Diodes-2 Engineering 43 Bruce Mayer, PE Registered Electrical & Mechanical Engineer

Engineering 43

Diodes-2

Bruce Mayer, PE

Registered Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Engineering-43: Engineering Circuit Analysis

1

Bruce Mayer, PE

BMayer@ChabotCollege.edu • ENGR-43_Lec-10b_Diode-2_SmallSignalAnalysis.pptx

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


2

Bruce Mayer, PE


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


3

Bruce Mayer, PE


Diode Models

 LoadLine Analysis works well when the ckt connected to a

SINGLE Diode can be “Thevenized”


4

 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

Bruce Mayer, PE


Diode Models

 Consider an Electrical Diode →

 We can MODEL the V-I

Behavior of this Device in

Several ways

I

V

REAL

Behavior

IDEAL

Model


5

OFFSET

Model

LINEAR

Model

Bruce Mayer, PE


Ideal Model (Ideal Rectifier)

Diode OFF

 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


6

Bruce Mayer, PE


Ideal Model (Ideal Rectifier)

Diode OFF

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


7

Bruce Mayer, PE


Example



Ideal Diode

 Find For Ckt Below find: I & d 1

• Use the

Ideal

Diode

Model

V o

I d 2





I d 1

A


8

Bruce Mayer, PE


Example



Ideal Diode

A



I d 1

I d 2



 Assume BOTH

Diodes are ON or

Conducting


9

 In this Case

V

D1

= V

D2

= 0

 Thus D2 Anode is connected to GND

 Then Find by Ohm

I d 2





10



0

10 k





V



1 mA

 Next use KCL at

I d 1

Node-A (in = out)



I d 2





0



  

9 .

9 k



Bruce Mayer, PE

V


Example



Ideal Diode

A



I d 1

I d 2



I

 Using I

D2 d 1





0



 

9 .

9 k



= 1 mA



V



1 mA


10

 Thus I d 1

 

0 .

0101 mA

 Now must Check that both Diodes are indeed conducting

 From the analysis

I d 1

 

10 µA I d 2

 

1 mA

 Thus the current thru both Diodes is positive which is consistent with the assumption 

Bruce Mayer, PE


Example



Ideal Diode

A



I d 1

I d 2



11

 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

V

D1

= 0 and

= 0 (by Short

Assumption) Find

Vo = GND+V

D2

+V

D1

= GND + 0 + 0 = 0

 Thus the Answer

I d 1

 

10 µA V o

Bruce Mayer, PE




0 V

Example



Ideal Diode

 Find For Ckt Below find: I & d 1

• Use the

Ideal

Diode

Model

V o



I d 1

I d 2



• Note the different values on

R1 & R2

– Swapped

B


12

Bruce Mayer, PE


Example



Ideal Diode

B



I d 1

I d 2



 Again Assume

BOTH Diodes are

ON, or Conducting


13

 As Before

V

D1

= V

D2

= 0

 Again V

B shorted to

GND thru D1

 Then Find by Ohm

I d 2





10



0



V

9 .

9 k





1 .

01 mA

 Now use KCL at

Node-B (in = out)

I d 1



I d 2





0



 

1 0 k



Bruce Mayer, PE



V


Example



Ideal Diode

B



I d 1

I d 2



I d 1

 Using I

D2





0



 

1 0 k



= 1.01 mA



V



1 .

01 mA


14

 Thus I d 1

 

0 .

01 mA

 Now must Check that both Diodes are indeed conducting

 From the analysis

I d 1

 

10 µA I d 2

 

1 .

01 mA

 We find and

INCONSISTENCY and our Assumption is WRONG 

Bruce Mayer, PE


Example



Ideal Diode

B



I d 1

I d 2



 In this Case D1 is an OPEN → I

D1

=0

 Current I

D2 must flow thru BOTH

Resistors

I

 Then Find by Ohm d 2





10



10





 

9 .

9





V k





1 .

005 mA  Must Iterate

 Assume

• D1 → OFF

D2 → ON


15

Bruce Mayer, PE


Example



Ideal Diode

B



I d 1

I d 2



 Must Check that D1 is

REVERSE Biased as it is assumed OFF

 By KVL & Ohm

V

B

 

10 V



10 kΩ



1 .

005 mA

V

B

 

0 .

