小信号模型与瞬态响应

任何实际器件都不是理想线性的,特别是在其整个动态范围内。如果将信号的范围限制在整个动态范围中相对较小且近似线性的范围内,剥离直流偏置取其微分特性,则就得到一个近似的线性模型——即小信号模型。瞬态响应分析与静态分析相比较是一种更符合实际工作状态的分析方法, 在标定工况转速下,探讨了求解缸盖瞬态应力的方法,研究了缸盖高应力梯度部位多轴应力随着时间的变化趋势,为缸盖的疲劳分析与寿命预测提供可靠的边界条件。
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1. Lecture #19 ANNOUNCEMENT • Quiz #4 (Thursday 4/3) to cover Chapters 10 & 11 OUTLINE • BJT transient response • BJT small-signal model, fT Reading: Chapter 12 Spring 2003 EE130 Lecture 19, Slide 1 BJT Switching - Qualitative Spring 2003 EE130 Lecture 19, Slide 2 1

2. Turn-on transient • We know: dQB = I BB − QB where IBB=VS/RS dt τB • The general solution is: QB (t ) = I BBτ B + Ae − t / τ B • Initial condition: QB(0)=0. since transistor is in cutoff     QB (t ) = I BBτ B (1 − e − t / τ B ) t r = τ B ln 1   VCC / RL   QB (t ) I BBτ B + Ae − t / τ B 1− I τ     τ = τt L0 ≤ t ≤ tr BB B iC (t ) =  t  VCC Lt ≥ t r  RL Spring 2003 EE130 Lecture 19, Slide 3 Turn-off transient • We know: dQB = −ξI BB − QB dt τB • The general solution is: QB (t ) = −ξI BBτ B + Ae − t / τ B • Initial condition: QB(0)=IBBτB [ QB (t ) = I BBτ B (1 + ξ )e − t / τ B − ξ ]      1+ ξ   I CC L0 ≤ t ≤ t sd t sd ≅ τ B ln   I CCτ t  iC (t ) =  QB (t ) I BBτ B [(1+ξ )e −t / τ B −ξ ]  I τ +ξ     τ = Lt ≥t BB B  t τt sd Spring 2003 EE130 Lecture 19, Slide 4 2

3. Small-Signal Model B C Forward-active mode, Common-emitter configuration: + Cπ vbe rπ gm vbe I C = α F I F e qVBE / kT − E E transconductance: dI C d gm ≡ = (α F I F e qVBE / kT ) At 300 K, for example, dVBE dVBE gm=IC /26mV. q = α F I F e qVBE / kT = I C /(kT / q ) kT g m = I C /(kT / q ) Spring 2003 EE130 Lecture 19, Slide 5 Small-Signal Model (cont.) 1 dI B 1 dI C g = = = m rπ dVBE β dc dVBE β dc β dc β dc rπ = rπ = gm gm dQF d (τ F I C ) Cπ = = = τ F gm dVBE dVBE This is the minority-carrier charge-storage capacitance, better known as the diffusion capacitance. Add the depletion-layer capacitance, CJBE : Cπ = τ F g m + CdBE Spring 2003 EE130 Lecture 19, Slide 6 3

4. Forward Transit Time τF QF τ F = τ E + τ BE + τ t + τ BC = IC where τ E = emitter delay time τ BE = emitter-base depletion region transit time τ t = base transit time τ BC = base-collector depletion-region transit time • To reduce the forward transit time, the emitter as well as the depletion layers must be kept thin. Spring 2003 EE130 Lecture 19, Slide 7 Example: Small-Signal Model Parameters A BJT is biased at IC = 1 mA and VCE = 3 V. βdc=90, τF=5 ps, and T = 300 K. Find (a) gm , (b) rπ , (c) Cπ . Solution: 1 mA mA (a) g m = I C /(kT / q) = = 39 = 39 mS (milli siemens) 26 mV V (b) rπ = βdc / gm = 90/0.039 = 2.3 kΩ c) Cπ = τ F g m = 5 ×10 −12 × 0.039 ≈ 1.9 ×10 −14 F = 19 fF(femto farad) Spring 2003 EE130 Lecture 19, Slide 8 4

5. Application of Small-Signal Model Once the model parameters have been determined, one can analyze circuits with arbitrary source and load impedance. B C The parameters are routinely + Signal C vbe rπ gm vbe Load determined through comprehensive source π - measurement of the BJT AC E E and DC characteristics. rb Cµ rc B C + Cπ rπ C dBC Full BJT equivalent circuit: vbe gm vbe ro − re E Spring 2003 EE130 Lecture 19, Slide 9 Cutoff Frequency fT B C + Signal Load 1 C source π vbe rπ gm vbe β ac = 1 at fT = - 2π (τ F + C JBE kT / qI C ) E E The load is a short circuit, and the signal source is a current source, ib , at frequency, f. At what frequency does the a.c. current gain fall to unity? ib ib vbe = = input admittance 1 / rπ + jωCπ ic = g m vbe ic gm 1 β (ω ) = = = ib 1 / rπ + jωCπ 1 / β F + jωτ F + jωCdBE kT / qI C Spring 2003 EE130 Lecture 19, Slide 10 5

6.For the full BJT 1 fT = equivalent circuit: 2π (τ F + (CdBE + CdBC )kT / (qI C ) + CdBC (re + rc )) SiGe HBT by IBM fT is commonly used as a metric for the speed of a transistor. Spring 2003 EE130 Lecture 19, Slide 11 Cutoff Frequency fT 1 fT = 2π (τ F + (CdBE + CdBC )kT / (qI C ) + CdBC (re + rc )) • To maximize ft: – Increase IC – Minimize CdBE, CdBC – Minimize re, rc – Minimize τF Spring 2003 EE130 Lecture 19, Slide 12 6

7.Base Widening at High IC: the Kirk Effect • At very high current densities (> 0.5 mA/µm2), base widening occurs*, so QB increases. → tt and τBC increase, so τF increases and fT decreases. *For an NPN BJT, the electron Top to bottom : density in the collector (n = NC) VCE = 0.5V, 0.8V, becomes insufficient to support 1.5V, 3V. the collector current even if the electrons move at the saturation velocity. I C = qAnvsat IC ρ dep ,C = qN C − qn = qN C − Avsat Eventually, ρ changes sign as IC increases (for fixed VBC), and the base width is effectively widened. Spring 2003 EE130 Lecture 19, Slide 13 BJT Structure for High Speed B E C P+polySi N+ polySi P+polySi p+ P base p+ Shallow N+ N collector trench Deep Deep trench N+ subcollector trench • Narrow base P− substrate • n+ poly-Si emitter • Self-aligned p+ poly-Si base contacts • Lightly-doped collector • Heavily-doped epitaxial subcollector • Shallow trenches and deep trenches filled with SiO2 for electrical isolation Spring 2003 EE130 Lecture 19, Slide 14 7