# DC & AC

DC voltage has a fixed polarity, and direct current flows in only one direction. Alternating voltage reverses in polarity and amplitude periodically with time. The resulting alternating current (AC) periodically reverses in direction and varies in amplitude with the changing values of voltage.

Both types of signals are important as most electronic circuits include AC and DC voltages and currents.

# The Sine Wave

The sine wave (sinusoidal wave or, simply, sinusoid) is the fundamental form of alternating current (AC) and voltage. The current reverses polarity over time. In one cycle, the polarity changes once. The time required for a given sine wave to complete one full cycle is called a period.

The number of cycles per second is the frequency (f), whose unit is the Hertz (Hz). One hertz is equal to one cycle per second. The frequency and period are reciprocal. More cycles per second results in a higher frequency and a shorter period.

# AC Voltage Sources

Sine waves are produced electro-magnetically by an AC generator or electronically by an oscillator circuit, which is used in a signal generator.

The figure shows a cross-section of an AC generator. A simplified model of this generator consists of a single loop of wire in a permanent magnetic field. Magnetic flux lines exist around the north and south poles of the magnet. When a conductor rotates through the magnetic field, a voltage is induced.

# AC Generator

In a horizontal starting position the loop does not induce a voltage because the conductors are not cutting across the magnetic flux lines. As the loop rotates through the first quarter of the cycle, it cuts through the flux lines producing the maximum induced voltage. During the second quarter of the cycle, the voltage decreases from its positive maximum back to zero. During the second half of revolution, the wire loop cuts through the magnetic field in the opposite direction. Thus, the induced voltage has the opposite polarity. After one complete revolution of the loop, one full cycle of the sinusoidal voltage has been completed.

# Measurement of Sine Waves

At any point in time on a sine wave, the voltage has an instantaneous value. As a cycle represents a continuous set of instantaneous values, other dimensions have been defined to enable comparing one wave to another. The peak value Up is the maximum value. It applies to either the positive or negative peak. The peak-to-peak value,Upp, is the voltage (or current) from the positive peak to the negative peak. Average value is an arithmetic average of all the values in a sine wave for one half-cycle, where Uavr = 0.637 Up

# RMS Value

To compare AC and DC voltage, the effective value of the AC voltage should be calculated using the root-mean-square (rms) value of the sinusoidal voltage. The (rms) value of a sinusoidal voltage or current is equal to the dc voltage and current that produces the same heating effect. The formula is Urms = 0.707 Up. The factor 0.707 for rms value is derived as the square root of the average (mean) of all the squares of the sine wave. To convert from rms to peak value, the formula Up = 1.414 Urms is used. Unless indicated otherwise, all sine wave ac measurements are in rms values.

# Phase Angle

The angular measurement of a sine wave can be related to the angular rotation of an AC generator, as shown in the diagram above. It is based on 360o of rotation for the complete cycle of a sine wave. The diagram shows angles in degrees over the full cycle of a sine wave. Since 360o = 2π rad, angles can be also expressed in radians using the formula in the illustration above.

The phase angle of a sine wave specifies the position of that sine wave relative to a reference. The illustration shows the phase shifts of a sine wave. There is a phase angle of 30o between sine wave A and sine wave B.

# Laws of Resistive AC Circuits

Ohm's Law and Kirchoff's Law apply to AC circuits in the same way they apply to DC circuits. If a sinusoidal voltage is applied across a resistor, there is a sinusoidal current. It is zero when the voltage is zero, and is max. when the voltage is max. The voltage and the current are in phase with each other.

In a resistive circuit that has an AC voltage source, the source voltage is the sum of all the voltage drops, just as in a DC circuit. Remember, both the voltage and the current must be expressed in the same way, i.e., both in rms, both in peak, etc.

# Superimposed DC and AC Voltages

Many practical circuits use both DC and AC voltages combined. For example, an amplifier needs DC voltages in order to conduct any current. When a small AC signal is applied to the input, the resulting output of an amplifier consists of DC with a superimposed AC signal.

