Hudgins, J L, Bogart, Jr, T.F., Mayaram, K, Kennedy, M. P, Kolumban, G. " Nonlinear Circuits The electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRc Press llc. 2000
Hudgins, J.L., Bogart, Jr., T.F., Mayaram, K., Kennedy, M.P., Kolumbán, G. “Nonlinear Circuits” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
5 Nonlinear circuits Jerry L Hudgins Limiting Circuits. Precision Rectifying Circuits Theodore F. Bogart, r 5.3 Distortion University of Southern Mississippi Method Three- Point Method Five- Point Method Kartikeya Mayaram Modulation.compressionandInterceptPointsCrossove Distortion. Failure-to-Follow Distortion. Frequency Distortion Michael Peter Kennedy PhaseDistortion.computerSimulationofDistortionComponents 5.4 Communicating with Chaos Elements of Chaotic Digital Communications Systems. Chaotic Digital Geza kolumban Modulation Schemes.Low-Pass Equivalent Models for Chaoti Technical University of Budapest Communications Systems. Multipath Performance of FM-DCSK 5.1 Diodes and rectifiers Jerry L. hudgins a diode generally refers to a two-terminal solid-state semiconductor device that presents a low impedance to current flow in one direction and a high impedance to current flow in the opposite direction. These properties allow the diode to be used as a one-way current valve in electronic circuits. Rectifiers are a class of circuits whose purpose is to convert ac waveforms(usually sinusoidal and with zero average value) into a waveform that has a significant non-zero average value(dc component). Simply stated, rectifiers are ac-to-dc energy converter circuits. Most rectifier circuits employ diodes as the principal elements in the energy conversion process; thus the almost inseparable notions of diodes and rectifiers. The general electrical characteristics of common diodes and some simple rectifier topologies incorporating diodes are discussed. Diodes Most diodes are made from a host crystal of silicon(Si) with appropriate impurity elements introduced to modify, in a controlled manner, the electrical characteristics of the device. These diodes are the typical Pn-junction (or bipolar)devices used in electronic circuits. Another type is the Schottky diode(unipolar), produced by placing a metal layer directly onto the semiconductor [Schottky, 1938; Mott, 1938. The metal emiconductor interface serves the same function as the pn-junction in the common diode structure. Other semiconductor materials such as gallium-arsenide( GaAs)and silicon-carbide(Sic) are also in use for new and ecialized applications of diodes. Detailed discussion of diode structures and the physics of their operation can be found in later paragraphs of this section. The electrical circuit symbol for a bipolar diode is shown in Fig. 5. 1. The polarities associated with the forward voltage drop for forward current flow are also included Current or voltage opposite to the polarities indicated in Fig. 5. 1 are considered to be negative values with respect to the diode conventions shown. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC 5 Nonlinear Circuits 5.1 Diodes and Rectifiers Diodes • Rectifiers 5.2 Limiters Limiting Circuits • Precision Rectifying Circuits 5.3 Distortion Harmonic Distortion • Power-Series Method • Differential-Error Method • Three-Point Method • Five-Point Method • Intermodulation Distortion • Triple-Beat Distortion • Cross Modulation • Compression and Intercept Points • Crossover Distortion • Failure-to-Follow Distortion • Frequency Distortion • Phase Distortion • Computer Simulation of Distortion Components 5.4 Communicating with Chaos Elements of Chaotic Digital Communications Systems • Chaotic Digital Modulation Schemes • Low-Pass Equivalent Models for Chaotic Communications Systems • Multipath Performance of FM-DCSK 5.1 Diodes and Rectifiers Jerry L. Hudgins A diode generally refers to a two-terminal solid-state semiconductor device that presents a low impedance to current flow in one direction and a high impedance to current flow in the opposite direction. These properties allow the diode to be used as a one-way current valve in electronic circuits. Rectifiers are a class of circuits whose purpose is to convert ac waveforms (usually sinusoidal and with zero average value) into a waveform that has a significant non-zero average value (dc component). Simply stated, rectifiers are ac-to-dc energy converter circuits. Most rectifier circuits employ diodes as the principal elements in the energy conversion process; thus the almost inseparable notions of diodes and rectifiers. The general electrical characteristics of common diodes and some simple rectifier topologies incorporating diodes are discussed. Diodes Most diodes are made from a host crystal of silicon (Si) with appropriate impurity elements introduced to modify, in a controlled manner, the electrical characteristics of the device. These diodes are the typical pn-junction (or bipolar) devices used in electronic circuits. Another type is the Schottky diode (unipolar), produced by placing a metal layer directly onto the semiconductor [Schottky, 1938; Mott, 1938]. The metalsemiconductor interface serves the