05 V

 

50 mV

 Thus D1 is INDEED

Reverse-Biased,

Thus the Ckt operation is

Consistent with our

Assumption 


16

Bruce Mayer, PE


Example



Ideal Diode

B



I d 1

I d 2



 Calculate Vo by noting that:

 D2 is ON → V

D2

D1 is OFF →

Current can only flow thru D2

= 0

 In this case Vo = V

B

 By the Previous

Calculation, Find

I d 1



0 A V o

 

50 mV


17

Bruce Mayer, PE


Offset & Linear Models

 The Offset Model  The Linear Model

 Better than Ideal, but no account of

Forward-Slope


18

 The model eqn: v

D



R b i

D

 Yet more



0 .

6 V accurate, but also does not account for

Rev-Bias Brk-Down

Bruce Mayer, PE


Point Slope Line Eqn

 When constructing multipiece-wise linear models, the

Point-Slope

Equation is extremely Useful

 y

 y

1



 m

 x

 x

1



• Where m





 y x

 y x

2

2

– (x

1

, y

1

) & (x

2



 y x

1

1

, y

2

) are

KNOWN Points


19

 Example: Find Eqn for line-segment:

18

(3,17)

16

14

12

10

8

6

(19,5)

4

2 4 m



16 6

 y

 x

8



5

10 12

 x

17

19



3

14

 

12

16

18 m

 

3

4

Bruce Mayer, PE


20

Point Slope Line Eqn

 Using the 2 nd Point

18

(3,17)

16

14

12

 Multiply by m, move

−5 to other side of =

 y



5



 

3

4

 x



19



10

8 y



5

 

3

4 x



57

4

20

6

(19,5)

4

2 4 6 8 10 12 14 16 18 x

 y



5



 

3  x



19



4

 Can easily convert to y = mx+b

20

Engineering-43: Engineering Circuit Analysis y

 

3

4 x



57

4



5 y

 

3

4 x



57

4



20

4 y

 

3 x



77

4


Slopes on vi Curve

 With Reference to the Point-Slope eqn v takes over for x , and

 If the curve is

NONlinear then the local conductance is the first Derivative i takes over for y m vi , Op

 di dv

Op



Amps

Volt



Siemens

 The Slope on a vi

Curve is a m vi conductance





I



V



G



Amps

Volt



S iemens m vi


21 m vi , Op

 di dv

Op

 g

 Recall the Op-Pt is also the Q-Pt m vi , Op

 m vi , Q

 di

 dv

Q

Bruce Mayer, PE g



18

Linear VI Curve


22

 Finally recall that conductance & resistance are

Inverses m vi



G



1

 Example: Find the

R

RESISTANCE of the device associated with the VI curve that follows

16

14

12

10

8

6

4

2 m vi

4 6





I



V

8



10 12 14 16



V (volts)

17



5



Amps



19



3



Volts

18



20 m vi



12 A

16 V



G



3

Siemens

4

Bruce Mayer, PE



18

Linear VI Curve

 Since R = 1/G Find the Device

R

Resistance as





V



I







19

17





3

5





Volts

Amps

16

14

12

10

1 m vi



R



1.333



8



4

3

Ohms

6 m vi



G



1

R



0.75

S

23



R



16 V

12 A

For a NONlinear vi

4

2 4 6 8 10 12 14 16

V (volts)

 The General Reln r

Op



1 g

 di

1 or alternativ ely dv

Op curve the local slope then: r = 1/g

Engineering-43: Engineering Circuit Analysis r

Op

 dv di

Op

 r

Q

 dv di

Q

Bruce Mayer, PE


18 20

Example PieceWise Linear Model

 Construct a

PieceWise

Linear

Model for the Zener vi curve shown at

Right


24

Bruce Mayer, PE


PieceWise Linear Zener

 Us Pt-Slp eqn with

(0.6V,0mA) for Pt-1 i

D

OR



0



100 mS

 v

D

: i

D



100 mS



 v

D

0 .

6 V





60 mA

 Segment- B is easy m

A



 m for Segment A

R

1

A





I



V







100

1.6





0



0.6

mA



V m

A



100 mS



1

10




25 i

D



0

Bruce Mayer, PE


PieceWise Linear Zener

 Us Pt-Slp eqn with

(−6V,0mA) for Pt-1 i

D



0



OR : i

D

83 .

