Electronics: Inductors and Transformers 1.5 1.6

Inductors

• Inductors are closely related to capacitors, with the rate of current change in an inductor being proportional to the voltage applied across it.
• The defining equation for an inductor is V = L di/dt, where L is the inductance measured in henrys (or mH, μH, nH, etc.).
• Putting a constant voltage across an inductor causes the current to rise as a ramp, with 1 V across 1 H producing a current that increases at 1 amp per second.
• The energy invested in ramping up the current in an inductor is stored internally in the form of magnetic fields, with the formula UL = 1/2 LI2, where UL is in joules (watt seconds) for L in henrys and I in amperes.
• The symbol for an inductor looks like a coil of wire, which is its simplest form, and its behavior comes from it being a magnetic device.
• Inductors store energy in the magnetic field, and the power associated with inductive current is not turned into heat, but is stored as energy in the inductor's magnetic field.
• When the current is interrupted, the energy is released back, similar to a capacitor.
• The basic inductor is a coil, which can be wound on various core materials, such as iron or ferrite, to increase the inductance.
• The inductance L of a coil is proportional to the square of the number of turns, with the formula L ≈ K d2n2 / (18d + 40l) μH, where K is a constant depending on the units of diameter and length.
• Inductors are used in radiofrequency (RF) circuits, serving as RF "chokes" and as parts of tuned circuits.

Inductors vs. Capacitors

• Inductors are, in a real sense, the opposite of capacitors.
• Capacitors are more widely used in electronic circuits due to their ideal performance, whereas practical inductors have limitations such as winding resistance, core losses, and self-capacitance.

Applications of Inductors

• Inductors are used in switching power converters, such as synchronous buck converters and synchronous boost converters, which can increase or decrease a dc input voltage efficiently.
• These converters are discussed extensively in Chapter 9, with examples of representative types listed in Table 9.5.

Transformers

• A transformer consists of two closely coupled coils: primary and secondary.
• An AC voltage applied to the primary appears across the secondary, with a voltage multiplication proportional to the turns ratio of the transformer.
• The current multiplication is inversely proportional to the turns ratio, and power is conserved.
• Transformers are efficient, with output power nearly equal to input power.
• A step-up transformer gives higher voltage at lower current.
• The impedance of a transformer increases by a factor of n^2, where n is the turns ratio.
• With an unloaded secondary, there is very little primary current.

Power Transformers

• Power transformers serve two important functions in electronic instruments:
  • They change the AC line voltage to a useful (usually lower) value for the circuit.
  • They isolate the electronic device from actual connection to the powerline, because the windings of a transformer are electrically insulated from each other.
• They come in a variety of secondary voltages and currents, from 1 volt to several thousand volts, and from a few milliamps to hundreds of amps.
• Typical power transformers for electronic instruments have secondary voltages from 10 to 50 volts, with current ratings of 0.1 to 5 amps or so.

Other Types of Transformers

• Transformers are used in electronic power conversion, where plenty of power is flowing, but typically as pulse or square waveforms, and at much higher frequencies (50 kHz to 1 MHz).
• Transformers for signals at audio frequencies and radio frequencies are also available.
• At radio frequencies, tuned transformers are used if only a narrow range of frequencies is present.
• There is an interesting class of transmission-line transformers.

Limitations and Imperfections

• Transformers for use at high frequencies must use special core materials or construction to minimize core losses.
• Low-frequency transformers (e.g., AC powerline transformers) are burdened by large and heavy cores.
• Inductances associated with the transformer can cause issues:
  • Magnetizing inductance causes a primary current even with no secondary load.
  • Leakage inductance causes a voltage drop that depends on load current.
• Other departures from ideal performance include:
  • Winding resistance
  • Core losses
  • Capacitance
  • Magnetic coupling to the outside world

Diodes and Diode Circuits

• Diodes are nonlinear devices, meaning that a doubling of the applied signal does not produce a doubling of the response.
• Diodes are passive, two-terminal devices, unlike transistors which are active devices.

Diode Characteristics

• A diode has a V-I curve, which shows the relationship between voltage and current.
• The diode's arrow (anode terminal) points in the direction of forward current flow.
• The forward voltage drop is approximately 0.6V, which is the voltage difference between the anode and cathode when current is flowing from anode to cathode.
• The reverse current is measured in the nanoamp range and is typically not of consequence until the reverse breakdown voltage (peak inverse voltage, PIV) is reached.
• The reverse breakdown voltage is typically around 75V for a general-purpose diode.
• Diodes can be treated as ideal one-way conductors, with a forward voltage drop of around 0.5-0.8V.
• Other important characteristics of diodes include maximum forward current, capacitance, leakage current, and reverse recovery time.

