Topic 3 Diodes and Diodes Circuits

Topic 3 Diodes and Diodes Circuits
ECE 271 Electronic Circuits I Topic 3 Diodes and Diodes Circuits NJIT ECE Dr. S. Levkov Chap 3 -1

Chapter Goals Develop electrostatics of the pn junction
Define regions of operation of the diode (forward bias, reverse bias, and reverse breakdown) Explore various diode models including the mathematical model, the ideal diode model, and the constant voltage drop model Apply the various types of models in circuit analysis Explore diode applications Explore different special types of diodes Understand the SPICE representation and model parameters for the diode. Practice simulating diode circuits using SPICE NJIT ECE Dr. S. Levkov Chap 3 -2

Diode Introduction A diode is formed by joining an n-type semiconductor with a p-type semiconductor. A pn junction is the interface between n and p regions. Study of diodes will consist of three sections: 1. PN-junction physics 2. Diode circuits analysis 3. Diode applications Diode symbol NJIT ECE Dr. S. Levkov Chap 3 -3

1. The pn junction A simple (or doped) semiconductor does not in itself possess properties that make it useful for electronic circuits. NJIT ECE Dr. S. Levkov Chap 3 -4

1. The pn junction A simple (or doped) semiconductor does not in itself possess properties that make it useful for electronic circuits. However, when a section of p-type material and a section of n-type material are brought in contact to form a pn junction, a number of interesting properties arise. NJIT ECE Dr. S. Levkov Chap 3 -5

1. The pn junction A simple (or doped) semiconductor does not in itself possess properties that make it useful for electronic circuits. However, when a section of p-type material and a section of n-type material are brought in contact to form a pn junction, a number of interesting properties arise. The major effect happens in the small section around the contact area – depletion region, where the holes and electrons recombine leaving no charge carriers. NJIT ECE Dr. S. Levkov Chap 3 -6

1. The pn junction A simple (or doped) semiconductor does not in itself possess properties that make it useful for electronic circuits. However, when a section of p-type material and a section of n-type material are brought in contact to form a pn junction, a number of interesting properties arise. The major effect happens in the small section around the contact area – depletion region, where the holes and electrons recombine leaving no charge carriers. NJIT ECE Dr. S. Levkov Chap 3 -7

pn Junction Electrostatics
Donor and acceptor concentration on either side of the junction. Concentration gradients give rise to diffusion current - ID NJIT ECE Dr. S. Levkov Chap 3 -8

Space-Charge (Depletion) Region at the pn Junction
If the diffusion process was to continue unabated, this would result eventually in the uniform concentration of e. & h. through the entire semiconductor and pn junction would disappear. This does not happen because the competing process appears that oppose the diffusion current. As holes diffuse from p-region they leave (-) ionized atoms of acceptors. As electrons diffuse from n-region they leave (+)ionized atoms of donors This creates the space-charge or depletion region with localized (-) and (+) charge  electric field E(x) drift current that oppose the diffusion current. NJIT ECE Dr. S. Levkov Chap 3 -9

Drift Currents Gauss’ Law: an electric field due to the charge distribution is Assuming constant permittivity* and one dimension Resulting electric field gives rise to a drift current IS of minority carriers. With no external circuit connections, drift and diffusion currents cancel: IS = ID . (There is no actual current, since this would imply power dissipation, rather the electric field cancels the diffusion current ‘tendency.’) The equilibrium condition is maintained by the barrier voltage V0, or , in another words, junction potential *In electromagnetism, permittivity is the measure of how much resistance is encountered when forming an electric field in a medium NJIT ECE Dr. S. Levkov Chap 3 -10

Potential Across the Junction
Charge Density Electric Field Potential Using (charge neutrality), it can be shown that NJIT ECE Dr. S. Levkov Chap 3 -11

Width of Depletion Region
From the previous expressions, the expression for the width of the space-charge region, or depletion region can be obtained. It is called the depletion region since the excess holes and electrons are depleted from the dopant atoms on either side of the junction. NJIT ECE Dr. S. Levkov Chap 3 -12

Example 1. Width of Depletion Region
Problem: Find built-in potential and depletion-region width for given diode Given data:On p-type side: NA = 1017/cm3 on n-type side: ND = 1020/cm3 Assumptions: Room-temperature operation with VT = V Analysis: with  NJIT ECE Dr. S. Levkov Chap 3 -13

Example 2. Diode Electric Field and p,n depletion width
Problem: Find the size of the individual depletion layers on either side of a pn junction and the max. value of the electric field for a given diode. Given data: On the p-type side: NA = 1017/cm3 on the n-type side: ND = 1020/cm3 from earlier example, Assumptions: Room-temperature operation From charge neutrality we have , and , and Thus Hence NJIT ECE Dr. S. Levkov Chap

Example 2. Diode Electric Field and p,n depletion width (cont.)
For the electric field, we have that from and the diagram on slide 3.8 it follows that is equal to the area under triangle. Since is the base of the triangle, it follows NJIT ECE Dr. S. Levkov Chap

The equilibrium in pn junction - summary
Result of diffusion created by the difference of concentration of holes and electrons in different parts of pn junction: holes will diffuse from left to the right and electrons from right to the left. This process results in diffusion current that creates the difference of potential at the depletion region. This is the majority carrier diffusion current. NJIT ECE Dr. S. Levkov Chap 3 -16

Result of diffusion created by the difference of concentration of holes and electrons in different parts of pn junction: holes will diffuse from left to the right and electrons from right to the left. This process results in diffusion current that creates the difference of potential at the depletion region. This is the majority carrier diffusion current. However, this difference of potential acts like a battery making an electric field that creates drift current. This is the minority carrier current also known as reverse saturation current NJIT ECE Dr. S. Levkov Chap 3 -17

Result of diffusion created by the difference of concentration of holes and electrons in different parts of pn junction: holes will diffuse from left to the right and electrons from right to the left. This process results in diffusion current that creates the difference of potential at the depletion region. This is the majority carrier diffusion current. However, this difference of potential acts like a battery making an electric field that creates drift current. This is the minority carrier current also known as reverse saturation current Under equilibrium conditions, the diffusion current is exactly balanced by the drift current so that the net current across the p–n junction is zero, NJIT ECE Dr. S. Levkov Chap 3 -18

