# DIODE CIRCUITS CHAPTER 2

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1 CHAPTER 2 DIODE CIRCUITS As discussed in the previous section, diodes are essentially one-way valves. They carry current in one direction, but block current in the other. In addition, in the forward conduction mode, the voltage must go above a threshold before significant conduction occurs. Diodes are used in a multitude of ways that utilize these characteristics. The objective of this course is not to look at hole and electron flows, but to see how the diodes can be used in circuits and how to analyze those circuits. For example, consider the circuit below, Figure 1. We are generally not interested in the holes and electrons in the diode, but rather, the current in the resistor or the voltage across the resistor. NUMERICAL METHOD Figure 1. A simple diode circuit The diode being a non-linear element means that the basic methods of circuit analysis learned in your circuits course cannot be used. Then, how does one go about analyzing the circuit? For example, we can write a loop equation V s = V d + V r = V d + I*R Unfortunately, there are two unknowns in this equation, I and V d. The relationship between voltage and current in the diode is given by VD ( ) V 0 1 T I = I ( e ) Where V T is 25.9 mv at 300 degrees K, and I O is the reverse saturation current. If we put the two equations together we get T V = V + RI ( e ) S d VD ( ) V 0 1 (1) This equation cannot be solved analytically. But it can be solved numerically. You can make a guess of the diode voltage and plug into the equation and see if you get a balance. Diode Circuits 1

2 If not, try another guess. This process can be carried out several ways. The simplest is to use a calculator, while more complex methods would use a computer. One interesting approach is to use a math solving program on a personal computer such as MathCAD. Literally all circuits could be solved this way, but that would be impractical, especially for more complex circuits. Secondly, the diode equation mentioned above only represents the hole-electron currents within the body of the diode. There is a considerable leakage on the surface of the semiconductor which makes the low current solutions in error. GRAPHICAL METHOD Instead of using the diode equation as the model, we can get quite satisfactory results by using simpler models, or by using graphical methods from the plotted V-I characteristic of the diode. Consider the circuit in Figure 2. The loop equation is V s = V d + I*R (2) a. b. Figure 2. Diode circuit to be solved graphically If we divide the circuit as shown in Figure 2b, the equation can be rewritten to solve for the voltage at the division. V d = 5 - I*R (3) This equation can be solved for current I Vd = 5 (4) R This equation represents the relationship between current and terminal voltage for the left-hand side of the circuit. The relationship between current and terminal voltage for the right-hand side is just the equation for diode. VD ( ) V 0 1 T I = I ( e ) (5) Diode Circuits 2

3 Each of these equations is just an I vs V d equation and can be plotted on a graph. Equation 4 is a linear equation and its plot is called the load line. It can easily be plotted by selecting values for V d and calculating I. For example, the two intercepts; V d = 0 and I = 0 are convenient choices. The equation 5 is the diode I vs V characteristic for the diode. It is usually obtained from a curve tracer in the laboratory. Both of these equations are plotted on the same graph in Figure 3. The operating point for the circuit is where the two plots cross. VD 5 V ( d = I0 e V ) T ( 1) (6) R Figure 3. Plots for graphical solution One problem with the graphical solution above is that the voltage scale is so large that precision in determining diode voltage is difficult. In many cases, the diode characteristic is plotted with a much lower maximum voltage than the voltage source in the circuit. In this case, the voltage intercept is off the scale and it is more difficult to plot the load line. This situation is easily handled by substituting in a fixed value of voltage in the load line equation and solving for the current. An example is shown in Figure 4. The maximum voltage on the diode characteristic plot is 2 Volts with the source voltage 5 Volts and a resistance of 1KΩ. Solving the load equation 5 I = 2 = 3 ma R which gives us the current at V d = 2 volts at the right hand end of the load line on the plot in Fig. 4. Figure 4. Diode Characteristic and Graphical Solution CIRCUIT MODELS The graphical method presented in the previous section is one possible method of solution of circuits with diodes. For obvious reasons, this method will get very tedious and time consuming for more complex circuits. What we would like to do is to find a way to solve the circuits analytically; with equations. We do this by using circuit models which are combinations of ideal circuit elements that behave the same or almost the same as the actual device. We then solve the circuit using standard methods of circuit analysis. Diode Circuits 3

