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FREQUENCY EFFECTS IN AMPLIFIERS
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[CHAP. 8
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VTh s2 C C RTh RC kRL s 1 gm C RC kRL 1 RC kRL sC gm VL
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For typical values, the coe cient of s2 on the right side of (8.36) is several orders of magnitude smaller than the other terms; by approximating this coe cient as zero (i.e., neglecting the s2 term), we neglect a breakpoint at a frequency much greater than !H . Doing so and using (8.32), we obtain the desired high-frequency voltage-gain ratio: Av s RC kRL sC gm VL r VS r rx s 1 gm C RC kRL 1 8:37
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HIGH-FREQUENCY FET MODELS
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The small-signal high-frequency model for the FET is an extension of the midfrequency model of Fig. 7-1. Three capacitors are added: Cgs between gate and source, Cgd between gate and drain, and Cds between drain and source. They are all of the same order of magnitude typically 1 to 10 pF. Figure 8-10 shows the small-signal high-frequency model based on the current-source model of Fig. 7-1(a). Another model, based on the voltage-source model of Fig. 7-1(b), can also be drawn.
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Fig. 8-10 High-frequency small-signal current-source FET model
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Example 8.8. For the JFET ampli er of Fig. 4-5(b), (a) nd an expression for the high-frequency voltage-gain ratio Av s and (b) determine the high-frequency cuto point. (a) The high-frequency small-signal equivalent circuit is displayed in Fig. 8-11, which incorporates Fig. 8-10. We rst nd a Thevenin equivalent for the network to the left of terminal pair a; a 0 . Noting that vgs vi , we see that the open-circuit voltage is given by sCgd gm gm V Vi sCgd i sCgd
VTh Vi
8:38
RG _
gm Lgs
RD RL
+ LL _
_ 0 S a
Fig. 8-11
CHAP. 8]
FREQUENCY EFFECTS IN AMPLIFIERS
If Vi is deactivated, Vi Vgs 0 and the dependent current source is zero (open-circuited). A driving-point source connected to a; a 0 sees only ZTh Vdp 1 Idp sCgd 8:39
Now, with the Thevenin equivalent in place, voltage division leads to VL Zeq sCgd gm 1 V Vi Zeq ZTh Th 1 ZTh =Zeq sCgd 8:40
where
1 1 1 1 Yeq sCds sCds gds GD GL Zeq rds RD RL
(8.41)
Rearranging (8.40) and using (8.41), we get Av s sCgd gm VL Vi s Cds Cgd gds GD GL 8:42
(b) From (8.42), the high-frequency cuto point is obviously !H gds GD GL Cds Cgd 8:43
Note that the high-frequency cuto point is independent of Cgs as long as the source internal impedance is negligible. (See Problem 8.40.)
MILLER CAPACITANCE
High-frequency models of transistors characteristically include a capacitor path from input to ouput, modeled as admittance YF in the two-port network of Fig. 8-12(a). This added conduction
YF IF Ii + Vi _ Yin + V1 _ Y1 I1 KF KR I2 Io + V2 _ Y2 Yo
Ii + Vi _ Yin
I1 YF (1 _ KF) KF KR
Io + V2 _
YF (1 _ KR)
Fig. 8-12
FREQUENCY EFFECTS IN AMPLIFIERS
[CHAP. 8
path generally increases the di culty of analysis; we would like to replace it with an equivalent shunt element. Referring to Fig. 8-12(a) and using KCL, we have Yin But Substitution of (8.45) into (8.44) gives Yin I1 V1 V2 YF Y1 1 KF YF V1 V1 8:46 Ii I IF 1 V1 V1 8:44
IF V1 V2 YF
(8.45)
where KF V2 =V1 is obviously the forward voltage-gain ratio of the ampli er. In a similar manner, Yo and the use of (8.45) in (8.47) gives us I V V2 Yo 2 1 YF V2 V2   Y2 KR 1 YF Y2 1 KR YF 8:48 Io I2 IF V2 V2 8:47
where KR V1 =V2 is the reverse voltage-gain ratio of the ampli er. Equations (8.46) and (8.48) suggest that the feedback admittance YF can be replaced with two shuntconnected admittances as shown in Fig. 8-12(b). When this two-port network is used to model an ampli er, the voltage gain KF usually turns out to have a large negative value, so that 1 KF YF % jKF j YF . Hence, a small feedback capacitance appears as a large shunt capacitance (called the Miller capacitance). On the other hand, KR is typically small so that 1 KR YF % YF .
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