Sinusoidal - Engineering Electrical Circuits - Lecture Slides, Slides of Electrical Circuit Analysis

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Sinusoidal

AC SteadySt

Outline – AC Steady State

• SINUSOIDS
  • Review basic facts about sinusoidal signals
• SINUSOIDAL AND COMPLEX FORCING

FUNCTIONS

• Behavior of circuits with sinusoidal independent

current & voltage sources

• Modeling of sinusoids in terms of

complex exponentials

Outline – AC SS cont.

• PHASOR DIAGRAMS
  • Representation of AC voltages and currents as

COMPLEX VECTORS

• BASIC AC ANALYSIS USING KIRCHHOFF’S LAWS
• ANALYSIS TECHNIQUES
  • Extension of node, loop, SuperPosition, Thevenin

and other KVL/KCL Linear-Circuit Analysis

techniques

Sinusoids

• Recall From Trig the Sine Function

 Where

• XM ≡ “Amplitude” or Peak or Maximum Value - Typical Units = A or V
• ω ≡ Radian, or Angular, Frequency in rads/sec
• ωt ≡ Sinusoid argument in radians (a pure no.)  The function Repeats every 2π; mathematically

 For the RADIAN Plot Above, The Functional

Relationship x (ω t + 2 π) = x ( ω t )

x ( t ) = XM sin ω t

Sinusoids cont.

• Now Define the Cyclic “Frequency”, f

 Quick Example

• USA Residential Electrical Power Delivered as a 115Vrms , 60Hz, AC sine wave

 Describes the Signal Repetition-Rate in Units of Cycles-Per-Second, or HERTZ (Hz)

• Hz is a Derived SI Unit

v t V ( t )

v t V t residence

residence = ⋅

( ) 162. 6 sin 376. 99

f ( )^2115 sin^2 π^60

T

f

2

2

1

⇒ =

= =

 Will Figure Out the √ 2 term Shortly

• RMS ≡ “ R oot (of the) M ean S quare”

Sinusoids cont.

• Now Consider the GENERAL Expression for a Sinusoid

 Where

• θ ≡ “Phase Angle” in Radians

 Graphically, for POSITIVE θ

x ( t ) = XM sin (ω t +θ )

"leads by θ "

"lags by θ "

Sinusoid Phase Difference

-1.

-1.

-1.

-0.

-0.

-0.

-0.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2. Time (S)

xi^

(V or A)

x1 (V or A) x2 (V or A)

file =Sinusoid_Lead-Lag_Plot_0311.xls

**Out of Phase

• 0.733 rads
• 42°
• 105 mS**

PARAMETERS

• T = 900 mS• f = 1.1111 Hz
• ω = 6.981 rad/s
• θ = 1.257 rads = 72°
• φ = 0.5236 rads = 30°

x1 LEADs x

For Different Amplitudes,Measure Phase Difference x2 LAGs x

• peak-to-peak
• valley-to-valley
• ZeroCross-to-ZeroCross

θ− φ= 105 mS ⋅^1900 PeriodmS ⋅ 1 Period^360 ° = 42 °

Useful Trig Identities

• To Convert sin↔cos

 To Make a Valid Phase-Angle Difference Measurement BOTH Sinusoids MUST have the SAME Frequency & Trig-Fcn (sin OR cos)

• Useful Phase-Difference ID’s 

sin sin( )

cos cos( ) ω ω π

ω ω π = − ±

= − ± t t

t t

= ^ −

= ^ +

sin cos

cos sin

π ω ω

π ω ω

t t

t t

 Additional Relations

α β α β α β

α β α β α β cos( ) cos cos sin sin

sin( ) sin cos cos sin

• = −

• = +

α β α β α β

α β α β α β cos( ) cos cos sin sin

sin( ) sin cos cos sin − = +

− = −

(rads) rads

(degrees)^180

2 radians 360

θ π

θ

π

= °

= °

Example – Phase Angles cont.

• Convert –6V Amplitude to Positive Value Using

 Then

 It’s Poor Form to Express phase shifts in Angles >180° in Absolute Value

cos(α )=−cos(α ± 180 ° )

6 cos( 1000 30 180 )

6 cos( 1000 30 ) 2

2 = + °+ °

v t

v t

 Next Convert cosine to sine using cos(α ) =sin(α + 90 ° )

6 sin( 1000 210 90 )

2 6 cos(^1000210 ) = + °+ °

= + ° t

v t

[ ( )] = [ − °]

= + °− °

= + °

6 sin 1000 60

6 sin 1000 300 360

2 6 sin(^1000300 )

t

t

v t

 So Finally

( ) 6 sin( 1000 60 )

( ) 12 sin( 1000 60 )

2

1 = − °

= + ° v t t

v t t

 Thus v1 LEADS v 2 by 120°

Sinusoid Phase Difference Example 7.

0

4

8

12

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0. Time (S)

Signal Level (V)

v1 (V) v2 (V)

file =Sinusoid_Lead-Lag_Plot_0311.xls

120°

rms Values Cont.

• For The Resistive Case, Define Ieff for the Avg Power Condition

 If the Current is DC, then i(t) = I (^) dc , so

 The Pav Calc For a Periodic Signal by Integ

 Now for the Time- Variable Current i(t) → I (^) eff , and, by Definition

i ( t )

R

P I R

p t i t R

av eff

2

2 ( ) ( )

=

=



 

 

 

 = (^) ∫ = ∫

• t + T

t

t T

t

Pav (^) T p t dt R T i t dt

0

0

0

0

(^1) ( ) (^12) ( )

2 2

2

0

0

0

0

1 ( )

dc

t T

t

dc

t T

t

av dc

t dt RI T

RI

I dt T

P R

∫

∫

Pav = RIeff = RIdc = P av 2 2

rms Values cont.

• In the Pwr Eqn (^)  Examine the Eqn for I (^) eff and notice it is Determined by - Taking the Square R OOT of the time-averaged, or M EAN, S QUARE of the Current  In Engineering This Operation is given the Short-hand notation of “rms”  So

 Equating the 1st^ & 3rd Expression for Pav find

2 2 2

0

0

dc eff

t T

t

Pav R T i t dt = RI = RI 

= (^) ∫

∫

=

t T

t

Ieff (^) T i t dt

0

0

( )

(^1 )

 This Expression Holds for ANY Periodic Signal (^) I (^) eff ≡ Irms

RMS Value for Sinusoid

• Sub cos^2 Trig ID into RMS integral
• Integrating Term by Term

( )

1 2

0

cos 2 2 2

1

2

1 1  

  













 = (^) ∫ + t + dt T

I I

T

rms M ω^ θ

( )

1 2

0 0

cos 2 2 2

1 1

2

1 1  

  

 = (^) ∫ + ∫ t + dt T

dt T

I I

T T

rms M ω^ θ

RMS Value for Sinusoid

• Rearranging a bit
• But the integral of a sinusoid over ONE

PERIOD is ZERO, so the 2 nd^ Term goes to Zero

leaving

( )

1 2

0 0

cos 2 2

1

2

1 1

1

2

1  

  

 = (^) ∫ + ∫ t + dt T

dt T

I I

T T

rms M ω^ θ

0

(^1212)

0

12

0

M M

T M

T rms M

T I

T

I

t

T

dt I

T

I I =

= (^) ∫