4 A model is what we use to represent an actual device. For example, we use a mathematical expression V=IR to represent the voltage across a resistor. This happens to be a very good model at low frequencies, below a few MHz. At higher frequencies, the inductive reactance becomes significant and the equation is no longer very accurate. Similarly, V = jωli and I = jωcv are good models for inductors and capacitors at relatively low frequencies. We also use models for voltage and current sources. For example, a typical voltage source is far from ideal. It usually has internal resistance and cannot supply unlimited current. Now let us take a look at what these ideal circuit models represent. If we draw a plot of the relationship between current and voltage of a resistor, we get the plot in Figure 5a. Similarly if we plot the I-V characteristic of the ideal voltage source, we get the plot in Figure 5b. The I-V characteristic of the voltage source in series with the resistor is shown in Figure 5c. a. Resistor b. Voltage source c. Series combination Figure 5. Models for a Resistor and a Voltage Source Circuit Models for Diodes. Now let's look at possible circuit models we can use for a diode. Clearly, a single straight line would not make a very good model. However, a crude approximation can be developed using two straight lines as shown in Figure 6a, a model of an ideal diode. The two straight lines are used only for a specific region as shown. The ideal diode is considered an open circuit if the voltage across it is less than zero, and is a short circuit if the forward current is greater than zero. These two regions are called the cutoff and Figure 6. Two Possible I-V Characteristic Models of a Diode conducting regions or the reverse biased and forward biased regions respectively. We will carry this nomenclature forward with us through more advanced models. The semiideal model in Figure 6b is a better model of the actual diode curve and is used Diode Circuits 4

5 extensively. In this model I = 0 for V < V γ, and V = V γ for I>0, where V γ is the threshold voltage. Another way to look at these models is to translate them into circuit elements. In fact, this is the way we will use these models for conceptualizing circuit operation. The circuit elements representing these models are shown in Figure 7. Figure 7. Circuit elements representing the models in Figure 6 SOLVING DIODE CIRCUITS USING DIODE CIRCUIT MODELS As a first example, let us try the solution of a simple rectifier circuit shown in Figure 8. We will do this first with an ideal diode model and then with the semi-ideal model. Solution for this circuit may mean finding the current or the voltage across the resistor or diode. Figure 8. Diode Rectifier Circuit Solution Using The Ideal Diode Model When attempting to solve this circuit assuming an ideal diode, the first thing you notice is that the diode has two regions of operation. How do you know which to use? The answer to that question comes more from experience than anything else. However, some quick mental analysis is often useful at arriving at a good starting point. If all else fails, simply guess and then check to see if the resulting solution is consistent with your original guess or assumption. For this example, a reasonable guess is that the diode is forward biased and is therefore conducting. In this region, for the ideal diode, V=0 and I>0. Putting this model into the circuit, as shown below, the resulting current is I = V/R = 2.5 ma Since I>0, our solution is consistent with the original assumption. Figure 9. Rectifier Circuit With Diode Replaced by Ideal Model (Conducting) Solution Using The Semi-Ideal Model Now let us use the semi-ideal diode model and solve the rectifier circuit again. The model equations are: V=0.70 for I>0 and I=0 for V<0.70. Diode Circuits 5

6 Again, let us assume that the diode is conducting, I>0. The circuit is redrawn below in Figure 10 with the diode replaced by the assumed model. Solving this circuit, I= ( )/2K = 2.15 ma. Again, since our solution is consistent with the circuit model, we can assume the solution is correct. Figure 10. Rectifier Circuit With Diode Replaced by Semi-Ideal Model (Conducting) We note that these two solutions are significantly different because we used significantly different models for the diode. The difference would have been smaller if the source voltage had been very large compared to V γ. Example 2, A More Complex Circuit As a second example, let us look at the circuit below. We will use the semi-ideal diode model for this problem. At first glance it is not clear whether the diode is conducting or not. Thus, we are not given any guidance in selecting the appropriate model. Let us guess that the diode is conducting. In that case, the circuit is redrawn with the diode replaced by its model in the conducting region. a. Original Circuit b. Circuit With Conducting Model Figure 11. Diode Circuit for Example 2 We can solve this circuit by simply determining the currents in the branches. I 1 = ( )/16K =.269mA I 2 =.70/2.2k.318mA I 3 (The diode current) = = ma The result implies that the current in the diode is in the reverse direction which is inconsistent with the diode model chosen. Our original guess was wrong. Let's try this problem again with the model for the non-conducting or cutoff region. The circuit is shown below. Figure 12. Circuit of Example 2 with Non-Conducting Model Diode Circuits 6