Result of diffusion created by the difference of concentration of holes and electrons in different parts of pn junction: holes will diffuse from left to the right and electrons from right to the left. This process results in diffusion current that creates the difference of potential at the depletion region. This is the majority carrier diffusion current. However, this difference of potential acts like a battery making an electric field that creates drift current. This is the minority carrier current also known as reverse saturation current Under equilibrium conditions, the diffusion current is exactly balanced by the drift current so that the net current across the p–n junction is zero, When no external current or voltage is applied to the p–n junction, the potential gradient (barrier voltage ) forms an energy barrier that prevents further diffusion of charge carriers across the junction. NJIT ECE Dr. S. Levkov Chap 3 -19

The pn junction under the applied voltage condition
When an external battery is connected across a pn junction, the amount of current flow is determined by the polarity of the applied voltage and its effect on the space–charge region. When the battery polarity is applied in the same direction as barrier voltage we say that a pn junction is in reverse bias. When the battery polarity is applied in the opposite direction to barrier voltage we say that a pn junction is in forward bias. NJIT ECE Dr. S. Levkov Chap 3 -20

The pn junction in reverse bias
- + + + + - - - The free electrons in the n–type material are attracted toward the positive terminal of the battery and away from the junction NJIT ECE Dr. S. Levkov Chap 3 -21

- + + + + - - - - The free electrons in the n–type material are attracted toward the positive terminal of the battery and away from the junction, creating new holes NJIT ECE Dr. S. Levkov Chap 3 -22

- + + + + - - - The free electrons in the n–type material are attracted toward the positive terminal of the battery and away from the junction, creating new holes and the positive charge on the right of depletion region becomes even more positive. NJIT ECE Dr. S. Levkov Chap 3 -23

- + + + + - - - + The free electrons in the n–type material are attracted toward the positive terminal of the battery and away from the junction, creating new holes and the positive charge on the right of depletion region becomes even more positive. The holes from the p–type material are attracted toward the negative terminal of the battery and away from the junction, NJIT ECE Dr. S. Levkov Chap 3 -24

- + + + + - - - + + The free electrons in the n–type material are attracted toward the positive terminal of the battery and away from the junction, creating new holes and the positive charge on the right of depletion region becomes even more positive. The holes from the p–type material are attracted toward the negative terminal of the battery and away from the junction, creative negatively charged ions NJIT ECE Dr. S. Levkov Chap 3 -25

- + + + + - - - + The free electrons in the n–type material are attracted toward the positive terminal of the battery and away from the junction, creating new holes and the positive charge on the right of depletion region becomes even more positive. The holes from the p–type material are attracted toward the negative terminal of the battery and away from the junction, creative negatively charged ions, and the negative charge on the left of depletion region becomes more negative. NJIT ECE Dr. S. Levkov Chap 3 -26

- + + + + - - - + The free electrons in the n–type material are attracted toward the positive terminal of the battery and away from the junction, creating new holes and the positive charge on the right of depletion region becomes even more positive. The holes from the p–type material are attracted toward the negative terminal of the battery and away from the junction, creative negatively charged ions, and the negative charge on the left of depletion region becomes more negative. As a result, the space–charge region at the junction becomes effectively wider, and the potential gradient increases until it approaches the potential of the external battery. NJIT ECE Dr. S. Levkov Chap 3 -27

- + + + + - - - + The free electrons in the n–type material are attracted toward the positive terminal of the battery and away from the junction, creating new holes and the positive charge on the right of depletion region becomes even more positive. The holes from the p–type material are attracted toward the negative terminal of the battery and away from the junction, creative negatively charged ions, and the negative charge on the left of depletion region becomes more negative. As a result, the space–charge region at the junction becomes effectively wider, and the potential gradient increases until it approaches the potential of the external battery. Current flow is then extremely small because no voltage difference ( electric field ) exists across either the p–type or the n–type region. NJIT ECE Dr. S. Levkov Chap 3 -28

The pn junction in forward bias
- The free electrons in the p–type material near the positive terminal of the battery break their electron–pair bonds and enter the battery, NJIT ECE Dr. S. Levkov Chap 3 -29

- - The free electrons in the p–type material near the positive terminal of the battery break their electron–pair bonds and enter the battery, creating new holes, NJIT ECE Dr. S. Levkov Chap 3 -30

- The free electrons in the p–type material near the positive terminal of the battery break their electron–pair bonds and enter the battery, creating new holes, which reduce the negative space charge on the left . NJIT ECE Dr. S. Levkov Chap 3 -31

- The free electrons in the p–type material near the positive terminal of the battery break their electron–pair bonds and enter the battery, creating new holes, which reduce the negative space charge on the left . The electrons from the negative terminal of the battery enter the n–type material NJIT ECE Dr. S. Levkov Chap 3 -32

- - The free electrons in the p–type material near the positive terminal of the battery break their electron–pair bonds and enter the battery, creating new holes, which reduce the negative space charge on the left . The electrons from the negative terminal of the battery enter the n–type material, diffuse toward the junction NJIT ECE Dr. S. Levkov Chap 3 -33

- The free electrons in the p–type material near the positive terminal of the battery break their electron–pair bonds and enter the battery, creating new holes, which reduce the negative space charge on the left . The electrons from the negative terminal of the battery enter the n–type material, diffuse toward the junction, and reduce the positive space charge on the right. NJIT ECE Dr. S. Levkov Chap 3 -34

The free electrons in the p–type material near the positive terminal of the battery break their electron–pair bonds and enter the battery, creating new holes, which reduce the negative space charge on the left . The electrons from the negative terminal of the battery enter the n–type material, diffuse toward the junction, and reduce the positive space charge on the right. As a result, the space charge region becomes effectively narrower, and the energy barrier decreases to an insignificant value. NJIT ECE Dr. S. Levkov Chap 3 -35

- The free electrons in the p–type material near the positive terminal of the battery break their electron–pair bonds and enter the battery, creating new holes, which reduce the negative space charge on the left . The electrons from the negative terminal of the battery enter the n–type material, diffuse toward the junction, and reduce the positive space charge on the right. As a result, the space charge region becomes effectively narrower, and the energy barrier decreases to an insignificant value. Excess electrons from the n–type material can then penetrate the space charge region, flow across the junction, and move by way of the holes in the p–type material toward the positive terminal of the battery, creating large diffusion current This electron flow continues as long as the external voltage is applied, with resulting diode current. NJIT ECE Dr. S. Levkov Chap 3 -36

