EE 105 Spring 2005 Discussion Notes written by Amin 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 R Vin Vin C Vout A t Figure 1 RC Circuit Figure 2 Step input resulting output voltage is given by the following t Vout A 1 e RC 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 sampleand 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 V R I or Q C V 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 V D 0 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 Page 1 EE 105 Spring 2005 Discussion Notes written by Amin 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 applied voltage Vd 0 3 2 1 0 5E 16 1E 15 1 5E 15 2E 15 depletion charge Q 4 2 5E 15 Figure 3 Depletion charge versus applied voltage 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 Q q j dQ dQ region VD region v d dV dV 2 dQ region has units of capacitance and can be dV viewed as the capacitance presented by a reverse biased pn junction in light of small voltage changes The proportionality constant in 2 i e Hence 2 can be re expressed as follows Q q j C j VD C j v d where C j 3 dQ region dV As presented in lecture C j0 Cj V 1 D 4 B Page 2 EE 105 Spring 2005 Discussion Notes written by Amin 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 V D 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 pnjunction 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 Page 3 EE 105 Spring 2005 Discussion Notes written by Amin this depletion layer alternately contracts and expands as the voltage across the pnjunction 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 reversebiased 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 small voltage signal it is as if the pn junction is alternately charging and discharging as the depletion layer expands and contracts In other words the small signal sees the pn junction acting as a capacitor and can be modeled as such This is pictorially depicted in Figure 4 n Cj p Figure 4 Small signal
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