EE 105 --- Spring 2005 --- Discussion Notes (written by Amin) Page 1 Monday, February 7, 2005 EE 105 Discussion Section 101 Announcements RC Circuit One of the circuits analyzed in EE 40 (and Lab 1 of this course) is depicted in Figure 1. It was further shown that if a step input with amplitude A is applied to a RC circuit, the resulting output voltage is given by the following: −⋅=−RCtouteAV 1 (1) The RC circuit above is often used in integrated circuits today with the resistor implemented using a MOSFET transistor. Such a circuit is often referred to as a sample-and-hold, and will be discussed later in the semester. For the time being, let’s examine a variant of the circuit depicted in Figure 1 that has the capacitor C replaced with a reversed-biased pn-junction. Reverse-Biased pn-Junction In general, devices in integrated circuits exhibit nonlinear behavior. In other words, a simple linear relationship (e.g. IRV⋅= or VCQ⋅=) can not be used to relate the electrical quantities of interest characterizing a particular device element. As an example, Figure 3 portrays the charge stored in the depletion region of a pn-junction as a function of the corresponding voltage applied across its terminals. Note that Figure 3 only considers the pn-junction under reverse bias conditions (i.e. 0<DV). It is evident from the aforementioned figure that a simple linear relationship does not exist between the stored charge and externally applied voltage. While this is true in general, a simple linear tinVC R inV outVAFigure 1. RC Circuit Figure 2. Step inputEE 105 --- Spring 2005 --- Discussion Notes (written by Amin) Page 2 relationship can be assumed if one is only interested in a very small region of the plot. Under such conditions, one is able to apply differentials from calculus to deduce a linear relationship between charge stored and voltage. More specifically, for a very small region of the plot depicted in Figure 3, the following relationship holds true: dregionDregionjvdVdQVdVdQqQ ⋅=∆⋅==∆ || (2) The proportionality constant in (2) (i.e. regiondVdQ| ) has units of capacitance, and can be viewed as the capacitance presented by a reverse-biased pn-junction in light of small voltage changes. Hence, (2) can be re-expressed as follows: djDjjvCVCqQ ⋅=∆⋅==∆ (3) where, regionjdVdQC|=. As presented in lecture, BDjjVCCφ−=10 (4) Figure 3. Depletion charge versus applied voltage -2.5E-15-2E-15-1.5E-15-1E-15-5E-160-4 -3 -2 -1 0applied voltage, Vddepletion charge, QEE 105 --- Spring 2005 --- Discussion Notes (written by Amin) Page 3 Two important things need to be emphasized from the above discussion: 1. The relationship depicted in (3) is only valid for small voltage changes across the reverse-biased pn-junction. While at first this may seem as a significant limitation, one needs to remember that we are primary interested in the design of amplifiers here. The reason why an amplifier is desired is because the signals under study are small and require amplification. In other words, as amplifier designers, we deal with small signals by definition (otherwise, why would we require an amplifier?). As a result, one has a justifiable reason in examining the performance of a device in response to small voltage (or current) changes. 2. The capacitance value of a reverse-biased pn-junction varies for different small region segments. For example, the derivative of the plot in Figure 3 to the far right is different in value to that present to the far left. This is indicated in (2) and (3) by mentioning that the derivative needs to be evaluated at the small region of interest. While at first this may seem as a nuisance, it is a powerful means of control to the circuit designer. Recall that the inputs being received by our amplifier from the outside world are small; hence, there is no way that one can operate in a small region far to the left in Figure 3 without any intervention. The circuit designer can get around this fact by electrically selecting the small region that the device operates within by somehow placing a DC voltage across the pn-junction. By doing so, one is essentially placing the pn-junction at a particular reverse-bias voltage; thereby, electrically setting its small-signal capacitance. This process of placing an electrical device in a certain region of operation through the application of a DC voltage is referred to as biasing. Essentially, one biases the device in a certain region such that the small-signal quantities of interest (here capacitance) end up being a desired value. In general, all of the small-signal parameters of an electrical device are controllable by the DC voltages across and DC currents flowing through that particular device. This is true for the reverse-biased pn-junction, for instance, as indicated by (4) where the small-signal capacitance varies with DV. Small-signal Model of Reverse-Biased pn-Junction As mentioned above, an amplifier designer inevitably deals with small signals. In order to analyze and design amplifiers then, one needs to have models that describe how the electrical devices forming the amplifier circuit operate in light of small signal changes. The process of developing for a device an equivalent circuit model that is valid for small signal changes is referred to as the development of a small-signal model for that particular device. Let’s next try to qualitatively develop the small-signal model for a pn-junction that is reverse biased. Recall from EE 40 that a reverse-biased pn-junction has negligible current flowing through it. This may lead one to think that a basic open circuit should be used to model a pn-junction that is reverse-biased. Although a reverse-biased pn-junction does not conduct current, it does store charge in its corresponding depletion layer. Furthermore,EE 105 --- Spring 2005 --- Discussion Notes (written by Amin) Page 4 this depletion layer alternately contracts and expands as the voltage across the pn-junction reduces or increases. In other words, the charge in the depletion layer changes in response to the voltage across the pn-junction. Such a phenomenon can best be characterized by using a capacitor rather than an open circuit in modeling the reverse-biased pn-junction. The value of this capacitance is given by (4) assuming that the voltage variations are small. From the point of the view of a
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