The pn junction in diode summary
- reverse saturation current of minority carriers NJIT ECE Dr. S. Levkov Chap 3 -37

Diode Junction Potential for Different Applied Voltages
NJIT ECE Dr. S. Levkov Chap 3 -38

Diode i-v Characteristics

Figure 3.9 Pspice model here NJIT ECE Dr. S. Levkov Chap 3 -42

Diode Equation where IS = reverse saturation current (A) vD = voltage applied to diode (V) q = electronic charge (1.60 x C) k = Boltzmann’s constant (1.38 x J/K) T = absolute temperature n = nonideality factor (dimensionless) VT = kT/q = thermal voltage (V) (25 mV at room temp.) IS is typically between and 10-9 A, and is strongly temperature dependent due to its dependence on ni2. The nonideality factor is typically close to 1, but approaches 2 for devices with high current densities. It is assumed to be 1 in this text. NJIT ECE Dr. S. Levkov Chap 3 -43

Diode Voltage and Current Calculations (Example)
Problem: Find diode voltage for diode at room-temperature dc operation for different specifications with VT = V. If IS = 0.1 fA, ID = 300 mA NJIT ECE Dr. S. Levkov Chap 3 -44

Problem: Find diode voltage for diode at room-temperature dc operation for different specifications with VT = V. If IS = 0.1 fA, ID = 300 mA If IS = 10 fA, ID = 300 mA NJIT ECE Dr. S. Levkov Chap 3 -45

Problem: Find diode voltage for diode at room-temperature dc operation for different specifications with VT = V. If IS = 0.1 fA, ID = 300 mA If IS = 10 fA, ID = 300 mA With IS = 0.1 fA, ID = 1 mA, NJIT ECE Dr. S. Levkov Chap 3 -46

Diode Current for Reverse, Zero, and Forward Bias
Reverse bias: for example: NJIT ECE Dr. S. Levkov Chap 3 -47

Reverse bias: for example: Zero bias: NJIT ECE Dr. S. Levkov Chap 3 -48

Reverse bias: for example: Zero bias: Forward bias: NJIT ECE Dr. S. Levkov Chap 3 -49

Semi-log Plot of Forward Diode Current and Current for Three Different Values of IS
Only small 60mV increase in the forward voltage is needed to raise the diode current by 10. NJIT ECE Dr. S. Levkov Chap 3 -50

Reverse Bias -details External reverse bias adds to the built-in potential of the pn junction. The shaded regions below illustrate the increase in the characteristics of the space charge region due to an externally applied reverse bias, vD. NJIT ECE Dr. S. Levkov Chap 3 -51

Reverse Bias (cont.) External reverse bias also increases the width of the depletion region since the larger electric field must be supported by additional charge. Since the reverse saturation results from thermal generation of electron-hole pairs in the depletion region, it is dependent on the volume of the space charge region. It can be shown that the reverse saturation current gradually increases with increased reverse bias: However, IS is approximately constant at IS0 under forward bias. NJIT ECE Dr. S. Levkov Chap 3 -52

Reverse Breakdown 2 V < VZ < 2000 V
Increased reverse bias eventually results in the diode entering the breakdown region, resulting in a sharp increase in the diode current. The voltage at which this occurs is the breakdown voltage, VZ. 2 V < VZ < 2000 V This is not destructive phenomenon, diode can be repeatedly operated in this condition. notes NJIT ECE Dr. S. Levkov Chap 3 -53

Reverse Breakdown Mechanisms
Avalanche Breakdown Si diodes with VZ greater than about 5.6 volts breakdown according to an avalanche mechanism. As the electric field increases, accelerated carriers begin to collide with fixed atoms. As the reverse bias increases, the energy of the accelerated carriers increases, eventually leading to ionization of the impacted atoms. The new carriers also accelerate and ionize other atoms. This process feeds on itself and leads to avalanche breakdown. NJIT ECE Dr. S. Levkov Chap 3 -54

Reverse Breakdown Mechanisms (cont.)
Zener Breakdown Zener breakdown occurs in heavily doped diodes. The heavy doping results in a very narrow depletion region at the diode junction. Reverse bias leads to carriers with sufficient energy to tunnel directly between conduction and valence bands  moving across the junction. Once the tunneling threshold is reached, additional reverse bias leads to a rapidly increasing reverse current. Quantum tunneling refers to the quantum mechanical phenomenon where a particle tunnels through a potential barrier that it classically could not surmount because its total energy kinetic is lower than the potential energy of the barrier. This tunneling plays an essential role in several physical phenomena, including radioactive decay, and has important application to modern devices such as flash memory, the tunneling diode and scanning tunneling microscope. Quantum tunneling is a consequence of the wave particle duality of matter and is often explained using the Heisenberg uncertainty principle: an elementary particle such as an electron has a nonzero probability of moving from one side of any physical barrier to the other, regardless of the height or width of the barrier, or in other words, regardless of whether the potential energy of the barrier is greater than the kinetic energy of the particle. Other names for this effect are quantum-mechanical tunneling, wave-mechanical tunneling, electron tunneling and tunnel effect. NJIT ECE Dr. S. Levkov Chap 3 -55

2. Diode Circuits Analysis
The diode is the most simple and fundamental example of nonlinear circuit elements. NJIT ECE Dr. S. Levkov Chap 3 -56

The diode is the most simple and fundamental example of nonlinear circuit elements. Like a resistor, a diode has two terminals. NJIT ECE Dr. S. Levkov Chap 3 -57

The diode is the most simple and fundamental example of nonlinear circuit elements. Like a resistor, a diode has two terminals. Unlike the resistor, which has a linear (straight line) relationship between current i and voltage v, the diode has a nonlinear i-v characteristics. NJIT ECE Dr. S. Levkov Chap 3 -58

The diode is the most simple and fundamental example of nonlinear circuit elements. Like a resistor, a diode has two terminals. Unlike the resistor, which has a linear (straight line) relationship between current i and voltage v, the diode has a nonlinear i-v characteristics. The main idea: diode conducts when it is forward biased and does not conduct when it is reverse biased. NJIT ECE Dr. S. Levkov Chap 3 -59

The diode is the most simple and fundamental example of nonlinear circuit elements. Like a resistor, a diode has two terminals. Unlike the resistor, which has a linear (straight line) relationship between current i and voltage v, the diode has a nonlinear i-v characteristics. The main idea: diode conducts when it is forward biased and does not conduct when it is reverse biased. To forward bias a diode, the anode (p) must be more positive than the cathode (n) or less negative. NJIT ECE Dr. S. Levkov Chap 3 -60

