3.3. Modeling the Diode Forward Characteristic

Transcription

1 3.3. Modeling the iode Forward Characteristic define a robust set of diode models iscuss simplified diode models better suited for use in circuit analysis and design of diode circuits: Exponential model Constant voltage-drop model Ideal diode model Small-signal (linearization) model 1

2 The Exponential Model Exponential diode model Most accurate Most difficult to employ in circuit analysis ue to nonlinear nature voltage across diode current through diode / V (eq4.6) 3.6) I I e T V I S V 2

3 The Exponential Model Q: How does one solve for I in circuit to right? V = 5V R = 1kOhm I = 0.7V A: Two methods exist graphical method iterative method Figure 3.10: A simple circuit used to illustrate the analysis of circuits in which the diode is forward conducting (eq4.7) (eq 3.7) I V V R 3

4 Graphical Analysis Using Exponential Model step #1: Plot the relationships of (3.6) and (3.7) on single graph step #2: Find intersection of the two load line and diode characteristic intersect at operating point voltage across diode current through diode / V (eq4.6) 3.6) I I e T V I (eq4.7) 3.7) I S V V V R Figure 3.11: Graphical analysis of the circuit in Fig using the exponential diode model. 4

5 Graphical Analysis Using Exponential Model Pro s Intuitive b/c of visual nature Con s Poor Precision Not Practical for Complex Analyses multiple lines required Figure 3.11: Graphical analysis of the circuit in Fig using the exponential diode model. 5

6 Iterative Analysis Using Exponential Method Ex 3.4 step #1: Start with initial guess of V. V (0) step #2: Use nodal / mesh analysis to solve I. step #3: Use exponential model to update V. V (1) = f(v (0) ) step #4: Repeat these steps until V (k+1) = V (k). Upon convergence, the new and old values of V will match. 6

7 Iterative Analysis Using Exponential Method Pro s High Precision Con s Not Intuitive Not Practical for Complex Analysis 10+ iterations may be required 7

8 The Need for Rapid Analysis Q: How can one analyze these diode-based circuits more efficiently? A: Find a simpler model. One example is assume that voltage drop across the diode is constant. Further refine or fine-tune later 8

9 The Constant Voltage-rop Model Voltage drop of a forwardconducting diode varies in a relatively narrow range ( V) The constant voltage-drop diode model assumes that the slope of I vs. V is 0.7V Not very different Employed in the initial phases of analysis and design Ex3.4: solution change if CVM is used? A: 4.262mA to 4.3mA Figure 3.12: evelopment of the diode constant-voltage-drop model: (a) the 9

10 Ideal iode Model When involving voltages much greater than the diode voltage drop Very quick analysis for a gross estimate For determine which diodes are on/off in a multidiode circuit The ideal diode model assumes that the slope of I vs. V is 0V Ex3.4: solution change if ideal model is used? A: 4.262mA to 5mA device symbol with two nodes mode #2: reverse bias = open ckt. mode #1: forward bias = short ckt 10

11 Small-Signal Model Operate at a dc biased point on the forward i-v characteristic and a small ac signal superimposed on dc c operating point (I, V ) by other model And then, modeled as variable resistor = inverse of the slope of the tangent to exponential i-v characteristic at the bias point Whose value is defined via linearization of exponential model Around bias point defined by constant voltage drop model V (0) = 0.7V 11

12 Small-Signal Model Q: How is the small-signal diode model defined? step #1: Consider the conceptual circuit of Figure 3.13(a). C voltage (V ) is applied to diode Upon V, arbitrary time-varying signal v d is superimposed 12

13 Small-Signal Model C only upper-case w/ upper-case subscript time-varying only lower-case w/ lower-case subscript total instantaneous lower-case w/ upper-case subscript C + time-varying 13

14 Small-Signal Model step #2: efine C current as in (3.8). step #3: efine total instantaneous voltage (v ) as composed of V and v d step #4: efine total instantaneous current (i ) as function of v (eq4.8) 3.8) (eq4.9) 3.9) (eq4.10) 3.10) I / V v ( t) V v ( t) v V v d S V d ( t) total instantaneous voltage across diode dc component of v ( t) ( t) time varying component of v ( t) i () t I e I e S / V note that this is different from (4.8) T v (3.8) T 14