Important Notes

• Diodes do not have a resistance and do not obey Ohm's law.
• Circuits with diodes do not have a Thévenin equivalent.

Diodes and Rectification

• A rectifier changes Alternating Current (AC) to Direct Current (DC), a fundamental application of diodes.
• The simplest rectifier circuit consists of a diode connected to an AC voltage source, typically provided by a transformer.

Rectifier Circuit Operation

• For a sinusoidal input voltage significantly larger than the forward drop (approximately 0.6 V for silicon diodes), the output voltage will resemble the waveform shown in Figure 1.57.
• Thinking of the diode as a one-way conductor can help understand the circuit's operation.

Diode Characteristics

• Various types of diodes have different characteristics, including maximum voltage ratings, typical current ratings, and capacitance values.
• Examples of diodes include PAD5, 1N4148, 1N4007, 1N5406, 1N6263, 1N5819, 1N5822, and MBRP40045.
• Schottky diodes have lower forward voltage and zero reverse-recovery time, but higher capacitance compared to other types of diodes.

Power-Supply Filtering

• The rectified waveforms need to be smoothed out to generate genuine dc.
• A large value capacitor is used to charge up to the peak output voltage during diode conduction.
• The capacitor stores energy (U = ½CV²) and provides the output current during discharging cycles.
• The capacitor value is chosen such that RloadC >> 1/f to ensure small ripple.

Ripple Voltage Calculation

• The load causes the capacitor to discharge, resulting in ripple voltage.
• The ripple voltage can be calculated using ΔV = Iload / fC (half-wave) or ΔV = Iload / 2fC (full-wave).
• The time constant for discharge should be much longer than the time between recharging.

Rectifier Configurations

• A full-wave bridge with an output storage capacitor can be used for power supplies.
• The diode drop can be eliminated using active switching or synchronous switching.

Important Notes

• Don't memorize formulas; instead, learn how to derive them.
• Consider component tolerances and design conservatively when building power supplies.

Rectifier Configurations for Power Supplies

• A dc power supply using a bridge circuit can be bought as a prepackaged module with current ratings ranging from 1 A to 25 A or more, and minimum breakdown voltages from 100 V to 1000 V.

Full-Wave Bridge Rectifier

• The full-wave bridge rectifier is the most efficient circuit in terms of transformer design.
• It produces a higher output voltage compared to a center-tapped full-wave rectifier.
• The power lost to heating in the transformer secondary windings is reduced by a factor of 2 for the bridge configuration.

Center-Tapped Full-Wave Rectifier

• The center-tapped full-wave rectifier circuit produces an output voltage that is half of what you get with a bridge rectifier.
• It is not the most efficient circuit in terms of transformer design, as each half of the secondary is used only half the time.
• To deliver the same output power, each half winding in the center-tapped circuit must supply the same current as the parallel pair in the bridge circuit, resulting in higher power loss due to I2R heating.

Split Supply

• The split supply circuit provides equal plus and minus voltages, which many circuits need.
• It is an efficient circuit, as both halves of the input waveform are used in each winding section.

Voltage Multipliers

• The voltage doubler circuit is a full-wave rectifier circuit that uses both halves of the input waveform.
• The ripple frequency is twice the ac frequency (120 Hz for 60 Hz line voltage in the United States).
• Variations of the voltage doubler circuit exist for voltage triplers, quadruplers, etc.
• These circuits can be extended to produce high dc voltages, such as in Cockcroft-Walton generators, used in applications like particle accelerators and image intensifiers.

Reducing Ripple Voltage

• Reducing ripple voltage to a desired level can be achieved by using sufficiently large capacitors.
• However, this approach has three disadvantages:
  • Required capacitors may be prohibitively bulky and expensive.
  • Very short interval of current flow during each cycle produces more I2R heating.
  • Even with reduced ripple, output voltage variations due to other causes still exist.

Voltage Multipliers

• Voltage multipliers can be configured to not require a floating voltage source (Figure 1.67).
• Examples of voltage multipliers include doubler, tripler, and quadrupler configurations.

Regulated DC Power Supply

• A better approach to power-supply design is to use enough capacitance to reduce ripple to low levels (perhaps 10% of the dc voltage).
• Then, use an active feedback circuit to eliminate the remaining ripple (Figure 1.68).
• This is known as a “linear regulated dc power supply”.
• Voltage regulators are used universally as power supplies for electronic circuits.
• Complete voltage regulators are available as inexpensive ICs (priced under $1).
• A power supply built with a voltage regulator can be made easily adjustable and self-protecting (against short circuits, overheating, etc.), with excellent properties as a voltage source.