The diode is the most simple and fundamental example of nonlinear circuit elements. Like a resistor, a diode has two terminals. Unlike the resistor, which has a linear (straight line) relationship between current i and voltage v, the diode has a nonlinear i-v characteristics. The main idea: diode conducts when it is forward biased and does not conduct when it is reverse biased. To forward bias a diode, the anode (p) must be more positive than the cathode (n) or less negative. To reverse bias a diode, the anode (p) must be less positive than the cathode (n) or more negative. NJIT ECE Dr. S. Levkov Chap 3 -61

A Diode Puzzle 1 Which lamps are alight? Some may not be full brightness. + + - -

A Diode Puzzle 2 Which lamps are alight? Some may not be full brightness. + + - -

Diode Models Since the diode has a nonlinear i-v characteristics, the circuit analysis is more complicated. Depending on the needs and type of the circuit (simple/complex) models of different degree of accuracy can be used: The full nonlinear model - graphical solution (load line analysis) or numerical solution The piecewise linear approximation at the Q-point (represent diode as a resistor and source) The ideal diode model. The constant voltage drop diode model (offset model) . The first two are “small signal models”, used for higher accuracy and simpler circuits. The last two are “large signal models” , used for more complicated circuits and less accuracy. NJIT ECE Dr. S. Levkov Chap 3 -66

Diode Circuit Analysis: Basics
The loop equation for the diode circuit is: This is also called the load line for the diode. We need also to add the diode equation: The solution to these equations can be found by different methods using models of different complexity mentioned earlier. V and R may represent the Thévenin equivalent of a more complex 2-terminal network. The objective of diode circuit analysis is to find the quiescent operating point for the diode. Q-Point = (ID, VD) Explanation here NJIT ECE Dr. S. Levkov Chap 3 -67

Nonlinear Model 1. Graphical Load-Line Analysis
This method uses the full nonlinear model of a diode. The loop equations for the Thévenin equivalent of a 2-terminal network with a diode are or (1) and (2) where ID and VD are to be found. Second of these two equations is nonlinear, so some specific methods have to be used to solve it. Load line analysis uses graphical method. Equation (1) is a line with a slope 1/R, equation (2) is the iv-curve of a diode. Their intersection represent the solution – a diode’s Q-point. NJIT ECE Dr. S. Levkov Chap 3 -68

Nonlinear Model 1. Graphical Load-Line Analysis -Example
Problem: Find diode Q-point Given data: V = 10 V, R = 10kW. Analysis: To define the load line we use two points: These points and the resulting load line are plotted. Q-point is given by intersection of load line and diode characteristic: Q-point = (0.95 mA, 0.6 V) NJIT ECE Dr. S. Levkov Chap 3 -69

Nonlinear Model 2. Numerical Solution
This method uses the full nonlinear model of a diode and numerical solution of nonlinear equation. Substituting diode equation (2) into the circuit equation (1) we get one nonlinear equation for variable VD Solving this equation gives diode’s Q-point. The solution is usually numerical using some iterative method. Can be obtained using software with a numerical solver (e.g. Matlab), Newton method using Excel, manual simple trial and error, manual iterations using Newton method. After obtaining solution, the linear two element diode model can be obtained. NJIT ECE Dr. S. Levkov Chap 3 -70

Nonlinear Model 2. Numerical Solution - Example
2) Make initial guess VD0 . 3) Evaluate f and its derivative f’ for this value of VD. 4) Calculate new guess for VD using 5) Repeat steps 3 and 4 till convergence. Using a spreadsheet we get : Q-point = ( mA, V) Since, usually we don’t have accurate saturation current values and significant tolerances exist for sources and passive components, we need answers precise to only 2or 3 significant digits. Given data: IS =10-13 A, VT = V. The solution is given by a transcendental equation. A numerical answer can be found by using Newton’s iterative method. 1) Create a function Spreadsheet example here NJIT ECE Dr. S. Levkov Chap 3 -71

Linearisation Two Element Model of Diode
After obtaining the exact solution, the two element linear diode model can be obtained. Differentiating expression for ID: Example (previous slide) NJIT ECE Dr. S. Levkov Chap 3 -72

Small Signal Diode Model
The last model is often called the “small signal model” – when the diode is biased to operate at a point on the forward i-v characteristic and a small ac signal is superimposed on the DC quantities. If the source signal is , then Find the diode’s Q-point (VD , ID ) for VD and find VD0 and rD Make two element small signal model Use it to find and VD0 Sketch here at the end by hand the “small signal “ circuit only. NJIT ECE Dr. S. Levkov Chap 3 -73

Ideal Diode Model vD =0 for iD >0 iD =0 for vD < 0
If an ideal diode is forward-biased, the voltage across the diode is zero. If an ideal diode is reverse-biased, the current through the diode is zero. Thus, the diode is assumed to be either on or off. vD =0 for iD >0 iD =0 for vD < 0 Thus, instead of one nonlinear equation vD =F(iD ) or iD =F(vD ) we have a nonlinear and non-unique “function” vD =0 for iD >0 iD =0 for vD < 0 NJIT ECE Dr. S. Levkov Chap 3 -74

Analysis using Ideal Model for Diode
Need to determine only the conduction state of an ideal diode. Assume a diode conduction state (on or off). NJIT ECE Dr. S. Levkov Chap 3 -75

Need to determine only the conduction state of an ideal diode. Assume a diode conduction state (on or off). 2. Substitute ideal diode circuit model into circuit (short circuit if “on,” open circuit if “off”). NJIT ECE Dr. S. Levkov Chap 3 -76

Need to determine only the conduction state of an ideal diode. Assume a diode conduction state (on or off). 2. Substitute ideal diode circuit model into circuit (short circuit if “on,” open circuit if “off”). 3. Solve for diode current and voltage, using linear circuit analysis techniques. NJIT ECE Dr. S. Levkov Chap 3 -77

Need to determine only the conduction state of an ideal diode. Assume a diode conduction state (on or off). 2. Substitute ideal diode circuit model into circuit (short circuit if “on,” open circuit if “off”). 3. Solve for diode current and voltage, using linear circuit analysis techniques. 4. If the solution is consistent with the assumption, then the initial assumption was correct; if not, the diode conduction state is opposite to that initially assumed. For example, if the diode has been assumed to be “off” but the diode voltage computed after replacing the diode with an open circuit is a forward bias, then it must be true that the actual state of the diode is “on.” NJIT ECE Dr. S. Levkov Chap 3 -78