The figure shows DC and AC sources in series. These two voltages add up algebraically. If Udc is greater than the peak value of the sinusoidal voltage, the combined AC and DC voltage is a sine wave that never reverses polarity. That is, the sine wave is riding on a DC level. If Udc is less than the Up, the sine wave will be negative during a portion of its lower half-cycle.

# Non-Sinusoidal AC Waveforms

The pulse and the triangular waveform are the other two major types of signals widely used in electronics. Any waveform that repeats itself at fixed intervals is periodic. The periodis denoted with a T. Triangular waveforms are formed by voltage or current ramps. A ramp is a linear increase or decrease in the voltage.

An ideal pulse consists of two equal but opposite steps separated by an interval of time called the pulse width. The duty cycle of the pulse is the ratio of the pulse width to the period and is usually expressed in a percentage.

# Capacitor & Capacitance

The capacitor is a device that can store an electrical charge (Q). It is composed of two conductive plates separated by an insulator (also called a dielectric). Connecting leads are attached to the parallel plates. The amount of charge that a capacitor can store per unit of voltage across its plates is called capacitance, (designated C). The greater the charge that a capacitor can store for a given voltage, the higher its capacitance value.

The Farad (F) is the basic unit of capacitance. Most capacitors that are used in electronics have capacitance values in microfarads (µF) and picofarads (pF).

# Charge Storage

In a neutral state, both plates of a capacitor have an equal number of free electrons. When a capacitor is connected to a DC voltage source through a resistor, the source moves electrons away from plate A through the circuit to plate B. As plate A loses electrons and plate B gains electrons, plate A becomes positive with respect to plate B. Remember that no electrons can flow through the insulator. The movements of electrons stop when the voltage across the capacitor equals the source voltage. The charge remains in the capacitor if it is disconnected from the source.

# Energy Storage

The opposite charges on the plates of a capacitor create many lines of force. They form an electrical field between the plates that store energy within the dielectric. The greater the forces between the charges on the plates of the capacitor, the greater the amount of energy stored.

The amount of stored charge is directly proportional to the voltage and the capacitance (Q = CU). The amount of stored energy depends on the square of the voltage across the plates of the capacitor.

# Types of Capacitors

Capacitors are normally classified according to the type of dielectric material used. The most common types are mica, ceramic, plastic-film, and electrolytic (aluminum oxide, tantalum oxide) capacitors. Capacitors are also available as surface-mounted components. They are called chip capacitors.

Capacitor values are indicated on the capacitor body either by numerical or alphanumerical labels or sometimes by color code. Capacitor labels indicate various parameters such as capacitance, voltage rating, and tolerance.

The illustration shows the basic construction of mica, ceramic, and plastic-film capacitors.

# Electrolytic Capacitors

Electrolytic capacitors offer much higher capacitance values than mica or ceramic capacitors and their voltage ratings are typically higher. Aluminum electrolytic capacitors are the most commonly used type. Tantalum capacitors have larger C, in smaller size, but they cost more than the aluminum type. Electrolytic capacitors are the only capacitors that require the observation of polarity when connecting to a circuit. Reversal of the voltage polarity can completely destroy a capacitor. The illustration shows typical electrolytic capacitors and a cut-away view of a teardrop-shaped tantalum capacitor.

# Variable Capacitors

Variable capacitors are used in a circuit where it is necessary to adjust the capacitance value, for example, in radio or TV tuners. The schematic symbol for a variable capacitor is shown above.

Adjustable capacitors that normally have slotted screw-type adjustments are called trimmers. They are used for very fine adjustments in a circuit. Ceramic or mica is a common dielectric in these capacitor types. Capacitance value is directly related to plate area A, and inversely related to plate separation d. For this reason, it is usually changed by adjusting one of these parameters.

# Series Capacitors

When the capacitors are connected in series, the total capacitance is less than the smallest capacitance value.

