terminal 1 looks like an equivalent current to ground where Z is transformed down by the Miller factor: 1 A I v Z v 1 M,1 Z 1 Z A From terminal 2, the situation is reciprocal v v v A v 1 A Z Z Z Z ZM,2 1 1 A v 2 I v2 v v v 17

18 Miller Equivalent Circuit Z + Z Z Note: M,1 M,2 Z M,1 Z Z ZM,1 1 1 A 1 A v v We can de-couple these terminals if we can calculate the gain A v across the impedance Z Often the gain Av is weakly dependent on Z The approximation is to ignore Z, calculate A, and then use the decoupled miller caps 18