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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 1 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Bruce Mayer, PE Registered Electrical & Mechanical Engineer Engineering 43 Diodes-2

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 2 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Learning Goals Understand the Basic Physics of Semiconductor PN Junctions which form most Diode Devices Sketch the IV Characteristics of Typical PN Junction Diodes Use the Graphical LOAD-LINE method to determine the “Operating Point” of Nonlinear (includes Diodes) Circuits

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 3 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Learning Goals Analyze diode-containing Voltage- Regulation Circuits Use various math models for Diode operation to solve for Diode-containing Circuit Voltages and/or Currents Learn The difference between LARGE-signal and SMALL-Signal Circuit Models IDEAL and PieceWise-Linear Models

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 4 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Diode Models LoadLine Analysis works well when the ckt connected to a SINGLE Diode can be “Thevenized” However, for NONLinear ckts, such as those containing multiple diodes, construction of the LOAD-Curve Eqn may be difficult, or even impossible. Many such ckts can be analyzed by Idealizing the diode

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 5 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Diode Models Consider an Electrical Diode → We can MODEL the V-I Behavior of this Device in Several ways V I REAL Behavior IDEAL Model OFFSET Model LINEAR Model

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 6 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Ideal Model (Ideal Rectifier) Analyze Ckts containing Ideal Diodes 1.Assume (or Guess) a “state” for each diode. Ideal Diodes have Two states 1.ON → a SHORT Ckt when Fwd Biased 2.OFF →an OPEN Ckt if Reverse Biased 2.Check the Assumed Opens & Shorts Should have Current thru the SHORTS Should have ∆V across the OPENS Diode ON Diode OFF

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 7 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Ideal Model (Ideal Rectifier) 3.Check to see if guesses for i-flow, ∆V, and BIAS-State are consistent with the Ideal-Diode Model 4.If i-flow, ∆V, and bias-V are consistent with the ideal model, then We’re DONE. If we arrive at even a SINGLE Inconsistency, then START OVER at step-1 Diode ON Diode OFF

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 8 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ideal Diode Find For Ckt Below find: Use the Ideal Diode Model

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 9 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ideal Diode Assume BOTH Diodes are ON or Conducting In this Case V D1 = V D2 = 0 Thus D2 Anode is connected to GND Then Find by Ohm Next use KCL at Node-A (in = out)

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 10 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ideal Diode Using I D2 = 1 mA Thus Now must Check that both Diodes are indeed conducting From the analysis Thus the current thru both Diodes is positive which is consistent with the assumption

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 11 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ideal Diode Since both Diodes conduct the Top of Vo is connected to GND thru D2 & D1 Another way to think about this is that since V D2 = 0 and V D1 = 0 (by Short Assumption) Find Vo = GND+V D2 +V D1 = GND = 0 Thus the Answer

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 12 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ideal Diode Find For Ckt Below find: Use the Ideal Diode Model Note the different values on R1 & R2 –Swapped

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 13 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ideal Diode Again Assume BOTH Diodes are ON, or Conducting As Before V D1 = V D2 = 0 Again V B shorted to GND thru D1 Then Find by Ohm Now use KCL at Node-B (in = out)

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 14 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ideal Diode Using I D2 = 1.01 mA Thus Now must Check that both Diodes are indeed conducting From the analysis We find and INCONSISTENCY and our Assumption is WRONG

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 15 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ideal Diode Must Iterate Assume D1 → OFF D2 → ON In this Case D1 is an OPEN → I D1 =0 Current I D2 must flow thru BOTH Resistors Then Find by Ohm

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 16 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ideal Diode Must Check that D1 is REVERSE Biased as it is assumed OFF By KVL & Ohm Thus D1 is INDEED Reverse-Biased, Thus the Ckt operation is Consistent with our Assumption

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 17 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ideal Diode Calculate Vo by noting that: D2 is ON → V D2 = 0 D1 is OFF → Current can only flow thru D2 In this case Vo = V B By the Previous Calculation, Find

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 18 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Offset & Linear Models The Offset Model Better than Ideal, but no account of Forward-Slope The Linear Model The model eqn: Yet more accurate, but also does not account for Rev-Bias Brk-Down

