3/28/2011 A mall ignal Analysis of a BJ lecture 1/12 A mall-ignal Analysis of a BJ he collector current i of a BJ is related to its base-emitter oltage as: i i e Jim tiles he Uni. of Kansas Dept. of EE
3/28/2011 A mall ignal Analysis of a BJ lecture 2/12 One messy result ay the current and oltage hae both D.. (, ) and small-signal ( i, ) components: c and i ( t) + i ( t) c ( t) + ( t) herefore, the total collector current is: i ( t) e + i ( t) e c ( t) + ( t) Jim tiles he Uni. of Kansas Dept. of EE
3/28/2011 A mall ignal Analysis of a BJ lecture 3/12 Apply the mall-ignal Approximation Q: Yikes! he exponential term is ery messy. s there some way to approximate it? A: Yes! he collector current i c is a function of base emitter oltage. Let s perform a small-signal analysis to determine an approximate relationship tween i c and. Note that the alue of ( t) + ( t) is always ery close to the D.. oltage for all time t (since ( t ) is ery small). We therefore will use this D.. oltage as the ealuation point (i.e., bias point) for our small-signal analysis. Jim tiles he Uni. of Kansas Dept. of EE
3/28/2011 A mall ignal Analysis of a BJ lecture 4/12 How fast it grows! We first determine the alue of the collector current i when the base emitter oltage is equal to the D alue : i s e e s Of course, the result is the D.. collector current. We now determine the change in collector current due to a change in baseemitter oltage (i.e., a first deriatie), ealuated at the D.. oltage : di d ( exp ) d d e e A Jim tiles he Uni. of Kansas Dept. of EE
3/28/2011 A mall ignal Analysis of a BJ lecture 5/12 A simple approximation hus, when the base-emitter oltage is equal to the D.. bias oltage, the collector current i will equal the D.. bias current. Likewise, this collector current will increase (decrease) by an amount of ( ) e ma for eery 1m increase (decrease) in. hus, we can easily approximate the collector current when the base-emitter oltage is equal to alues such as: Respectiely, the answers are: + 1m + 3m 2m 05m. i + ( ) e (1) ma i + ( ) e (3) ma i + ( ) e (-2) ma i + ( ) e (-0.5) ma where we hae assumed that scale current is expressed in ma, and thermal oltage is expressed in m. Jim tiles he Uni. of Kansas Dept. of EE
3/28/2011 A mall ignal Analysis of a BJ lecture 6/12 he small signal approximation Recall that the small-signal oltage ( t ) represents a small change in ( t ) from its nominal (i.e., bias) oltage. For example, we might find that the alue of ( t ) at four different times t are: ( t ) 1 m 1 ( t ) 3m 2 3 4 ( t ) 2 m ( t ) 0. 5 m hus, we can approximate the collector current using the small-signal approximation as: ( ) i ( t) + e ( t) where of course e. his is a ery useful result, as we can now explicitly determine an expression for the small-signal current ic( t )! Jim tiles he Uni. of Kansas Dept. of EE
3/28/2011 A mall ignal Analysis of a BJ lecture 7/12 he small-signal collector current Recall i ( t) + i ( t), therefore: c ( ) i ( t) + i ( t) + e ( t) c ubtracting the D.. current from each side, we are left with an expression for the small-signal current ic( t ), in terms of the small-signal oltage ( t ) : ( ) i ( t) e ( t) c We can simplify this expression by noting that e, resulting in: and thus: ( ) e e ic( t) ( t) Jim tiles he Uni. of Kansas Dept. of EE
3/28/2011 A mall ignal Analysis of a BJ lecture 8/12 ransconductance: A small signal parameter We define the alue as the transconductance g m : g m A and thus the small-signal equation simply comes: i ( t) g ( t) c m Jim tiles he Uni. of Kansas Dept. of EE
3/28/2011 A mall ignal Analysis of a BJ lecture 9/12 How transistors got their name Let s now consider for a moment the transconductance g m. he term is short for transfer conductance: conductance cause its units are amps/olt, and transfer cause it relates the collector current to the oltage from base to emitter the collector oltage is not releant (if in actie mode)! Note we can rewrite the small-signal equation as: ( t) 1 i ( t) g c m he alue (1 g m ) can thus considered as transfer resistance, the alue describing a transfer resistor. ransfer Resistor we can shorten this term to ransistor (this is how these deices were named)! Jim tiles he Uni. of Kansas Dept. of EE
3/28/2011 A mall ignal Analysis of a BJ lecture 10/12 We can summarize our results as: ummarizing e D.. Equation i ( t) g ( t) mall-ignal Equation c m i ( t) + g ( t) mall-ignal Approximation m Note that we know hae two expressions for the total (D.. plus small-signal) collector current. he exact expression: i ( t) e + ( t) and the small-signal approximation: i ( t) + g ( t) m Jim tiles he Uni. of Kansas Dept. of EE
3/28/2011 A mall ignal Analysis of a BJ lecture 11/12 Accurate oer a small region Let s plot these two expressions and see how they compare: i Exact mall-signal alidity Regions g m t is eident that the small-signal approximation is accurate (it proides nearly the exact alues) only for alues of i near the D. bias alue, and only for alues of near the D. bias alue. he point (, ) is alternately known as the D.. bias point, the transistor operating point, or the Q-point. Jim tiles he Uni. of Kansas Dept. of EE
3/28/2011 A mall ignal Analysis of a BJ lecture 12/12 hange the D bias, change the transconductance Note if we change the D.. bias of a transistor circuit, the transistor operating point will change. he small-signal model will likewise change, so that it proides accurate results in the region of this new operating point: i Exact mall-signal alidity Regions g m Jim tiles he Uni. of Kansas Dept. of EE