7 Solving this circuit, we find that the voltage across the diode is: V d = 5.0x 2.2k/(2.2K + 16K) = V (<0.70 V) This result is consistent with our model and we can assume it is the correct result. What would have happened if we had chosen the non-conducting model first? We might not have tried the other model. This raises the question of whether both models will give answers that are consistent with the assumption. Actually, this can happen only in cases where there is feedback and the loop gain is greater than unity which can only happen with an amplifier in the circuit. In these cases, there are two stable states. A typical application is a flip-flop. These cases are easy to recognize. As an exercise, modify the previous example and determine the value of R2 (originally the 2.2k resistor) that will ensure that the diode is conducting at 0.01mA. Example 3, A Clipping Circuit As a more advanced example, let us look at a clipping circuit shown in Figure 13. This is not a steady state case as before, but uses a time varying input signal. Let us assume the input signal is a sine wave. What is the output signal? Figure 13. Example 3, Diode Clipping Circuit We can do a simple calculation as we did before and discover that at the most negative peak, the diode is cutoff, and at the most positive peak, the diode is conducting. At Vin = -5, Vo = -5 Volts. At Vin = +5, Vo = = 1.70 Volts Obviously, some place in between, the diode switches states. One of the tricks is to find that transition point. Let us start with the case of the diode non-conducting as shown below. It is easy to see that Vout = Vin as long as the diode is an open circuit. The diode will remain in this state as long as the diode voltage is less than 0.70 volts. As long as Vout (and Vin) is less than 1.70 Volts, the diode will be off. Thus Vout = Vin for Vin < 1.70 Figure 14. Clipping Circuit With Diode Non-Conducting For Vin > 1.70, The diode will be conducting. The circuit is shown below. For this case, Vout = 1.70 for Vin > Diode Circuits 7

8 Figure 15. Clipping Circuit With Diode Conducting If we combine the two regions of operation, and draw the waveforms generated by this circuit, we will get the results shown in Figure 16. Actually, the diode begins to conduct when the forward voltage is in the neighborhood of 0.60 volts and fully turns on at about 0.70 volts. In between is some sort of smooth transition. The sharp transitions of the waveforms in Figure 16 would actually be rounded. For the purposes of this course, this transition is not very important. Figure 16. Output Waveform of the Diode Clipping Circuit With A Sinusoidal Input Signal Advanced Models The models in Figure 6 are obviously only crude approximations to the actual curve. A more advanced model will take into consideration that the diode current is not zero below the knee of the curve. Looking just at that portion of the curve, it looks almost resistive, with a rather high resistance. In fact, this region is often modeled that way with a resistor on the order of 1 megohm. The fact of the matter is that a significant portion of the current in this region is due to surface leakage, (resistive). For the conducting portion of the curve, the curve is nearly vertical, but not quite. Again, this suggests another resistance, but in series with a voltage source. This resistance is mainly due to the body resistance of the bulk material. The result is a two-region linear model that is piece-wise continuous. The models and mathematical expressions are shown in Figure 17. Figure 17. Two-Resistor Model of a Diode Diode Circuits 8

9 The circuit model shown in Figure 17 is a much better representation than the earlier ones, but even this one differs greatly from the actual characteristic in the region of the knee. More sophisticated models would use three or more regions for the model. Choosing an Appropriate Model The model for the ideal diode is useful for many purposes that do not require great precision in circuit solution. The two-resistor model is rather complex and is sometimes used where relatively accurate analysis is required. However, this model is more difficult to use than the other two. As a compromise, we will use the semi-ideal model for most of the analysis in this course. It will give a reasonably good solution to digital switching circuits used while not adding unnecessarily to the circuit complexity. For silicon diodes, the value of V γ is usually taken between 0.60 and 0.70 volts. There are physical justifications for the threshold voltage assumed, but the value used is often selected from observations of the typical voltage measured across the diode with normal operating current. For integrated circuit diodes, the cross sectional area of the diode body is kept rather small and the current density is often rather high. Thus, for integrated circuits, 0.70 volts is often used. For discrete diodes, the current densities are often somewhat lower, allowing a lower value, usually 0.65 volts. In both integrated circuits and low-power discrete circuits, any reverse voltage applied during operation is quite low, allowing the reverse current to be neglected. In power diodes, the current densities are often very high, approaching the upper device limit, and the reverse voltages also often approach the device limits. In these cases, the body resistance during forward conduction and the reverse leakage current during reverse biasing cannot be neglected. Thus, in this case, the two-resistor model is more appropriate and is often used. Germanium diodes are usually used only for low power applications, and usually used as discrete devices as germanium is rarely used in IC's. In practice, V γ is usually taken between 0.2 and 0.3 volts. We will use a slight modification to the semi-ideal model for most of this course where we are talking about switching circuits. The model we will actually use is: We will not operate where 0.60<V<0.70 Diode Circuits 9