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 19 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Point Slope Line Eqn When constructing multipiece-wise linear models, the Point-Slope Equation is extremely Useful Where –(x 1, y 1 ) & (x 2, y 2 ) are KNOWN Points Example: Find Eqn for line-segment: (3,17) (19,5)

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 20 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Point Slope Line Eqn Using the 2 nd Point Can easily convert to y = mx+b Multiply by m, move −5 to other side of = (3,17) (19,5)

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 21 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Slopes on vi Curve With Reference to the Point-Slope eqn v takes over for x, and i takes over for y The Slope on a vi Curve is a conductance If the curve is NONlinear then the local conductance is the first Derivative Recall the Op-Pt is also the Q-Pt

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 22 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Slopes on vi Curve Finally recall that conductance & resistance are Inverses Example: Find the RESISTANCE of the device associated with the VI curve that follows

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 23 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Slopes on vi Curve Since R = 1/G Find the Device Resistance as For a NONlinear vi curve the local slope then: r = 1/g The General Reln

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 24 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example PieceWise Linear Model Construct a PieceWise Linear Model for the Zener vi curve shown at Right

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 25 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis PieceWise Linear Zener m for Segment A Us Pt-Slp eqn with (0.6V,0mA) for Pt-1 Segment- B is easy

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 26 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis PieceWise Linear Zener m for Segment C Us Pt-Slp eqn with (−6V,0mA) for Pt-1 Thus the PieceWise Model for the Zener

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 27 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example PieceWise Linear Model Alternatively in terms of Resistances ADVICE: remember the Pt-Slope Line-Eqn

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 28 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Half-Wave Rectifier Ckt Consider an Sinusoidal V-Source, such as an AC socket in your house, supplying power to a Load thru a Diode Power InputLoad Voltage

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 29 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis HalfWave Rectifier Note that the Doide is FWD-Biased during only the POSITIVE half-cycle of the Source Using this simple ckt provides to the load ONLY positive-V; a good thing sometimes However, the positive voltage comes in nasty PULSES which are not well tolerated by positive-V needing loads

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 30 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Smoothed HalfWave Rectifier Adding a Cap to the Circuit creates a Smoothing effect In this case the Diode Conducts ONLY when v s >v C and v C =v L This produces v L (t) and i L (t) curves Note that i L (t) is approx. constant

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 31 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Smoothed HalfWave Rectifier The change in Voltage across the Cap is called “Ripple” Often times the load has a Ripple “Limit” from which we determine Cap size From the i L (t) curve on the previous slide note: Cap Discharges for Almost the ENTIRE Cycle time, T (diode Off) The Load Current is approx. constant, I L Recall from EARLY in the Class Ripple

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 32 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Smoothed HalfWave Rectifier Also from Cap Physics (chp3) In the Smoother Ckt the Cap charges during the “Ripple” portion of the curve Equating the Charge & Discharge “Q’s find Note that both these equations are Approximate, but they are still useful for initial Ckt Design Solving the equations for the Cap Value needed for a given V r ChargeDischarge

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 33 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Smoothed HalfWave Rectifier Find the Approximate Average Load Voltage V L,hi V L,lo

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 34 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Capacitor-Size Effect Any load will discharge the capacitor. In this case, the output will depend on how the RC time constant compares with the period of the input signal. The plots at right consider the various cases for the simple circuit above with a 1kHz, 5V sinusoidal input

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 35 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Full Wave Rectifier The half-wave ckt will take an AC- Voltage and convert it to DC, but the rectified signal has gaps in it. The gaps can be eliminated thru the use of a Full-Wave rectifier ckt The Diodes are Face-to-Face (right) Butt-to-Butt (left) This rectified output has NO Gaps

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 36 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Full Wave Rectifier Operation D1 Supplies V to Load D4 Supplies V to Load

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 37 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Full Wave Rectifier Smoothing The Ripple on the FULL wave Ckt is about 50% of that for the half-wave ckt Since the Cap DIScharges only a half-period compared to the half-wave ckt, the size of the “smoothing” cap is then also halved:

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 38 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Small Signal Models Often we use NonLinear Circuits to Amplify, or otherwise modify, non-steady Signals such as ac-sinusoids that are small compared to the DC Operating Point, or Q-Point of the Circuit. Over a small v or i range even NonLinear devices appear linear. This allows us to construct a so-called small signal Linear Model