Diode Applications

• A diode can be used to make a waveform of one polarity only, known as a signal rectifier.
• The easiest way to achieve this is to rectify the differentiated wave.
• The circuit will not give an output for square waves smaller than 0.6 V pp due to the diode's forward drop of approximately 0.6 V.
• To circumvent this limitation, hot carrier diodes (Schottky diodes) can be used, which have a forward drop of about 0.25 V.
• Another solution is to use a diode to provide bias to hold the diode at the threshold of conduction, compensating for the forward drop.

Signal Rectifier

• A signal rectifier can be used to produce a waveform of one polarity only.
• The circuit can be used to produce a train of pulses corresponding to the rising edge of a square wave.

Diode Gates

• Diodes can be used to pass the higher of two voltages without affecting the lower.
• An example of this is battery backup, where a diode is used to keep a circuit running even when the device is switched off.

Diode Clamps

• Diodes can be used to limit the range of a signal to prevent it from exceeding certain voltage limits.
• The circuit will prevent the output from exceeding about +5.6 V, with no effect on voltages less than that.
• The series resistor limits the diode current during clamping action.

Voltage Divider Providing Clamping Voltage

• A voltage divider can be used to provide the reference voltage for a clamp.
• The resistance looking into the voltage divider (Rvd) must be small compared to R.

Symmetrical Clamp

• A symmetrical clamp confines a signal to the range −5.6 to +5.6 V.
• Exercise 1.22: Design a symmetrical clamp.

Diode Limiter

• A diode limiter limits the output "swing" to one diode drop in either polarity, roughly ±0.6 V.
• This circuit is often used as input protection for a high-gain amplifier.

Diodes as Nonlinear Elements

• The forward current through a diode is proportional to an exponential function of the voltage across it at a given temperature.
• Diodes can be used to generate an output voltage proportional to the logarithm of a current.
• The diode's nonlinear V-I curve can be exploited to create a logarithmic converter.

Logarithmic Converter

• A logarithmic converter can be created using a diode and a resistor.
• The output voltage is proportional to the logarithm of the input current.
• The diode drop compensation method can be used to improve the accuracy of the logarithmic converter.
• The circuit can be designed to be insensitive to changes in temperature.

Inductive Loads and Diode Protection

• When a switch is opened that is providing current to an inductor, the voltage across the inductor rises abruptly to keep the current flowing.
• This is because inductors have the property V = L dI/dt, making it impossible to suddenly turn off the current without implying an infinite voltage across the inductor's terminals.
• The inductor tries to maintain the current flow by forcing a high positive voltage across its terminals, which can damage the switch or transistor.
• A diode can be used to protect the switch from inductive kick by placing it across the inductor, which becomes forward-biased when the switch is turned off.
• The diode must be able to handle the initial current that was flowing through the inductor, and a 1N4004 diode is suitable for most cases.
• The disadvantage of this protection circuit is that it lengthens the decay of current through the inductor.

Alternative Protection Methods

• For applications where the current must decay quickly, a resistor can be used instead of a diode, chosen so that Vsupply + IR is less than the maximum allowed voltage across the switch.
• A zener with a series diode can also be used to provide a linear-like ramp-down of current.
• For inductors driven from an AC source (e.g., transformers, AC relays), a diode protection will not work, and an RC "snubber" network can be used instead.
• A bidirectional zener-like voltage-clamping element, such as a TVS zener or metal-oxide varistor (MOV), can also be used for transient voltage protection.

Additional Protection Methods

• An RC snubber network can be used to protect against inductive kick in instruments running from the AC power line.
• A transient suppressor, such as a TVS zener or MOV, can be included across the AC powerline terminals to prevent inductive spike interference and protect the instrument from powerline spikes.

Inductors as Friends

• Inductors can be useful in certain circuits, such as charging capacitors, where they can provide advantages over traditional resistors.
• In a capacitor charging circuit, using an inductor instead of a resistor can provide a lossless charge, whereas a resistor would lose half the energy as heat.

Resonant Charging

• Resonant charging uses an inductor and capacitor in a resonant circuit to charge the capacitor.
• The output-voltage waveform is a sinusoidal half-cycle at the resonant frequency f = 1/2π√LC.
• Charging is complete after a half-cycle of the resonant frequency, and the capacitor is charged to twice the input voltage.

Mechanical Analogy

• A mechanical analogy for resistive charging is dropping packages onto a conveyor belt, where 50% of the energy is lost as the packages accelerate to the belt speed.
• A mechanical analogy for reactive charging is using a conveyor belt with springs to accelerate packages to twice the belt speed, with no energy lost.

Applications of Resonant Charging

• Resonant charging is used in high-voltage supplies for flashlamps and stroboscopes, allowing for full charge between flashes and no current immediately after discharge.

Frequency Analysis

• Frequency analysis is used to visualize and understand the frequency components of a signal, such as in audio equalization.
• The audible frequency range is typically considered to be between 20 Hz and 20 kHz.