Ideal Model for Diode Example 1 (simple) – Find Q-point

Since source appears to force positive current through diode, assume diode is on. NJIT ECE Dr. S. Levkov Chap 3 -80

Since source appears to force positive current through diode, assume diode is on. Our assumption is correct Q-point = (1 mA, 0 V) NJIT ECE Dr. S. Levkov Chap 3 -81

Since source appears to force positive current through diode, assume diode is on. Our assumption is correct Q-point = (1 mA, 0 V) NJIT ECE Dr. S. Levkov Chap 3 -82

Since source appears to force positive current through diode, assume diode is on. Since source is forcing current backward through diode assume diode is off. Our assumption is correct Q-point = (1 mA, 0 V) NJIT ECE Dr. S. Levkov Chap 3 -83

Since source appears to force positive current through diode, assume diode is on. Since source is forcing current backward through diode assume diode is off. Hence ID = 0 . Loop equation is: Our assumption is correct. Q-point = (0 mA, -10 V) Our assumption is correct Q-point = (1 mA, 0 V) NJIT ECE Dr. S. Levkov Chap 3 -84

Ideal Model for Diode Example 2 - average
KCL better: NJIT ECE Dr. S. Levkov Chap 3 -85

Ideal Model for Diode Example 2 - average
KCL better: NJIT ECE Dr. S. Levkov Chap 3 -86

Ideal Model for Diode Example 3 - two-diode circuit
Find diodes Q-points using the ideal diode model. NJIT ECE Dr. S. Levkov Chap 3 -87

Ideal Model for Diode Example 3 - two-diode circuit
Analysis: The ideal diode model is chosen. Assumption 1. Since the 15-V source appears to force positive current through D1 and D2, and the -10-V source is forcing positive current through D2, assume both diodes are on. Since the voltage at node D is zero due to the short circuit of ideal diode D1, But, ID1 < 0 is not allowed by the diode, so try again. NJIT ECE Dr. S. Levkov Chap 3 -88

Ideal Model for Diode Example 3 - two-diode circuit (cont.)
Since the current in D1 is zero, ID2 = I1, Q-Points are D1 : (0 mA, V):off D2 : (1.67 mA, 0 V) :on Now, the results are consistent with the assumptions. Assumption 2. Since current in D2 is valid, but that in D1 is not, the second guess is D1 off and D2 on. NJIT ECE Dr. S. Levkov Chap 3 -89

Constant Voltage Drop (CVD) Model for Diode (Offset Model)
vD = Von for iD >0 iD = 0 for vD < Von. Typically, if Von = 0.6V: Ideal diode = CVD with Von =0 NJIT ECE Dr. S. Levkov Chap 3 -90

Constant Voltage Drop Model for Diode Example 1 - simple
Since the 10-V source appears to force positive current through the diode, assume diode is on. NJIT ECE Dr. S. Levkov Chap 3 -91

Since the 10-V source appears to force positive current through the diode, assume diode is on. NJIT ECE Dr. S. Levkov Chap 3 -92

Since the 10-V source appears to force positive current through the diode, assume diode is on. Solution shows the current is positive, so assumption is correct, diode is on. Q-point = (0.94 mA, 0.6 V) NJIT ECE Dr. S. Levkov Chap 3 -93

Constant Voltage Drop Model for Diode Example 2 - average
Find the lowest value of v1 such that D1 conducts if VB = 2V: a) use the CVD diode model; b) compare with ideal diode model. NJIT ECE Dr. S. Levkov Chap 3 -94

Constant Voltage Drop Model for Diode Example 2 - average
Find the lowest value of v1 such that D1 conducts if VB = 2V: a) use the CVD diode model; b) compare with ideal diode model. a) Assume the diode D1 is off. Then there is no current. i = 0  KVL: -v1 + vD = 0  vD1 = v1 – Diode conducts if vD1 > 0, so v1 > 2.6 is the value of v1 required. b) The diagram will look similarly but without the 0.6V battery. Assume D1 is off. Similarly to (a), we have vD1 = v1 – 2  v1 > 2.0 is the required v1 . NJIT ECE Dr. S. Levkov Chap 3 -95

Constant Voltage Drop Model for Diode Example 3 – multiple diodes circuit
Find the Q-point for D1 , D2, D3 . NJIT ECE Dr. S. Levkov Chap 3 -96

Constant Voltage Drop Model for Diode Example 3 – multiple diodes circuit (cont. 1)
a) Assume all diodes are off. Calculating the voltages in the circuit on the left (zero currents) we see that all diodes have large forward biases. Assumption does not hold. b) Assume all diodes are on. Going from right to left, we can calculate the node voltages: VC = VB = = 0 VA = = Then And OK No OK Assumption does not hold for D2, since its current is negative. NJIT ECE Dr. S. Levkov Chap 3 -97

Constant Voltage Drop Model for Diode Example 3 – multiple diodes circuit (cont 2)
c) Assume D1 , D3 - on, D2 - off. 1) KVL for the loop from +10V source to -20V source: 2) From KVL for the loop from -10V source to the ground: 3) KVL for the loop from the ground to +10V source: NJIT ECE Dr. S. Levkov Chap 3 -98

Models comparison We can compare the four models on the simple example that was used to illustrate all of them All four results are quite similar. Even the simplest ideal model overestimates the current only by 10%. We see that the current is quite insensitive to the to the actual choice of the actual choice of the diode voltage. This is a result of the exponential dependence for the current and quite large source voltage. NJIT ECE Dr. S. Levkov Chap 3 -99

3. Diode Applications NJIT ECE Dr. S. Levkov Chap

Rectifier Circuits A basic rectifier converts an ac voltage to a pulsating dc voltage. A filter then eliminates ac components of the waveform to produce a nearly constant dc voltage output. Rectifier circuits are used in virtually all electronic devices to convert the 120-V 60-Hz ac power line source to the dc voltages required for operation of electronic devices. In rectifier circuits, the diode state changes with time and a given piecewise linear model is valid only for a certain time interval. NJIT ECE Dr. S. Levkov Chap

Half-Wave Rectifier Circuit with Resistive Load
NJIT ECE Dr. S. Levkov Chap

For the positive half-cycle of the input, the source forces positive current through the diode, the diode is on, and vO = vS. NJIT ECE Dr. S. Levkov Chap

