SYLLABUS AC Fundamentals AC Analysis AC power Three phase circuit AC sinusoids AC response (reactance, impedance) Phasors and complex numbers AC Analysis RL, RC, RLC circuit analysis Mesh and Nodal analysis AC power Average power, Reactive power, Complex power Power triangle Three phase circuit Y and Delta connection Line and Phase voltages

ALTERNATING CURRENT (AC) SINUSOIDS

OBJECTIVES Explain the difference between alternating (AC) and direct current (DC). Express angular measure in both degrees and radians. Compute the peak, peak-peak, and instantaneous values of a waveform. Define and solve for the RMS value of a waveform. Define cycle, period, and frequency of a sinusoid. Given the analytical expression, sketch and explain the graph of a sinusoid. Determine the relative phase of a sinusoidal waveform.

OBJECTIVES (cont) Determine the total voltages and currents that have DC and AC components. Apply Ohm’s Law, KCL, and KVL to analyze a simple AC circuit. Write the time domain equation for any sinusoidal waveform with a DC component.

SINE WAVES Voltage can be produced such that, over time, it follows the shape of a sine wave The magnitude of the voltage continually changes. Polarity may or may not change. When it does not change, the current does not change direction. When polarity does change, the current changes direction. When graphing a sinusoidal voltage, the polarity changes only when the magnitude alternates between “+” and “-” values.

AC SINEWAVE 1 cycle Voltage is positive Polarity change t voltage + -

OTHER ACs SINE WAVE TRIANGLE WAVE SQUARE WAVE

HOW IS A SINE WAVE GENERATED ? Electromagnetic Induction. (Ship AC generators produce sine wave voltages through electromagnetic induction): magnetic field conductor relative motion between the two. Electronic Signal Generators Function Generators: multi-waveforms.

GENERATING AC VOLTAGES One way to generate ac voltage is to rotate a coil of wire at constant angular velocity in a fixed magnetic field

FARADAY’S LAW “ Voltage is induced in a circuit whenever the flux linking (i.e. passing through) the circuit is changing.. and that the magnitude of the voltage is proportional to the rate of change of the flux linkages”

DC vs AC DC Source: voltage POLARITY of the source and current DIRECTION do not change over time.

AC SOURCE AC source: Voltage polarity changes therefore the current changes direction.

PERIOD AND FREQUENCY Period: Time to complete one complete cycle Symbol: T Frequency: Number of cycles in one second Symbol: f Measured in hertz (Hz) t V

FREQUENCY Definition: the number of cycles per second of a waveform Denoted by the lower case letter f Its unit is the hertz (Hz)

Ex. 1 cycle f=1 Hz 1 second

Ex. 1 cycle 1 cycle f=2 Hz 1 second

Ex. 60 cycles 1 cycle 1 second

PERIOD Definition: the duration of one cycle. It is the inverse of frequency. Denoted by the upper case letter T Measured in second, s

The period of a waveform can be measured between any two corresponding point. Often it is measured between zero points because they are easy to establish on an oscilloscope trace

T (between peaks) (between zero points) (Any two identical points) t

Ex. Determine the period and frequency of the waveform of the figure above. T2 = 10 ms T1 = 8 ms

Solution Time interval T1 does not represent a period as it is not measured between corresponding points. Interval T2, however, is. Thus, T = 10 ms and,

PEAK VALUES (VP, IP) Max Voltage (Current) Symbol VM ( IM ) The maximum value of V (I) measured from the point of inflection (“baseline or DC offset”) From the graph: VM - VDC Also called “Amplitude”

baseline VM or Amplitude VDC t V

PEAK TO PEAK VALUES (VPP, IPP) Peak to Peak Voltage (Current) Symbol VPP ( IPP ) The difference between the maximum value of V (I) and the minimum value of V (I) From the graph: VMAX – VMIN Equals twice peak value VPP = 2VP

VPP VMIN VMAX t V

ROOT-MEAN-SQUARE (VRMS, IRMS ) Named for the mathematical process by which the value is calculated. “Effective Voltage (VEFF)” “ The RMS value of a sine wave is equal to the value of an equivalent DC circuit that would produce the same heating effect or power in a load as the given sine wave.” Most meters read in RMS The voltage accessed at electrical wall sockets is RMS.

ROOT-MEAN-SQUARE (VRMS, IRMS )

COMPATIBILITY OF VALUES When Peak voltages are used as source values, current calculations will also be in Peak values. Likewise, an RMS source produces answers in RMS. When solving a problem make sure all values are expressed ONE way (peak, peak to peak, or RMS)! VM Vrms Vpp

VOLTAGE & CURRENT VALUES Ohm’s Law still applies: V=IR If current changes with time and R is a constant, voltage will also change with time Voltage will be proportional to current

VOLTAGE & CURRENT VALUES A graph of current and voltage in a resistor produces identical waveforms: Peak at the same time Cross the same baseline, at the same time Differ only in amplitude: IP is 1/R of VP

INSTANTANEOUS VALUES Instantaneous Values ( v, i ) value of voltage and current at any: instant in time or at at any angle Mathematically expressed 2 ways:

ANGULAR DOMAIN We can identify points on the sine wave in terms of an angular measurement (degrees or radians). The instantaneous value of the sine wave can be related to the angular rotation of the generator, (1 rotation = 360°=2 radians)

Sine Wave Angles: Degrees & Radians 2p radians = 360o 1 radian = 57.3o

TIME DOMAIN Because the time to complete a cycle is frequency dependent, we can also identify points on the sine wave in terms of time. To convert between the time domain and angular domain remember:

PHASE ANGLE Symbol is q (theta). It is expressed as an angle Phase angle specifies the lateral shift in the position of a sine wave from a reference wave. Examine the same event, on each wave: Two events occurring at the same angle or at the same time are in phase. Events occurring at different angles or at different times are out of phase.

PHASE ANGLE (angular domain) Wave A is the reference wave: Wave B is 90° out of phase.

PHASE ANGLE (Time domain) Wave A is the reference wave. Compare the positive peak events: Wave A peaks at 30ms; Wave B at 60ms T=120ms q /360º = Dt/T = (60ms-30ms)/120ms. q = 90º Take note: Graf dlm domain masa

LEADING & LAGGING Since wave B peaked after the reference wave peaked, we say it LAGS the reference wave by 90º ; q = - 90º If wave B was the reference, wave A would peak before the reference wave (B). We would say it LEADS the reference wave; q = + 90º Note: Because it is the reference wave, q for ANY reference wave is 0 º

Ex: t = 1 ms/div Compute the phase angle if: V1(t) is the reference wave V2 (t) is the reference wave t = 1 ms/div

Ex: V2 is the reference. Write the equations. t = 1 ms/div

SUPERIMPOSED DC & AC A circuit can have both a DC voltage source and an AC We say that the “AC rides on the DC” The graph of the voltage is displaced vertically from 0, to the DC voltage level. Algebraically:

REVIEW QUIZ The difference between DC and AC ? 3 items required for electromagnetic induction. Frequency is equal to ? Name 3 different Sine wave values. How many radians in 360 degrees ? If the peak value is 170 V, the RMS value = ? What type of shift does a phase angle represent?