Both capacitors store the same amount of charge. The voltage across each one depends on its capacitance value (U = Q/C). By Kirchoff's voltage law, the sum of the capacitor voltages equals the source voltage (Us = U1 + U2). Since U = Q/C and Q = QT = Q1 = Q2 the relationship for two capacitors in series is derived. It can be extended to any number of capacitors in series as shown in the diagram.

# Parallel Capacitors

When capacitors are connected in parallel, the total capacitance is the sum of the individual capacitances. When the switch is closed, part of the total charge is stored by C1 and part is stored by C2. The portion of the total charge that is stored by each capacitor depends on its capacitance value (Q = CU). Since the voltage across both capacitors is the same, the larger capacitor stores more charge. The charges stored by both capacitors equals the total charge that was delivered from the source (QT = Q1 + Q2). Because all the voltages are the same, the CT is the sum of both capacitances.

# Capacitors in DC Circuit (Charging)

Charging and discharging are the main effects of capacitors. A capacitor charges when it is connected to a DC voltage source through a resistor. Initially, when the switch is open, there is no voltage flowing across the capacitor's plate. When the switch is closed, the current jumps to its maximum value, and the capacitor begins to charge. As the charging process continues, the current decreases, and the voltage across the capacitor increases until it equals the source voltage. When the capacitor is fully charged, there is no current. A capacitor blocks constant DC. A capacitor charges following a nonlinear curve as shown above.

# Capacitors in DC Circuit (Discharging)

The capacitor discharges when a conducting path is provided across the plates without any applied voltage. A capacitor can serve as a voltage source, temporarily, by producing a discharge current. When the switch is closed, the capacitor begins to discharge. Initially, the current jumps to a maximum. The direction of the current during discharge is opposite to that of the charging current. During discharging, the current and capacitor voltages decrease. When the capacitor has fully discharged, the current and the capacitor voltage are zero. The discharging curve is shown in the illustration above.

# RC Time Constant

A certain time is required for the capacitor to fully charge or discharge. The rate at which the capacitor charges or discharges is determined by the RC time constant of the circuit. It is symbolized by Greek letter τ (Tau), where τ = RC. When R is increased, the charging current is reduced, thus increasing the charging time of the capacitor. When C is increased, the amount of charge increases, thus, more time is required to charge capacitor for the same current. In one time constant, the capacitor voltage changes approximately 63%. It reaches its final value at approximately 5τ.

# Capacitor Testing

A capacitor can be checked with an ohmmeter. First, it is removed from a circuit. Next, the capacitor leads are shorted to fully discharge it. The ohmmeter (set on a high ohms range such as R x 1Mohm) is connected across the capacitor. For a good capacitor, the meter pointer moves quickly towards near zero ohms. The pointer move slowly toward infinity as the capacitor charges from the ohmmeter's battery. When a capacitor is fully charged, the pointer is at infinity. If the capacitor is internally shorted, the meter will go to zero and stay there. If it is open, the meter will stay at infinity.

# Capacitor Testing - Leakage

After the charging voltage is removed, a perfect capacitor would keep its charge indefinitely. However, there is no perfect insulator. The dielectric of any capacitor will conduct very small amounts of leakage current. Thus, after some period of time, the capacitor will loose its charge. An equivalent circuit for a non-ideal capacitor is shown in the diagram. Such capacitors may cause trouble in high-resistance circuits.

During testing, if the capacitor shows charging, but the final resistance reading is appreciably less than normal, the capacitor is leaky.

# Capacitor in AC Circuit

The most important property of a capacitor is its ability to block a steady DC voltage, while passing AC signals.

In the illustration above, the capacitor is connected to a sinusoidal voltage source. Current is always leading the capacitor voltage by 90o. If the source voltage has a constant amplitude value and its frequency is increased, the amplitude of the current increases accordingly. Further, when the frequency of the source decreases, the current amplitude decreases. Therefore, the capacitor offers opposition to current, which varies inversely with frequency.