For the positive half-cycle of the input, the source forces positive current through the diode, the diode is on, and vO = vS. During the negative half cycle, negative current can’t exist in the diode. The diode is off, current in resistor is zero, and vO =0 . NJIT ECE Dr. S. Levkov Chap

For the positive half-cycle of the input, the source forces positive current through the diode, the diode is on, and vO = vS. During the negative half cycle, negative current can’t exist in the diode. The diode is off, current in resistor is zero, and vO =0 . This is for the ideal diode model. What will be for CVD? NJIT ECE Dr. S. Levkov Chap

Half-Wave Rectifier Circuit with Resistive Load (cont.)
Using the CVD model, during the on-state of the diode vO = (VP sinwt)- Von. The output voltage is zero when the diode is off. VD = Von. NJIT ECE Dr. S. Levkov Chap

Using the CVD model, during the on-state of the diode vO = (VP sinwt)- Von. The output voltage is zero when the diode is off. Often a step-up or step-down transformer is used to convert the 120-V, 60-Hz voltage available from the power line to the desired ac voltage level as shown. VD = Von. NJIT ECE Dr. S. Levkov Chap

Using the CVD model, during the on-state of the diode vO = (VP sinwt)- Von. The output voltage is zero when the diode is off. Often a step-up or step-down transformer is used to convert the 120-V, 60-Hz voltage available from the power line to the desired ac voltage level as shown. The rectifier output signal is very far from being DC. How to improve? VD = Von. NJIT ECE Dr. S. Levkov Chap

Using the CVD model, during the on-state of the diode vO = (VP sinwt)- Von. The output voltage is zero when the diode is off. Often a step-up or step-down transformer is used to convert the 120-V, 60-Hz voltage available from the power line to the desired ac voltage level as shown. The rectifier output signal is very far from being DC. How to improve? Time-varying components in the rectifier output are removed using a filter capacitor. VD = Von. NJIT ECE Dr. S. Levkov Chap

Rectifier with Capacitor Load Peak Detector Circuit
To understand better the operation of a filter, first consider the peak detector circuit. NJIT ECE Dr. S. Levkov Chap

To understand better the operation of a filter, first consider the peak detector circuit. As the input voltage rises, the diode is on, and the capacitor (initially discharged) charges up to the input voltage minus the diode voltage drop. NJIT ECE Dr. S. Levkov Chap

To understand better the operation of a filter, first consider the peak detector circuit. As the input voltage rises, the diode is on, and the capacitor (initially discharged) charges up to the input voltage minus the diode voltage drop. At the peak of the input voltage, diode current tries to reverse, and the diode cuts off. The capacitor has no discharge path and retains a constant voltage providing a constant output voltage: Vdc = VP - Von NJIT ECE Dr. S. Levkov Chap

Half-Wave Rectifier Circuit with RC Load

As the input voltage rises during the first quarter cycle, the diode is on and the capacitor (initially discharged) charges up to the peak value of the input voltage. Vdc = NJIT ECE Dr. S. Levkov Chap

As the input voltage rises during the first quarter cycle, the diode is on and the capacitor (initially discharged) charges up to the peak value of the input voltage. At the peak of the input, the diode current tries to reverse, the diode cuts off, and the capacitor discharges exponentially through R. Discharge continues till the input voltage exceeds the output voltage which occurs near the peak of next cycle. Vdc = NJIT ECE Dr. S. Levkov Chap

As the input voltage rises during the first quarter cycle, the diode is on and the capacitor (initially discharged) charges up to the peak value of the input voltage. At the peak of the input, the diode current tries to reverse, the diode cuts off, and the capacitor discharges exponentially through R. Discharge continues till the input voltage exceeds the output voltage which occurs near the peak of next cycle. This process then repeats once every cycle. Vdc = NJIT ECE Dr. S. Levkov Chap

As the input voltage rises during the first quarter cycle, the diode is on and the capacitor (initially discharged) charges up to the peak value of the input voltage. At the peak of the input, the diode current tries to reverse, the diode cuts off, and the capacitor discharges exponentially through R. Discharge continues till the input voltage exceeds the output voltage which occurs near the peak of next cycle. This process then repeats once every cycle. This circuit can be used to generate negative output voltage if the top plate of capacitor is grounded instead of bottom plate. In this case, Vdc = -(VP - Von) Vdc = NJIT ECE Dr. S. Levkov Chap

The following diagram shows the input and output voltage for the real diode (or CVD model), when NJIT ECE Dr. S. Levkov

Ripple Voltage and Conduction Interval
The output voltage is not constant as in an ideal peak detector, but has a ripple voltage Vr. Simpler expression for VR : Ocharge =Qdischarge  VR *C = Idc*T NJIT ECE Dr. S. Levkov Chap

The output voltage is not constant as in an ideal peak detector, but has a ripple voltage Vr. The diode conducts for a short time DT called the conduction interval during each cycle, and its angular equivalent is called the conduction angle c. Simpler expression for VR : Ocharge =Qdischarge  VR *C = Idc*T NJIT ECE Dr. S. Levkov Chap

The output voltage is not constant as in an ideal peak detector, but has a ripple voltage Vr. The diode conducts for a short time DT called the conduction interval during each cycle, and its angular equivalent is called the conduction angle c. It’s usually assumed that Vr << VDC and DT <<T, where VDC = VP - Von is the ideal DC voltage in the absence of ripple. How to find DT? Simpler expression for VR : Ocharge =Qdischarge  VR *C = Idc*T NJIT ECE Dr. S. Levkov Chap

The output voltage is not constant as in an ideal peak detector, but has a ripple voltage Vr. The diode conducts for a short time DT called the conduction interval during each cycle, and its angular equivalent is called the conduction angle c. It’s usually assumed that Vr << VDC and DT <<T, where VDC = VP - Von is the ideal DC voltage in the absence of ripple. How to find DT Simpler expression for VR : Ocharge =Qdischarge  VR *C = Idc*T NJIT ECE Dr. S. Levkov Chap

Half-Wave Rectifier Analysis Example
Problem: Find the dc output voltage, output current, ripple voltage, conduction interval, and conduction angle for a half-wave rectifier. Given data: secondary voltage Vrms = 12.6 (60 Hz), R = 15 W, C = 25,000 mF, Von = 1 V. Analysis: Using discharge interval T=1/60 s, NJIT ECE Dr. S. Levkov Chap