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 39 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Small Signal Analysis Small signal Analysis is usually done in Two Parts: 1.Large-Signal DC Operating Point (Q-Pt) 2.Linearize about the Q-Pt using calculus Recall from Calculus This approximation become more accrate as ∆y & ∆x become smaller

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 40 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Small Signal Analsyis Now let y→i D, and x→ v D Use a DC power Supply to set the operating point on the diode curve as shown at right This could be done using LoadLine methods From Calculus Next Take derivative about the Q-Pt

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 41 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Small Signal Analysis About Q-Pt Now if we have a math model for the vi curve, and we inject ON TOP of V DQ a small signal, ∆v D find The derivative is the diode small-signal Conductance at Q

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 42 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Small Signal Analysis In the large signal Case: R = 1/G By analogy In the small signal case: r = 1/g Also since small signal analysis is associated with small amounts that change with time… Define the Diode’s DYNAMIC, small- signal Conductance and Resistance

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 43 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Small Signal Analysis Note Units for r d Recall the approximation for i D Change Notation for Small Signal conditions Find r d for a “Shockley” Diode in majority FWD-Bias Recall Shockley Eqn Then the Large- signal Operating Point at v D = V DQ

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 44 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Small Signal Analysis Taking the derivative of the Shockely Eqn Recall from last sld Sub this Reln into the Derivative Eqn Recall Subbing for di D /dv D

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 45 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Notation: Large, Small, Total V DQ and I DQ are the LARGE Signal operating point (Q-Pt) DC quantities These are STEADY-STATE values v D and i D are the TOTAL and INSTANTANEQOUS quantities These values are not necessarily steady- state. To emphasize this we can write v D (t) and i D (t)

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 46 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Notation: Large, Small, Total v d and i d are the SMALL, AC quantities These values are not necessarily steady- state. To emphasize this we can write v d (t) and i d (t) An Example for Diode Current notation

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 47 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Effect of Q-Pt Location From Analysis

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 48 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis DC Srcs SHORTS in Small-Signal In the small-signal equivalent circuit DC voltage-sources are represented by SHORT CIRUITS; since their voltage is CONSTANT, they exhibit ZERO INCREMENTAL, or SIGNAL, voltage Alternative Statement: Since a DC Voltage source has an ac component of current, but NO ac VOLTAGE, the DC Voltage Source is equivalent to a SHORT circuit for ac signals

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 49 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Setting Q, Injecting v Consider this ckt with AC & DC V-srcs Sets Q Sets v d

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 50 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Large and Small Signal Ckts Recall from Chps 3 and 5 for Caps: OPENS to DC SHORTS to fast AC Thus if C 1 is LARGE it COUPLES v in (t) with the rest of the ckt Similarly, Large C 2 couples to the Load To Find the Q-point DEcouple v in and v o to arrive at the DC circuit

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 51 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Large and Small Signal Ckts Finding the Large signal Model was easy; the Caps acts as an OPENS The Small Signal Ckt needs more work Any DC V-Supply is a SHORT to GND The Diode is replaced by r d (or g d ) The Caps are Shorts Thus the Small Signal ckt for the above

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 52 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example: Small Signal Gain Find the Small Signal Amplification (Gain), A v, of the previous circuit Using the Small Signal Circuit Note that R C, r d, and R L are in Parallel And v o (t) appears across this parallel combination The equivalent ckt

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 53 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example: Small Signal Gain Thus for this Ckt the Large, Small, and small-Equivalent ckts Then the Amplification (Gain) by Voltage Divider

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 54 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis All Done for Today Small Signal BJT Amp Common Collector Amplifier LARGE Signal Model SMALL Signal Model

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 55 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Bruce Mayer, PE Registered Electrical & Mechanical Engineer Engineering 43 Appendix

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 56 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 57 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Small Signal Analysis In the large signal Case: R = 1/G By analogy In the small signal case: r = 1/g Also since small signal analysis is associated with small amounts that change with time… Define the Diode’s DYNAMIC Conductance and Resistance

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 58 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis P10.67 Graph v o vs. v i for v i : −5V to +5V

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 59 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 60 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 61 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

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ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx 62 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

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