# Capacitive Reactance

The opposition to sinusoidal current in a capacitor is called capacitive reactance. The symbol is Xc, and its unit is the ohm. Xc varies inversely not only with frequency but with capacitance as well. When a sinusoidal voltage with a fixed amplitude and fixed frequency is applied to a capacitor with given value, there is a constant amount of AC current. When the capacitance value is increased, the current increases. The formula for Xc is shown above. Ohm's Law applies to capacitive circuits as follows: U = I Xc.

# Series RC Circuit

In a series RC circuit, the current is the same through both the resistor and capacitor. Thus, the resistor voltage (UR) is in phase with the current (I), and the capacitor voltage (UC) lags the current by 90o. Therefore, there is a phase difference of 90o between UR and UC as shown above. From Kirchoff's voltage law, the sum of the voltage drops must equal the source voltage, Us. Since UR and UC are 90o out of phase, the magnitude of the source voltage can be expressed by using the Pythagorean theorem, as shown in the diagram.

# Capacitive Impedance

The impedance Z of an RC circuit is the complete opposite to sinusoidal current. Its unit is the ohm. The phase angle is the phase difference between the total current and the source voltage. In a purely resistive circuit, the impedance is equal to total resistance. The phase angle is zero. In a purely capacitive circuit, the impedance is the total capacitive reactance. The phase angle is 90o, with the current leading the voltage. The impedance, Z, of a series RC circuit, depends on both the R and the C reactance values. It is determined by the impedance triangle shown. The phase angle is between zero and 90o.

# ZC Frequency Dependence

Capacitance reactance Xc varies inversely with frequency. Impedance Z changes in the same way as Xc. Therefore, in RC circuits, Z is inversely related to frequency. The diagram illustrates how Z and Xc change with frequency, with the source voltage held at a constant value. As the frequency increases, Xc decreases. Less voltage is dropped across the capacitor since Uc = I Xc. Also, Z decreases as Xc decreases, causing the current to increase. An increase in I causes more voltage across R as UR = IR.

# Analysis of Series RC Circuit

Ohm's Law and Kirchoff's Law are used in the analysis of a series RC circuit. Ohm's Law, when applied to a series RC circuit, involves the use of quantities of Z, U, and I. The three equivalent forms of Ohm's Law are shown above. From Kirchoff's voltage law, the sum of the voltage drops equals the source voltage (Us). Since UR and UC are 90o out of phase, the magnitude of the source voltage is expressed by the voltage triangle as shown in the illustration.

# Analysis of Parallel RC Circuit

The source voltage appears across both the resistive and the capacitive branches. Therefore, Us, UR and Uc are all in phase and of the same magnitude. In a parallel circuit, each branch has its individual current. The resistive branch current IR is in phase with Us, but the capacitive branch current Ic leads Us by 90o. By Kirchoff's current law, the total current is the phasor sum of the two branch currents. The impedance of a parallel circuit equals the applied voltage divided by the total current Zeq = Us / I.

A permanent magnet has a magnetic field around it, which consists of lines of force, or flux lines Φ, going from the north pole (N) to the south pole (S). Moving a magnet relative to a coil of wire and thus cutting across the flux lines induces a current through the coil.

Faraday's Law states: The induced voltage uind is directly proportional to the rate of change of the magnetic field with respect to the coil and the number of turns in the coil. A coil with more turns (loops), produces a greater voltage. The faster the magnet is moved, the greater the induced voltage.

# Basic Inductor

A coil of wire forms a basic inductor. Current through the coil produces an electromagnetic field, which creates a north (N) and a south (S) pole. The more lines of force, the greater the flux, and the stronger the magnetic field.

Constant current has an associated constant magnetic field and there is no induced voltage. An increase in current expands the field. A decrease in current reduces it. As the field expands and collapses with current change, the flux Φ is effectively in motion. Hence, a varying current can produce induced voltage without magnetic motion.

# Inductance

Inductance is the ability of a conductor to produce induced voltage when the current varies. Conductors that introduce a definite inductance into the circuit are called inductors or coils. The symbol for inductance is L, and the unit is the Henry (H). The inductance is one Henry when the current, changing at the rate of 1A per second, induces 1V across the coil.