15 Small-Signal Model step #5: Redefine (3.10) as function of both V and v d step #6: Split this exponential in two step #7: Redefine total instant current in terms of C component (I ) and time-varying voltage (v d ) (eq4.11) 3.11) i () t I e (eq4.11) 3.11) (eq (eq4.12) 3.12) S action: split this exponential using appropriate laws V v / V V / VT v i() t ISe e i () t I e I v d T / V T d / V T 15

16 Small-Signal Model step #8: Apply power series expansion to (3.12). step #9: Because v d /V T << 1, certain terms may be neglected example: (eq a) 2 e x x x x 1 x 2! 3! 4! action: apply power series expansion to (4.12) because vd / VT1, these terms are assumed to be negligible 2 3 v d vd 1 vd 1 i( t) I1 VT VT 2! VT 3! action: eliminate negligible terms v (eq4.14) 3.14) i( t) I1 V d T power series expansion of e vd / VT 16

17 Small-Signal Model small signal approximation Shown to right for exponential diode model total instant current (i ) small-signal current (i d. ) small-signal resistance/ incremental resistance (r d. ) Valid for for v d < 5mV amplitude (not peak to peak) I i () t I v VT d i () t I i i r d d d 1 r d V I T v d i d 17

18 Small-Signal Model Assuming that the signal amplitude is sufficiently small such that the excursion along the i-v curve is limited to a short almost-linear segment r d = 1 i v i =I This method may be used to approximate any function y = f(x) around an operating point (x 0, y 0 ). 1 x y y( t) y x( t) x 0 0 x yy 18

19 3.3.7: Small-Signal Model Q: How is small-signal resistance r d defined? A: From steady-state current (I ) and thermal voltage (V T ) as below. Note this approximation is only valid for smallsignal voltages v d < 5mV. After dc analysis, r d V T Eliminating all dc sources (short-circuiting dc voltage sources and open-circuiting dc current sources) Replacing the diode by its small-signal resistance I 19

20 Example 3.5: Small-Signal Model Consider the circuit shown in Figure 3.14(a) for the case in which R = 10kOhm. The power supply V+ has a dc value of 10V over which is super-imposed a 60Hz sinusoid of 1V peak amplitude (known as the supply ripple) Q: Calculate both amplitude of the sine-wave signal observed across the diode. A: v d. (peak) = 2.68mV Assume diode to have 0.7V drop at 1mA current. 20

21 Figure 3.14: (a) circuit for Example 3.5. (b) circuit for calculating the dc operating point. (c) small-signal equivalent circuit. 21

22 C AC C = + AC Figure 3.14: (a) Circuit for Example 3.5. (b) Circuit for calculating the dc operating point. (c) Small-signal equivalent circuit. 22

23 Small-Signal Model Q: How is the small-signal diode model defined? A: The total instantaneous circuit is divided into steady-state and time varying components, which may be analyzed separately and solved via algebra. In steady-state, diode represented as CVM. In time-varying, diode represented as resistor. 23 Neither of these circuits employ the exponential model simplifying the solving process.

24 When to use these models? exponential model low voltages less complex circuits emphasis on accuracy over practicality constant voltage-drop mode: medium voltages = 0.7V more complex circuits emphasis on practicality over accuracy ideal diode model high voltages >> 0.7V very complex circuits cases where a difference in voltage by 0.7V is negligible small-signal model 24

25 iode Forward rop in Voltage Regulation Voltage regulator Provide a constant dc voltage between its output terminals To remain output as constant as possible in spite of changes in supply and load Q: What characteristic of the diode facilitates voltage regulation? A: The approximately constant voltage drop across it (0.7V). 25

26 Example 3.6: iode-based Voltage Regulator Consider circuit shown in Figure A string of three diodes is used to provide a constant voltage of 2.1V. Q: What is the change in this regulated voltage caused by (a) a +/- 10% change in supply voltage and (b) connection of 1kOhm load resistor. 26 Figure 3.15: Circuit for Example 3.6.