same function as the pn-junction in the common diode structure. Other semiconductor materials such as gallium-arsenide (GaAs) and silicon-carbide (SiC) are also in use for new and specialized applications of diodes. Detailed discussion of diode structures and the physics of their operation can be found in later paragraphs of this section. The electrical circuit symbol for a bipolar diode is shown in Fig. 5.1. The polarities associated with the forward voltage drop for forward current flow are also included. Current or voltage opposite to the polarities indicated in Fig. 5.1 are considered to be negative values with respect to the diode conventions shown. Jerry L. Hudgins University of South Carolina Theodore F. Bogart, Jr. University of Southern Mississippi Kartikeya Mayaram Washington State University Michael Peter Kennedy University College Dublin Géza Kolumbán Technical University of Budapest
The characteristic curve shown in Fig. 5. 2 is representative of the current voltage dependencies of typical diodes. The diode conducts forward current with a small forward voltage drop across the device, simulating a closed switch. The relationship between the forward current and forward voltage is approximately given by the Shockley diode equation Shockley, 1949: FIGURE5.1 Circuit symbol for a diode indicating ip=I exp (5.1) ity associated with the oltage and current directions. where I is the leakage current through the diode, q is the electronic charge, n is a correction factor, k Boltzmanns constant, and Tis the temperature of the semiconductor. Around the knee of the curve in Fig 5.2 is a positive voltage that is termed the turn-on or sometimes the threshold voltage for the diode. This valu an approximate voltage above which the diode is considered turned"on"and can be modeled to first degree as a closed switch with constant forward drop. Below the threshold voltage value the diode is considered weakly conducting and approximated as an open switch. The exponential relationship shown in Eq (5.1)means that the diode forward current can change by orders of magnitude before there is a large change in diode voltage, thus providing the simple circuit model during conduction. The nonlinear relationship of Eq (5.1)also provides a means of frequency mixing for applications in modulation circuits Reverse voltage applied to the diode causes a small leakage current(negative according to the sign convention to flow that is typically orders of magnitude lower than current in the forward direction. The diode can withstand reverse voltages up to a limit determined by its physical construction and the semiconductor material used. Beyond this value the reverse voltage imparts enough energy to the charge carriers to cause large increases in current. The mechanisms by which this current increase occurs are impact ionization(avalanche)[McKay, 1954] and a tunneling phenomenon(Zener breakdown)[Moll, 1964 ]. Avalanche breakdown results in large power dissipation in the diode, is generally destructive, and should be avoided at all times. Both breakdown regions are superimposed in Fig 5.2 for comparison of their effects on the shape of the diode characteristic curve. Avalanche breakdown occurs for reverse applied voltages in the range of volts to kilovolts depending on ne exact design of the diode. Zener breakdown occurs at much lower voltages than the avalanche mechanisn Diodes specifically designed to operate in the Zener breakdown mode are used extensively as voltage regulators During forward conduction the power loss in the diode can become excessive for large current flow. Schottky diodes have an inherently lower turn-on voltage than pn-junction diodes and are therefore more desirable the energy losses in the diodes cant(such as output rectifiers in switching power supplies). Other considerations such as recovery characteristics from forward conduction to reverse blockin iD(A) Vo ((not to scale) 00 FIGURE 5.2 a typical diode dc characteristic curve showing the current dependence on e 2000 by CRC Press LLC
© 2000 by CRC Press LLC The characteristic curve shown in Fig. 5.2 is representative of the currentvoltage dependencies of typical diodes. The diode conducts forward current with a small forward voltage drop across the device, simulating a closed switch. The relationship between the forward current and forward voltage is approximately given by the Shockley diode equation [Shockley, 1949]: (5.1) where Is is the leakage current through the diode, q is the electronic charge, n is a correction factor, k is Boltzmann’s constant, and T is the temperature of the semiconductor. Around the knee of the curve in Fig. 5.2 is a positive voltage that is termed the turn-on or sometimes the threshold voltage for the diode. This value is an approximate voltage above which the diode is considered turned “on” and can be modeled to first degree as a closed switch with constant forward drop. Below the threshold voltage value the diode is considered weakly conducting and approximated as an open switch. The exponential relationship shown in Eq. (5.1) means that the diode forward current can change by orders of