An inductor stores energy in the magnetic field created by the current. The energy stored is proportional to the inductance and the square of the current. The energy is supplied by the voltage source that produces the current.

# Lenz's Law

When the current through a coil changes, a voltage is induced. Lenz's Law states that the polarity of the induced voltage always opposes the change in current that caused it. The diagram above illustrates this law. When the switch closes, the current tries to increase, and the magnetic field starts expanding. The expanding magnetic field induces a voltage, which opposes an increase in current. So, at the instant of switching, the current remains the same. When the rate of expansion decreases, the induced voltage decreases, allowing the current to increase. As the current reaches a constant value, there is no induced voltage.

# Lenz's Law

The diagram illustrates the direction of induced voltage when the current is switched off. In a steady-state condition, the current has a constant value. There is no induced voltage because the magnetic field is unchanging. If the switch is opened, the current tries to reduce, and the magnetic field begins to collapse. At the time of switching, the induced voltage has a direction that prevents any decrease in current. The current remains the same as prior to the switch opening. When the rate of collapse decreases, induced voltage decreases, allowing current to decrease to zero value.

# Types of Inductors

An inductor is basically a coil of wire. The material around which the coil is formed is called the core. Both fixed and variable inductors are classified according to the type of core material used. Three common types are: the air core, the iron core, and the ferrite core. Each has a unique symbol, shown above. Inductors are made in a variety of shapes and sizes. Some are shown above. Small fixed inductors are encapsulated in an insulating material and have the appearance of a small resistor. Variable inductors usually have a screw type adjustment, to allow inductance to be changed.

# Series and Parallel Inductors

When inductors are connected in series, the total inductance, LT, is the sum of the individual inductors. The formula is similar to total resistance in series and total capacitance in parallel.

When inductors are connected in parallel the total inductance is less than the smallest inductance. The reciprocal of the total inductance is equal to the sum of the reciprocals of the individual inductances. The formula is similar to the formula for total parallel resistance and total series capacitance.

# Inductors in DC Circuits - The Time Constant

An inductor will energize when it is connected to a DC voltage source. When there is constant direct current in an inductor, there is no induced voltage. The inductance itself appears as a short to DC. The induced voltage always opposes any current change. Thus, a certain amount of time is required for the current to change from one value to another. The rate at which the current changes is determined by the RL time constant, τ. It depends on both the inductance and the resistance of a circuit. Current in an inductor changes exponentially by 63% within one time constant. It reaches its final value at approximately 5τ.

# Testing Inductors

An inductor is made of wire material that has a certain resistance per unit of length. Thus, the coil, with a certain number of turns, has the inherent resistance called the DC resistance or the winding resistance. The value of the winding resistance can be from one ohm to several hundreds ohms.

Inductors can be tested with an ohmmeter. If the coil is good the ohmmeter will show the winding resistance. The most common failure in an inductor is an open coil. If there is an open coil, an ohmmeter check will indicate infinite resistance.

# Inductors in AC Circuits

Inductors are able to pass a DC current and block an AC current. In the illustration above, an inductor is connected to a sinusoidal voltage source. The current lags the inductor voltage by 90o. The source voltage is held at a constant amplitude. If the frequency increases, the rate of change also increases, and more voltage is induced across the inductor in a direction opposite to the current. This causes the current to decrease in amplitude when the frequency increases.

Similarly, a decrease in frequency will cause an increase in current. Thus, the inductor offers opposition to the current, and that opposition varies directly with the frequency.

# Inductive Reactance

The opposition to sinusoidal current in an inductance is called inductive reactance. The symbol is XL, and its unit is the ohm. XL varies directly, not only with frequency, but with inductance as well. If a sinusoidal voltage with a fixed amplitude and fixed frequency is applied to an inductor with a certain inductance, there is a constant amount of AC current. When the inductance value is increased, the current decreases. So XL is directly proportional to fL. Ohm's law applies to an inductive circuit as follows: U = I XL.

