AC SINUSOIDSAC SINUSOIDS

Lecture 6 (I)

SCOPESCOPE

Explain the difference between AC and 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 Define cycle, period, and frequency Given the analytical expression, sketch and

explain the graph of a sinusoid. Determine the relative phase of a sinusoidal

waveform.

OBJECTIVESOBJECTIVES (cont) (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 WAVESSINE 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 SINEWAVEAC SINEWAVE

1 cycle1 cycle

Voltage is positive

Voltage is negative

Polarity change

t

voltage

+

-

0

voltage

+

-

0

Voltage is

Polarity change

t

voltage

+

-

0

voltage

+

-

0

OTHER ACsOTHER ACs

SINE WAVE

TRIANGLE WAVE

SQUARE WAVE

HOW IS A SINE WAVE HOW IS A SINE WAVE GENERATED ?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 GENERATING AC VOLTAGESVOLTAGES

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

FARADAY’S LAWFARADAY’S LAW

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

DC vs ACDC vs AC

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

V1 ohm

I

Voltage

time

AC SOURCEAC SOURCEAC source: Voltage polarity changes

therefore the current changes direction.

V(1.25s)= +2v

1 ohm

I

0 time(sec)

2v

-2v

1 2 3 4

V(3.75s)= -2v

1 ohmI

PERIOD AND FREQUENCYPERIOD AND FREQUENCYPeriod: Time to complete one

complete cycleSymbol: T

Frequency: Number of cycles in one secondSymbol: fMeasured in hertz (Hz)

Tf

1

t

V

FREQUENCYFREQUENCY

Definition: the number of cycles per second of a waveform

Denoted by the lower case letter f Its unit is the hertz (Hz)

secondper cycle 1 hertz 1

1 cycle

1 second

f=1 Hz

Ex. Ex.

f=2 Hz

Ex. Ex.

1 cycle

1 second

1 cycle

60 cycles1 cycle

1 second

?

Ex. Ex.

PERIODPERIOD

Definition: the duration of one cycle.

It is the inverse of frequency. Denoted by the upper case

letter TMeasured in second, s

Hz)(T

1 f and )s(

f

1T

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)

T (between zero points)

T (Any two identical points)

t

Ex. Ex.

Figure shows an oscilloscope trace of a square wave. Each horizontal division represents 50 μs. Determine the frequency.

SolutionSolution

Since the wave repeats itself every 200 μs, its period (T) is 200 μs and,

kHz 510200

16

sf

Ex. Ex.

Determine the period and frequency of the waveform of the figure above.

T2 = 10 ms

T1 = 8 ms

SolutionSolution

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,

Hz1001010

13

sf

PEAK VALUES (VPEAK VALUES (VPP, I, IPP))

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”

baselinebaseline

VVMM oror Amplitude Amplitude

VVDCDC

t

V

PEAK TO PEAK VALUES PEAK TO PEAK VALUES (V(VPP, PP, IIPPPP))

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

VVPPPP

VVMINMIN

VVMAXMAX

t

V

ROOT-MEAN-SQUARE (VROOT-MEAN-SQUARE (VRMSRMS, , IIRMS RMS ))

Named for the mathematical process by which the value is calculated. “Effective Voltage (VEFF)”

PPRMS V0.707V2

2V

COMPATIBILITY OF VALUESCOMPATIBILITY 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)!

VVMMVVrmsrms

Vpp

VOLTAGE & CURRENT VOLTAGE & CURRENT VALUESVALUES

Ohm’s Law still applies: V=IRIf current changes with time and R is a

constant, voltage will also change with time

Voltage will be proportional to current

VOLTAGE & CURRENT VOLTAGE & CURRENT VALUESVALUES

A graph of current and voltage in a resistor produces identical waveforms:Peak at the same timeCross the same baseline, at the same time

Differ only in amplitude:IP is 1/R of VP

INSTANTANEOUS INSTANTANEOUS VALUESVALUES

Instantaneous Values ( v, i )value of voltage and current at any:

instant in time or at at any angle

Mathematically expressed 2 ways:

sin 2

sin

M

M

v(t) V ( ft )

v( ) V ( )

ANGULAR DOMAINANGULAR 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 = 1 rotation = 360°=2360°=2 radians radians)

deg180

rad rad

180

deg

Sine Wave Angles: Degrees & Radians2 radians = 360o 1 radian =

57.3o

TIME DOMAINTIME 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:

sin 2Mv(t) V ( ft )

2 ft t

PHASE ANGLEPHASE ANGLE

Symbol is (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 PHASE ANGLE (angular domain)domain)

Wave A is the reference wave:Wave B is 90° out of phase.

PHASE ANGLE (Time domain)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 /360º = t/T = (60ms-30ms)/120ms. = 90º

LEADING & LAGGINGLEADING & LAGGING

Since wave B peaked after the reference wave peaked, we say it LAGS the reference wave by 90º ; = - 90º

If wave B was the reference, wave A would peak before the reference wave (B). We would say it LEADS the reference wave;

= + 90º Note: Because it is the reference wave, for ANY reference wave is 0 º

Ex:Ex:Compute the phase angle if:

V1(t) is the reference wave

V2 (t) is the reference wave

V 1(t)

V 2(t)

t = 1 ms/divt = 1 ms/div

Ex:Ex:

V 1(t)

V 2(t)

t = 1 ms/divt = 1 ms/div

V2 is the reference. Write the equations.

sin 2

sin

dc M

dc M

v(t) V V ( ft )

v( ) V V ( )

SUPERIMPOSED DC & ACSUPERIMPOSED 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 QUIZREVIEW 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?