magnitude before there is a large change in diode voltage, thus providing the simple circuit model during conduction. The nonlinear relationship of Eq. (5.1) also provides a means of frequency mixing for applications in modulation circuits. Reverse voltage applied to the diode causes a small leakage current (negative according to the sign convention) to flow that is typically orders of magnitude lower than current in the forward direction. The diode can withstand reverse voltages up to a limit determined by its physical construction and the semiconductor material used. Beyond this value the reverse voltage imparts enough energy to the charge carriers to cause large increases in current. The mechanisms by which this current increase occurs are impact ionization (avalanche) [McKay, 1954] and a tunneling phenomenon (Zener breakdown) [Moll, 1964]. Avalanche breakdown results in large power dissipation in the diode, is generally destructive, and should be avoided at all times. Both breakdown regions are superimposed in Fig. 5.2 for comparison of their effects on the shape of the diode characteristic curve. Avalanche breakdown occurs for reverse applied voltages in the range of volts to kilovolts depending on the exact design of the diode. Zener breakdown occurs at much lower voltages than the avalanche mechanism. Diodes specifically designed to operate in the Zener breakdown mode are used extensively as voltage regulators in regulator integrated circuits and as discrete components in large regulated power supplies. During forward conduction the power loss in the diode can become excessive for large current flow. Schottky diodes have an inherently lower turn-on voltage than pn-junction diodes and are therefore more desirable in applications where the energy losses in the diodes are significant (such as output rectifiers in switching power supplies). Other considerations such as recovery characteristics from forward conduction to reverse blocking FIGURE 5.2 A typical diode dc characteristic curve showing the current dependence on voltage. FIGURE 5.1 Circuit symbol for a bipolar diode indicating the polarity associated with the forward voltage and current directions. i I qV nkT D s D = Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ exp – 1
PoV(not to scale IGURE 5.3 The effects of temperature variations on the forward voltage drop and the avalanche breakdown voltage in Iso make one diode type more desirable than another. Schottky diodes condu charge carrier and are therefore inherently faster to turn off than bipolar diodes. However, one of the limitations of Schottky diodes is their excessive forward voltage drop when designed to support reverse biases above about 200 V. Therefore, high-voltage diodes are the pn-junction ty polar diode are many. The forward voltage drop uring conduction will decrease over a large current range, the reverse leakage current will increase, and the reverse avalanche breakdown voltage(Vap) will increase as the device temperature climbs. A family of static haracteristic curves highlighting these effects is shown in Fig. 5. 3 where T3> T2>T. In addition, a major effect on the switching characteristic is the increase in the reverse recovery time during turn-off. Some of the key parameters to be aware of when choosing a diode are its repetitive peak inverse voltage rating, VRRM(relates to the avalanche breakdown value), the peak forward surge current rating, IEsM(relates to the maximum allowable transient heating in the device), the average or rms current rating, Io(relates to the steady-state heating in the device), and the reverse recovery time, tr, (relates to the switching speed of the device) Rectifiers This section discusses some simple uncontrolled rectifier circuits that are commonly encountered. The term uncontrolled refers to the absence of any control signal necessary to operate the primary switching elements(diodes) in the rectifier circuit. The discussion of controlled rectifier circuits, and the controlled switches themselves, is more appropriate in the context of power electronics applications [Hoft, 1986]. Rectifiers are the fundamental building block in dc power supplies of all types and in dc power transmission used by some electric utilities. A single-phase full-wave rectifier circuit with the accompanying input and output voltage waveforms is show half-cycles of the input voltage. The forward drop across the diodes is ignored on the output graph, whid.a in Fig. 5.4. This topology makes use of a center-tapped transformer with each diode conducting on opposi a valid approximation if the peak voltages of the input and output are large compared to 1 V. The circuit changes a sinusoidal waveform with no dc component(zero average value)to one with a dc component of 2 Vpeak/. The rms value of the output is 0. 