# Series RL Circuit

In a series RL circuit, the current is the same through both the resistor and the inductor. Thus, the resistor voltage UR is in phase with the current I, and the inducted voltage UL leads the current by 90o. Therefore, there is a phase difference of 90o between UR and UL as shown above. From Kirchoff's voltage law the sum of the voltage drops must equal the source voltage, Us. Since UR and UL are 90o out of phase, the magnitude of the source voltage can be expressed by using the Pythagorean theorem, as shown above.

# Inductive Impedance

The impedance Z of an RL circuit is the total opposition to sinusoidal current. Its unit is the ohm. The phase angle is the phase difference between the total current and the source voltage. In a purely resistive circuit, the impedance is equal to total resistance. The phase angle is zero. In a purely inductive circuit, the impedance is the total inductive reactance. The phase angle is 90o, and inductor voltage leads the current. The impedance Z of a series RL circuit depends on both the R and the L reactance values. It is determined by the impedance triangle as shown. The phase angle is between zero and 90o.

# ZL Frequency Dependence

Inductive reactance XL varies directly with frequency. Impedance Z changes in the same way as XL. Therefore, in RL circuits, Z is directly dependent on frequency. The diagram illustrates how Z and XL change with frequency, with the source voltage held at a constant value. As the frequency increases, XL increases. More voltage is dropped across the inductor, since UL = IXL. Also, Z increases as XL increases, causing the current to decrease. An decrease in I causes less voltage across R as UR = IR.

# Analysis of a Series RL Circuit

Ohm's Law and Kirchoff's law are used in the analysis of a series RL circuit. When Ohm's law is applied to a series RL circuit, it involves the use of quantities of Z, U, and I. The three equivalent forms of Ohm's law are shown above. From Kirchoff's voltage law, the sum of the voltage drops equals the source voltage Us. Since UR and UL are 90o out of phase, the magnitude of the source voltage is expressed by the voltage triangle as shown in the diagram.

# Analysis of Parallel RL Circuit

The source voltage appears across both the resistive and the capacitive branches. Thus, Us, UR and UL are all in phase and of the same magnitude. In parallel circuits, each branch has an individual current. The resistive branch current IR is in phase with Us, but the inductive branch current IL lags Us and the resistor current by 90o. By Kirchoff's current law, the total current is the phasor sum of the two branch currents. The impedance of a parallel circuit equals the applied voltage divided by the total current Zeq = Us / I.

# Troubleshooting

When there is an open coil in a series RL circuit, there is no path for the current. Therefore, the resistor voltage is zero, and the total source voltage appears across the open inductor. When the resistor is open, there is no current, and the inductor voltage is zero. The total source voltage appears across the open resistor.

In a parallel RL circuit, an open resistor or inductor will cause the total current to decrease, because the total impedance will increase. The branch with an open component will have zero current.

# Practice Sine Wave Values

Determine Up, Upp, Urms and the half-cycle Uavr for the sine wave illustrated on the scope screen display. Use the oscilloscope settings for the volts/division and sec/division, which are indicated under the screen.

Change the amplitude of the signal from the signal generator. Using the new oscilloscope settings, calculate the sine wave values mentioned above. # Practice Peak & RMS Values

Observe the instrument readings. Explain why, for the same voltage (source voltage), instruments read two different values? What is the period of the sine wave ? # Practice AC Signals (Period & Frequency)

Determine the peak-to-peak values (period and frequency) of each sine wave from the oscilloscope screen displays and the settings for volts/division and sec/division that are indicated under the screen. What is the duty cycle for the pulse signal ? # Troubleshooting Series RC Circuits

Observe the instrument readings. Determine the type of failure in the circuit. Decide which component has failed. # Troubleshooting Series RL Circuits

Observe the instrument readings. Determine the type of failure in the circuit. Decide which component has failed.

First, make sure that the correct source voltage is applied to the input (measure amplitude and frequency from the scope screen). Next, check resistors for correct values. Disconnect the circuit from the source voltage and check to see if the resistors have such a small a value that the voltage across it could be negligible. Next, connect the source voltage to the circuit and measure the voltage drops across each inductor. Analyze obtained results. 