3 mS

 v

D



83 .

3 mS



 v

D





6 V

 



500 mA m

C



 m for Segment C

1

R

C





I



V







0





6









100





7 .

2

 mA

 

V m

C



100 mA

1.2V



83 .

33 mS



1

12



 Thus the PieceWise

Model for the Zener i

D







100 mS

8 3.3mS



 v

D

0 v

D



60 mA



500 mA if if if

 v

D



0 .

6 V

6 V v

D

 v

D





0



6 V

.

6 V


26

Bruce Mayer, PE


Example PieceWise Linear Model

 Alternatively in terms of i

D





 v

D

10



Resistances



60 mA if

0 if v

D



6 V







 v

12

D





500 mA if v

D v

D



0 .

6 V





6 V

0 .

6 V

 ADVICE: remember the

Pt-Slope

27

Line-Eqn

Engineering-43: Engineering Circuit Analysis Bruce Mayer, PE


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


28

Power Input Load Voltage

Bruce Mayer, PE


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


29

Bruce Mayer, PE


Smoothed HalfWave Rectifier

 Adding a Cap to the

Circuit creates a

Smoothing effect

 This produces v

L

(t) and i

L

(t) curves

 In this case the

Diode Conducts

ONLY when v s

>v

C and v

C

=v

L


30 i

C i

D



C



 dv

L

 i

C

 i

L dt



 Note that i

L

(t) is approx. constant

Bruce Mayer, PE



 The change in

Voltage across the

Cap is called “Ripple”

Ripple

 Often times the load has a Ripple “Limit” from which we determine Cap size


31

 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

Charge



Current



Time

Or, symbolical ly : Q



I



T

Bruce Mayer, PE



 Also from Cap

Physics

(chp3)



Q cap



C

 

V cap

 Note that both these equations are

Approximate, but they are still useful

32

 In the Smoother Ckt the Cap charges during the “Ripple” portion of the curve

Engineering-43: Engineering Circuit Analysis for initial Ckt Design

 Equating the Charge

& Discharge “Q’s find



Q cap



Charge Discharge

C



V r



I

L



T

 Solving the equations for the

Cap Value needed for a given V r



T

C



I

L

V r

Bruce Mayer, PE



 Find the Approximate Average Load

Voltage

V

L,lo

V

L,hi

V

L , avg

33

V

L , avg



V m



V

L , hi



V

L , lo

2




V m



V r

2

Bruce Mayer, PE


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

 

RC T



1 mS


34

Bruce Mayer, PE


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


35

 The Diodes are

• Face-to-Face (right)

• Butt-to-Butt (left)

 This rectified output has

NO Gaps

Bruce Mayer, PE


Full Wave Rectifier Operation

D1 Supplies V to Load

D4 Supplies V to Load


36

Bruce Mayer, PE


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:

C



I

L



T

2 V r


37

Bruce Mayer, PE


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 Non Linear devices appear linear .

This allows us to construct a so-called

38 small signal Linear Model



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 dy

Calculus dx



 y

 x

 This approximation become more accrate as ∆y & ∆x become smaller


39

Bruce Mayer, PE


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 di

D dv

D

  i

D

 v

D

 units of A/V or Siemens



 Next Take derivative about the Q-Pt


40

Bruce Mayer, PE



 About Q-Pt di

D dv

D

Q



 i

D

 v

D near Q

 Now if we have a math model for the vi curve, and we inject ON TOP of

V

DQ

∆v

D a small signal, find


41

 i

D







 di

D dv

D

Q







 

D

 g d

  v

D

 The derivative is the diode small-signal

Conductance at Q

Bruce Mayer, PE



 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, smallsignal Conductance and Resistance r g d d



 di

D dv

D

Q

1 g d







 di

D dv

D

Q









1

 dv

D di

D

Q


42

Bruce Mayer, PE


Small Signal Analysis r d



 Note Units for r d





 di

D dv

D

Q









1







Amp

Volt







1



Volt

Amp

 Recall the

  approximation for i

 i

D







 di

D dv

D

Q







 