707Vpeak The dc value can be increased further by adding a low-pass filter in cascade with the output. The usual form of this filter is a shunt capacitor or an LC filter as shown in Fig. 5.5. The resonant frequency of the LC filter should be lower than the fundamental frequency of the rectifier output for effective performance. The ac portion of the output signal is reduced while the dc and rms values are increased by adding the filter. The remaining c portion of the output is called the ripple. Though somewhat confusing, the transformer, diodes, and filter are often collectively called the rectifier circuit. Another circuit topology commonly encountered is the bridge rectifier. Figure 5.6 illustrates single-and three-phase versions of the circuit. In the single-Phase circuit diodes DI and D4 half-cycle of the input while D2 and D3 conduct on the negative half-cycle of the input. Alternate pairs of diodes conduct in the three-phase circuit depending on the relative amplitude of the source signals. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC may also make one diode type more desirable than another. Schottky diodes conduct current with one type of charge carrier and are therefore inherently faster to turn off than bipolar diodes. However, one of the limitations of Schottky diodes is their excessive forward voltage drop when designed to support reverse biases above about 200 V. Therefore, high-voltage diodes are the pn-junction type. The effects due to an increase in the temperature in a bipolar diode are many. The forward voltage drop during conduction will decrease over a large current range, the reverse leakage current will increase, and the reverse avalanche breakdown voltage (VBD) will increase as the device temperature climbs. A family of static characteristic curves highlighting these effects is shown in Fig. 5.3 where T3 > T2 > T1. In addition, a major effect on the switching characteristic is the increase in the reverse recovery time during turn-off. Some of the key parameters to be aware of when choosing a diode are its repetitive peak inverse voltage rating, VRRM (relates to the avalanche breakdown value), the peak forward surge current rating, IFSM (relates to the maximum allowable transient heating in the device), the average or rms current rating, IO (relates to the steady-state heating in the device), and the reverse recovery time, trr (relates to the switching speed of the device). Rectifiers This section discusses some simple uncontrolled rectifier circuits that are commonly encountered. The term uncontrolled refers to the absence of any control signal necessary to operate the primary switching elements (diodes) in the rectifier circuit. The discussion of controlled rectifier circuits, and the controlled switches themselves, is more appropriate in the context of power electronics applications [Hoft, 1986]. Rectifiers are the fundamental building block in dc power supplies of all types and in dc power transmission used by some electric utilities. A single-phase full-wave rectifier circuit with the accompanying input and output voltage waveforms is shown in Fig. 5.4. This topology makes use of a center-tapped transformer with each diode conducting on opposite half-cycles of the input voltage. The forward drop across the diodes is ignored on the output graph, which is a valid approximation if the peak voltages of the input and output are large compared to 1 V. The circuit changes a sinusoidal waveform with no dc component (zero average value) to one with a dc component of 2Vpeak/p. The rms value of the output is 0.707Vpeak. The dc value can be increased further by adding a low-pass filter in cascade with the output. The usual form of this filter is a shunt capacitor or an LC filter as shown in Fig. 5.5. The resonant frequency of the LC filter should be lower than the fundamental frequency of the rectifier output for effective performance. The ac portion of the output signal is reduced while the dc and rms values are increased by adding the filter. The remaining ac portion of the output is called the ripple. Though somewhat confusing, the transformer, diodes, and filter are often collectively called the rectifier circuit. Another circuit topology commonly encountered is the bridge rectifier. Figure 5.6 illustrates single- and three-phase versions of the circuit. In the single-phase circuit diodes D1 and D4 conduct on the positive half-cycle of the input while D2 and D3 conduct on the negative half-cycle of the input. Alternate pairs of diodes conduct in the three-phase circuit depending on the relative amplitude of the source signals. FIGURE 5.3 The effects of temperature variations on the forward voltage drop and the avalanche breakdown voltage in a bipolar diode
Conducting Diode D1 I D2 I DI I D2 I D1 ID2 FIGURE 5.4 A single-Phase full-wave rectifier circuit using a center-tapped transformer with the associated input and Filter FIGURE 5.5 A single-Phase full-wave rectifier with the addition of an output filter. 本D本D3 Filter D2 AD4 D32D5 te D4本D6 FIGURE 5.6 Single- and three-phase bridge rectifier circuits. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC FIGURE 5.4 A single-phase full-wave rectifier circuit using a center-tapped transformer with the associated input and output waveforms. FIGURE 5.5 A single-phase full-wave rectifier with the addition of an output filter. FIGURE 5.6 Single- and three-phase bridge rectifier circuits. Vin L C C + – Filter Load