## Introduction |

## Components |

## Teaching Aims |

## DC & AC |

Both types of signals are important as most electronic circuits include |

## The Sine Wave |

The The number of cycles per second is the |

## AC Voltage Sources |

Sine waves are produced electro-magnetically by an signal generator. The figure shows a cross-section of an |

## 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 peak-to-peak value,U, is the voltage (or current) from the positive peak to the negative peak. _{pp}Average value is an arithmetic average of all the values in a sine wave for one half-cycle, where U = 0.637 _{avr}U _{p} |

## RMS Value |

To compare AC and DC voltage, the ( value of a sinusoidal voltage or current is equal to the rms)dc voltage and current that produces the same heating effect. The formula is U = 0.707 _{rms}U. The factor 0.707 for _{p}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 U = 1.414 _{p}U is used. Unless indicated otherwise, all sine wave _{rms}ac measurements are in rms values. |

## Phase Angle |

The angular measurement of a sine wave can be related to the angular rotation of an The |

## Laws of Resistive AC Circuits |

In a |

## Superimposed DC and AC Voltages |

Many practical circuits use both The figure shows AC and DC voltage is a sine wave that never reverses polarity. That is, the sine wave is riding on a DC level. If U is less than the _{dc}U, the sine wave will be negative during a portion of its lower half-cycle. _{p} |

## Non-Sinusoidal AC Waveforms |

The An ideal |

## Capacitor & Capacitance |

The The |

## 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 |

## 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 ( |

## Types of Capacitors |

Capacitors are normally classified according to the type of dielectric material used. The most common types are 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 |

## 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 |

## Series Capacitors |

When the capacitors are connected Both capacitors store the same amount of charge. The voltage across each one depends on its capacitance value ( U + _{1}U). Since _{2}U = Q/C and Q = Q = _{T}Q = _{1}Q 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. _{2} |

## Parallel Capacitors |

When capacitors are connected C. The portion of the total charge that is stored by each capacitor depends on its capacitance value (_{2}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 (Q = _{T}Q + _{1}Q). Because all the voltages are the same, the _{2}C is the sum of both capacitances. _{T} |

## Capacitors in DC Circuit (Charging) |

Charging and discharging are the main effects of capacitors. A capacitor charges when it is connected to a |

## Capacitors in DC Circuit (Discharging) |

The capacitor discharges when a conducting path is provided across the plates |

## 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. 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 |

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

## Capacitor in AC Circuit |

The most important property of a capacitor is its ability to passing .AC signalsIn the illustration above, the capacitor is connected to a sinusoidal voltage source. Current is always opposition to current, which varies inversely with frequency. |

## Capacitive Reactance |

The opposition to sinusoidal current in a capacitor is called X 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 _{c}AC current. When the capacitance value is increased, the current increases. The formula for X is shown above. Ohm's Law applies to capacitive circuits as follows: _{c}U = I X. _{c} |

## Series RC Circuit |

In a I), and the capacitor voltage (U) lags the current by 90_{C}^{o}. Therefore, there is a phase difference of 90^{o} between U and _{R}U as shown above. From Kirchoff's voltage law, the sum of the voltage drops must equal the source voltage, _{C}U. Since _{s}U and _{R}U are 90_{C}^{o} 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 |

## Z |

Capacitance reactance X. Therefore, in _{c}RC circuits, Z is inversely related to frequency. The diagram illustrates how Z and X change with frequency, with the source voltage held at a constant value. As the frequency increases, _{c}X decreases. Less voltage is dropped across the capacitor since _{c}U = _{c}I X. Also, Z decreases as _{c}X decreases, causing the current to increase. An increase in _{c}I causes more voltage across R as U = _{R}IR. |

## Analysis of Series RC Circuit |

Ohm's Law and Kirchoff's Law are used in the analysis of a series U and _{R}U are 90_{C}^{o} 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, U and _{R}U are all in phase and of the same magnitude. In a parallel circuit, each branch has its individual current. The resistive branch current _{c}I is in phase with _{R}U, but the capacitive branch current _{s}I leads _{c}U by 90_{s}^{o}. 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 Z_{eq} = U. _{s} / I |

## Faraday's Law |

A permanent magnet has a magnetic field around it, which consists of lines of force, or flux lines Φ, going from the north pole (
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 |

## Inductance |

An inductor |

## Lenz's Law |

When the current through a coil changes, a voltage is induced. Lenz's Law states that the |

## 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 |

## Series and Parallel Inductors |

When inductors are connected in series, the total inductance, 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 |

## 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 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 |

## Inductors in AC Circuits |

block an current. In the illustration above, an inductor is connected to a sinusoidal voltage source. The current AClags the inductor voltage by 90. 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. ^{o}Similarly, a decrease in frequency will cause an increase in current. Thus, the inductor offers opposition to the current, and that |

## Inductive Reactance |

The opposition to sinusoidal current in an inductance is called X 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 _{L}AC current. When the inductance value is increased, the current decreases. So X is directly proportional to _{L}fL. Ohm's law applies to an inductive circuit as follows: U = I X. _{L} |

## Series RL Circuit |

In a I, and the inducted voltage U leads the current by 90_{L}^{o}. Therefore, there is a phase difference of 90^{o} between U and _{R}U as shown above. From Kirchoff's voltage law the sum of the voltage drops must equal the source voltage, _{L}U. Since _{s}U and _{R}U are 90_{L}^{o} out of phase, the magnitude of the source voltage can be expressed by using the Pythagorean theorem, as shown above. |

## Inductive Impedance |

The |

## Z |

Inductive reactance X. Therefore, in _{L}RL circuits, Z is directly dependent on frequency. The diagram illustrates how Z and X change with frequency, with the source voltage held at a constant value. As the frequency increases, _{L}X increases. More voltage is dropped across the inductor, since _{L}U = _{L}IX. Also, Z increases as _{L}X increases, causing the current to decrease. An decrease in _{L}I causes less voltage across R as U = _{R}IR. |

## Analysis of a Series RL Circuit |

Ohm's Law and Kirchoff's law are used in the analysis of a series U and _{R}U are 90_{L}^{o} 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, U and _{R}U are all in phase and of the same magnitude. In parallel circuits, each branch has an individual current. The resistive branch current _{L}I is in phase with _{R}U, but the inductive branch current _{s}I lags _{L}U and the resistor current by 90_{s}^{o}. 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 Z = _{eq}U. _{s} / I |

## Troubleshooting |

When there is an In a parallel |

## Practice Sine Wave Values |

Determine U, _{pp}U and the half-cycle _{rms}U 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._{avr}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. |