Peak Diode Current In rectifiers, nonzero current exists in the diode for only a very small fraction of period T, yet an almost constant dc current flows out of the filter capacitor to load. The total charge lost from the filter capacitor in each cycle is replenished by the diode during a short conduction interval causing high peak diode currents. If the repetitive current pulse is modeled as a triangle of height IP and width DT, then, equating the charge supplied during conduction interval to the charge lost during the whole period Using the values from the previous example: NJIT ECE Dr. S. Levkov Chap

Surge Current In addition to the peak diode currents, there is an even larger current through the diode called the surge current that occurs when power is first turned on. During first quarter cycle, current through diode is approximately Peak values of this initial surge current occurs at t = 0+: using values from previous example. Actual values of surge current won’t be as large as predicted above because of the neglected series resistances associated with the rectifier diode and transformer. NJIT ECE Dr. S. Levkov Chap

Peak Inverse Voltage Rating
The peak inverse voltage (PIV) rating of the rectifier diode is the diode breakdown voltage. When the diode is off, the reverse-bias across the diode is Vdc - vS. When vS reaches negative peak, The PIV value corresponds to the minimum value of Zener breakdown voltage required for the rectifier diode. NJIT ECE Dr. S. Levkov Chap

Diode Power Dissipation
Average power dissipation in a diode is given by The simplification is done by assuming a triangular approximation for the diode current and that the voltage across diode is constant at Von. Average power dissipation in the diode series resistance is given by This power dissipation can be reduced by minimizing peak current through the use of a proper size of filter capacitor or by using full-wave rectifiers. NJIT ECE Dr. S. Levkov Chap

Full-Wave Rectifiers Full-wave rectifiers utilize both have periods of the input sin signal and reduce power dissipation. All specifications are the same as for half-wave rectifiers. Reversing polarity of the diodes gives a full-wave rectifier with negative output voltage. NJIT ECE Dr. S. Levkov Chap

Full-Wave Rectifiers Full-wave rectifiers cut capacitor discharge time in half and require half the filter capacitance to achieve a given ripple voltage. All specifications are the same as for half-wave rectifiers. NJIT ECE Dr. S. Levkov Chap

Full-Wave Rectifier Equations

Vr and discharge time is one half of those in half-wave rectifier. NJIT ECE Dr. S. Levkov Chap

Vr and discharge time is one half of those in half-wave rectifier. Conduction interval and peak current are also reduced. PIV rating is the same for each diode. For negative output voltage – reverse the polarity of both diodes. NJIT ECE Dr. S. Levkov Chap

Full-Wave Bridge Rectifier
The requirement for a center-tapped transformer in the full-wave rectifier is eliminated through use of 2 extra diodes. All other specifications are the same as for a half-wave rectifier except For negative output voltage – reverse the polarity of both diodes. NJIT ECE Dr. S. Levkov Chap

Rectifier Topology Comparison
Filter capacitors are a major factor in determining cost, size and weight in design of rectifiers. NJIT ECE Dr. S. Levkov Chap

Filter capacitors are a major factor in determining cost, size and weight in design of rectifiers. For a given ripple voltage, a full-wave rectifier requires half the filter capacitance as that in a half-wave rectifier. Reduced peak current can reduce heat dissipation in diodes. Benefits of full-wave rectification outweigh increased expenses and circuit complexity (an extra diode and center-tapped transformer). NJIT ECE Dr. S. Levkov Chap

Filter capacitors are a major factor in determining cost, size and weight in design of rectifiers. For a given ripple voltage, a full-wave rectifier requires half the filter capacitance as that in a half-wave rectifier. Reduced peak current can reduce heat dissipation in diodes. Benefits of full-wave rectification outweigh increased expenses and circuit complexity (an extra diode and center-tapped transformer). The bridge rectifier eliminates the center-tapped transformer, and the PIV rating of the diodes is reduced. Cost of extra diodes is negligible. NJIT ECE Dr. S. Levkov Chap

Rectifier Topology Comparison and Design Tradeoffs

Rectifier Design Analysis
Problem: Design a rectifier with specifications: Vdc = 15 V, Vr < 0.15 V, Idc = 2 A Assumptions: Von = 1 V, Vr << VDC , DT<<T , 60Hz Analysis: Use a full-wave bridge rectifier that needs a smaller value of filter capacitance, smaller diode PIV rating, and no center-tapped transformer. NJIT ECE Dr. S. Levkov Chap

Diode Logic Gates Diodes together with resistors can be used to implement the digital logic functions. Consider a positive logic system with low (logical 0) as 0V and high (logical 1) as 5V. Circuit (a) represent logical OR function: Circuit (b) represent logical AND function NJIT ECE Dr. S. Levkov Chap

Limiter Circuits For the input in the range the limiter acts as a linear circuit: vo = Kvi (if K<1 – passive limiter). If vi is beyond that range, the output is limited by (L-, L+ ) Transfer characteristic for a limiter circuit NJIT ECE Dr. S. Levkov Chap

Diode Limiter Circuits Examples

Diode Demodulator and Envelop Detector

4. Special Diodes and Applications

Zener Diode As could be seen on diode’s iv-characteristics, in the reverse breakdown region the curve is very steep (almost vertical). This suggests using it in the design of voltage regulators. NJIT ECE Dr. S. Levkov Chap

Zener Diode As could be seen on diode’s iv-characteristics, in the reverse breakdown region the curve is very steep (almost vertical). This suggests using it in the design of voltage regulators. Diodes designed to operate in reverse breakdown are called Zener diodes and use the indicated symbol. NJIT ECE Dr. S. Levkov Chap

Zener Diode As could be seen on diode’s iv-characteristics, in the reverse breakdown region the curve is very steep (almost vertical). This suggests using it in the design of voltage regulators. Diodes designed to operate in reverse breakdown are called Zener diodes and use the indicated symbol. In breakdown, the diode is modeled with a voltage source, VZ, and a series resistance, RZ. RZ models the slope of the i-v characteristic. NJIT ECE Dr. S. Levkov Chap

Breakdown Region Diode Model
IZK – knee current, below it iv-curve is almost straight line. NJIT ECE Dr. S. Levkov Chap

IZK – knee current, below it iv-curve is almost straight line. VZ is usually specified at the test current IZT . 6V Zener diode will exhibit 6V at specified test current of, say, 10ma. VZ can be within the range of a few volts to few hundreds. NJIT ECE Dr. S. Levkov Chap