D



 v r d

D

D

 Change Notation for

Small

Signal conditions

 i

D

 v

D



 i d v d


43

 Find r d for a

“Shockley” Diode in majority FWD-Bias

 Recall Shockley Eqn i

D



I s e



 v

D nV

T







1

 Then the Largesignal Operating

I

DQ

Point at v

D



I s e





V

DQ nV

T







1



= V

DQ

I s e





V

DQ nV

T





Bruce Mayer, PE



 Taking the derivative of the

Shockely Eqn di

D dv

D

Q



I

S e



 v

D nV

T



 1 nV

T



1 nV

T







I

S e



 v

D nV

T

 Recall from last sld











I s e





V

DQ nV

T







I

DQ

 Sub this Reln into the Derivative Eqn di

D dv

D

Q



1 nV

T

I

 

DQ



I

DQ nV

T

 Recall r d







 di

D dv

D

Q









1

 Subbing for di

D

/dv

D r d







 di

D dv

D

Q









1







I

DQ nV

T







1

 nV

T

I

DQ


44

Bruce Mayer, PE


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 steadystate. To emphasize this we can write v

D

(t) and i

D

(t)


45

Bruce Mayer, PE


Notation: Large, Small, Total

 v d and i d are the SMALL, AC quantities

• These values are not necessarily steadystate. To emphasize this we can write v d

(t) and i d

(t)

 An

Example for

Diode

Current notation


46

Bruce Mayer, PE


Effect of Q-Pt Location

 From i d

Analysis

 v d r d i d 2 , pp i d 1 pp and r d

 nV

T

I

DQ v d 1 pp v d 2 pp


47

Bruce Mayer, PE


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

48

SHORT circuit for ac signals



Setting Q, Injecting v

 Consider this ckt with AC & DC V-srcs

Sets Q

Sets v d


49

Bruce Mayer, PE


Large and Small Signal Ckts

 Recall from Chps 3 and 5 for Caps:

• OPENS to DC

• SHORTS to fast AC

Z

C



1

 j



C



 Thus if C

1 it COUPLES v in

(t) with the rest of the ckt is LARGE

 Similarly, Large C

2 couples to the Load


50

 To Find the Q-point

DE couple v in and v o to arrive at the DC circuit

Bruce Mayer, PE


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


51

 Thus the Small Signal ckt for the above

Bruce Mayer, PE


Example: Small Signal Gain

 Find the Small

Signal Amplification

(Gain), A v

, of the previous circuit

 Using the Small

Signal Circuit


52

 Note that R

C

, r d

, and

R

L are in Parallel

1

R p



1

R

C



1

R d



1

R

L

 And v o

(t) appears across this parallel combination

 The equivalent ckt

Bruce Mayer, PE v o



 




Example: Small Signal Gain

 Thus for this Ckt the

Large, Small, and small-Equivalent ckts

1

R p



1

R

C



1

R d



1

R

L v o



 



 Then the

Amplification (Gain) by Voltage Divider

A v

 v v in o



R

R p



R p


53

Bruce Mayer, PE


All Done for Today

Small

Signal

BJT Amp

LARGE

Signal

Model

 Common Collector

Amplifier


54

SMALL

Signal

Model

Bruce Mayer, PE


Engineering 43

Appendix

Bruce Mayer, PE

Registered Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu


55

Bruce Mayer, PE


v o



 

 v o



 




56

Bruce Mayer, PE



 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 r g d d



 di

D dv

D

Q

1 g d







 di

D dv

D

Q









1

 dv

D di

D

Q


57

Bruce Mayer, PE


P10.67

 Graph v o vs. v i for v i

: −5V to +5V

6

-6 -5 -4 -3 -2 -1

0

-1

0

-2

5

4

3

2

1

1

-3

-4

-5

-6 file =XY_Plot_0211.xls

2 3 4 5 6


58

Bruce Mayer, PE



59

Bruce Mayer, PE



60

Bruce Mayer, PE



61

Bruce Mayer, PE



62

Bruce Mayer, PE


Diodes-2 Engineering 43 Bruce Mayer, PE Registered Electrical & Mechanical Engineer

Engineering 43

Diodes-2

All Done for Today

Small

Signal

BJT Amp

Engineering 43

Appendix

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Diodes-2 Engineering 43 Bruce Mayer, PE Registered Electrical &amp; Mechanical Engineer

Engineering 43

Diodes-2

All Done for Today

Small

Signal

BJT Amp

Engineering 43

Appendix

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Diodes-2 Engineering 43 Bruce Mayer, PE Registered Electrical & Mechanical Engineer