10 This model makes some sense in that a close look at a typical diode curve reveals that the current is usually negligible with the voltage below 0.60 V. Once the current reaches an appreciable amount, the voltage across the diode has risen to about 0.70 V. In between, the I-V characteristic is a smooth curve. In digital switching circuits we find the diode will either be in the cutoff region or the forward conducting region with substantial current. We simply will not operate in the region in between. Diode Circuits 10

11 DESIGN EXERCISE (Suggested by Dr. W. L. Cooley) This design exercise is an example of an application that illustrates the use of diodes in circuits. This circuit utilizes the offset voltage of the diode. It assumes a semi-ideal characteristic: I=0 for V<V γ and V= V γ for I>0. In this case, V γ = 0.7 volts. The application is to make a non-linear voltmeter that will measure the available capacity of a 10 volt Ni-Cad battery. The Ni-Cad battery voltage at full charge is 10 volts. As the battery is discharged, the terminal voltage drops slowly down to about 9 volts. Below 9 volts, the battery capacity is nearly used up and further discharge will cause the terminal voltage to drop rapidly. What we would like to have is a meter that will show full scale when the battery is 10 volts, and dropping linearly to 10% of full scale when the battery voltage drops to 9 volts. Below 9 volts, the meter reading drops linearly to zero at zero terminal voltage. Above 10 volts input, the meter reading is to be limited to full scale. This characteristic is shown graphically in Figure 18 below. Figure 18. Graphical representation of desired meter characteristic. The meter to be used has 1Kohm resistance and a full scale deflection of 1 microamp. Solution: An example solution is given in the EE 56 Web site under Additional Information. Diode Circuits 11

12 EXERCISES 1. The symbol for a diode is shown along with a pn junction. a. Draw the polarity that will cause forward current in both devices. b. Draw an arrow showing the direction of forward current for both devices 2. Find the current in the following circuit. The diode I-V characteristic is given below. I = 3. Solve again, if the voltage source is 3 volts. I = 4. Solve again, if the voltage source is 3 volts and the resistor is changed to 150 ohms. I = 5. Solve Exercise #2 analytically using an ideal diode model. Diode Circuits 12

13 I = 6. Solve Exercise #2 analytically, using the following model for the diode. I F = 0 if V D < 0.60 volts V D = 0.70 volts, if I F > 0 I = Diode Circuits 13

14 Problems 1. A diode has a reverse saturation current of 10 pa and V T = 25.9 mv. What is the applied voltage to get a forward current of 100 ma? 2. A silicon diode was found to have a reverse saturation current of.001 microamp. What would the current be if a forward voltage of 0.50 volts was applied? 3. Use numerical methods to solve for the current in the circuit and the voltage across the 1NWV56 diode. Continue the iterative process until the error in the diode voltage is less than 0.1 mv. The reverse saturation current of the diode is 0.01 na. The temperature is 27 o C so that V T = 25.9 mv. Use the equation below for the diode current. I = I 0(e V D/ V T -1) 4. Solve the circuit above using graphical techniques. Assume the 1NWV56 diode has the characteristic given on the following page. Find the current in the circuit and the voltage across the diode. Diode Circuits 14

15 5. Repeat Problem 4, but with the voltage source at 10 volts and the resistor at 5 KΩ. Diode Circuits 15

16 6. Solve the circuit below for currents I 1 and I 2. Demonstrate that your answer is consistent with the assumptions you made. Diode Model: Forward Conducting Region Non-Conducting Region V=0.70 volts for I>0 I=0 for V<0.70 volts Circuit for Problems 6 and 7 7. Work Problem #6 again, but with a source voltage of 12 volts. 8. For the circuit given, draw the current waveform and the waveform for the voltages across the resistor and the diode. Use the diode model given in Problem #6. Carefully determine the breakpoints (abrupt changes in slope). Write the equations for each segment of each waveform showing the range of values of V in for which the equations are valid. 9. Do a computer simulation of problem 8. Use the circuit models from problem 6. Show the waveforms. 10. Do a computer simulation for problem 8, but this time use the built-in model for a silicon diode. Show the waveforms. 11. Determine the output voltage for the circuit shown. The diode model is: V D = for I>0 I = 0 for V D < A device found in the Martian ruins has a model described by the following equations. Equations: i. V x = 2.5 V for I > 25 ma ii. V x = 100 I for -5.0 < V x < 2.5 V Diode Circuits 16

17 iii. I = -5.0 ma for V x < -5 V a. Draw a circuit model for each region of operation. b. Draw the I-V characteristic for the device Diode Circuits 17

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