IZK – knee current, below it iv-curve is almost straight line. VZ is usually specified at the test current IZT . 6V Zener diode will exhibit 6V at specified test current of, say, 10ma. VZ can be within the range of a few volts to few hundreds. rZ = DV/DI - dynamic resistance, is in the range of few ohms to few tens. NJIT ECE Dr. S. Levkov Chap

Analysis of Zener Diode
Using load-line analysis: NJIT ECE Dr. S. Levkov Chap

Using load-line analysis: Choose 2 points: (0V, -4mA) and NJIT ECE Dr. S. Levkov Chap

Using load-line analysis: Choose 2 points: (0V, -4mA) and (-5V, -3mA) NJIT ECE Dr. S. Levkov Chap

Using load-line analysis: Choose 2 points: (0V, -4mA) and (-5V, -3mA) to draw the load line. NJIT ECE Dr. S. Levkov Chap

Using load-line analysis: Choose 2 points: (0V, -4mA) and (-5V, -3mA) to draw the load line. It intersects the i-v characteristic at the Q-point: (-2.9 mA, -5.2 V). NJIT ECE Dr. S. Levkov Chap

Using load-line analysis: Choose 2 points: (0V, -4mA) and (-5V, -3mA) to draw the load line. It intersects the i-v characteristic at the Q-point: (-2.9 mA, -5.2 V). Using the piecewise linear model: VZ = 5V, rZ =100 ohm, NJIT ECE Dr. S. Levkov Chap

Voltage Regulator Using the Zener Diode

The Zener diode keeps the voltage across load resistor RL constant. For Zener breakdown operation, IZ > 0. NJIT ECE Dr. S. Levkov Chap

For proper regulation, Zener current must be positive. If the Zener current < 0, the Zener diode no longer controls the voltage across the load resistor and the voltage regulator is said to have “dropped out of regulation”. The Zener diode keeps the voltage across load resistor RL constant. For Zener breakdown operation, IZ > 0. NJIT ECE Dr. S. Levkov Chap

Voltage Regulator Using a Zener Diode: Example Including Zener Resistance
Problem: Find the output voltage and Zener diode current for a Zener diode regulator.. What is the variation in the load voltage if the source voltage varies by 25%? Given data: R = 5 kW, RZ = 0.1 kW, VZ = 5 V NJIT ECE Dr. S. Levkov Chap

Voltage Regulator Using a Zener Diode: Example Including Zener Resistance
Nominal value: Worst case analysis. VS=15V: VS=25V: 25% change in VS  2% change in VL Problem: Find the output voltage and Zener diode current for a Zener diode regulator.. What is the variation in the load voltage if the source voltage varies by 25%? Given data: R = 5 kW, RZ = 0.1 kW, VZ = 5 V NJIT ECE Dr. S. Levkov Chap

Photo Diodes and Photodetectors
If the reverse biased depletion region of a pn junction diode is illuminated with light with sufficiently high frequency, photons can provide enough energy to cause electrons to jump the semiconductor bandgap to generate electron-hole pairs and create the electric field and photocurrent. h =Planck’s constant = x J-s  = frequency of optical illumination l = wavelength of optical illumination c = velocity of light = 3 x 108 m/s Photon-generated current can be used in photodetector circuits to generate an output voltage NJIT ECE Dr. S. Levkov Chap

Solar Cells and Light-Emitting Diodes
Without the reverse bias, the illuminated photodiode functions as a solar cell. In solar cell applications, optical illumination is constant, and dc current IPH is generated. The goal is to extract power from the cell, and the i-v characteristics are plotted in terms of cell current and cell voltage. For a solar cell to supply power to an external circuit, the ICVC product must be positive, and the cell should be operated near the point of maximum output power Pmax. Light-Emitting Diodes (LEDs) use recombination processes in the forward-biased pn junction diode to produce light. When a hole and electron recombine, an energy equal to the bandgap of the semiconductor is released as a photon. NJIT ECE Dr. S. Levkov Chap

Diode Spice Model Rs represents the inevitable series resistance of a real device structure. The current controlled current source models the ideal exponential behavior of the diode. Capacitor C includes depletion-layer capacitance for the reverse-bias region as well as diffusion capacitance associated with the junction under forward bias. Typical default values: Saturation current IS = 10 fA, Rs = 0 W, transit time TT = 0 seconds, N = 1 NJIT ECE Dr. S. Levkov Chap

Schottky Barrier Diode
One semiconductor region of the pn junction diode can be replaced by a non-ohmic rectifying metal contact. A Schottky contact is easily formed on n-type silicon. The metal region becomes the anode. An n+ region is added to ensure that the cathode contact is ohmic. Current is conducted by major carriers only – electrons Schottky diodes turn on at a lower voltage than pn junction diodes and have significantly reduced internal charge storage under forward bias. Uses: bipolar logic circuits, fast switching applications. NJIT ECE Dr. S. Levkov Chap

Diode Layout NJIT ECE Dr. S. Levkov Chap

PN-junction Diode Capacitance
Typical capacitance in diodes is quite small (tens of pF- few nF ). However it exists and influences the dynamic switching properties if a diode. Reverse Bias Capacitance (cont.) Changes in voltage lead to changes in depletion width and charge. This leads to a capacitance that we can calculate from the charge-voltage dependence. Where Cj0 is the zero bias junction capacitance per unit area NJIT ECE Dr. S. Levkov Chap

Forward Bias Capacitance
In forward bias operation, additional charge is stored in neutral region near edges of space charge region. tT is called diode transit time and depends on size and type of diode. Additional diffusion capacitance, associated with forward region of operation is proportional to current and becomes quite large at high currents. NJIT ECE Dr. S. Levkov Chap

Dynamic Switching Behavior of Diodes
The non-linear depletion-layer capacitance of the diode prevents the diode voltage from changing instantaneously and determines turn-on and recovery times. Both forward and reverse current overshoot the final values when the diode switches on and off as shown NJIT ECE Dr. S. Levkov Chap

Diode Temperature Coefficient
Diode voltage under forward bias: Taking the derivative with respect to temperature yields Assuming iD >> IS, IS  ni2, and VGO is the silicon bandgap energy at 0K. For a typical silicon diode NJIT ECE Dr. S. Levkov Chap

Topic 3 Diodes and Diodes Circuits

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