1. Chapter 7 – AC Steady-State Analysis
We want to analyze the behavior of circuits to x(t) = XMsin(t); why?
(1) This is the format of signals generated by power companies such as AEP.
(2) Using a tool called Fourier Series, we can represent other signals (i.e. digital signals) as a sum of sinusoidal signals.
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ENGR201 AC Steady-State Analysis

2. ENGR201 AC Steady-State AnalysisXM
-XM
/2 
3/2 2
x(t) = XMsin(t);
The function is periodic with period T  x[(t+T)] = x(t ) T
 t
Frequency, f, is a measure of how many periods (or cycles) the signal completes per second and is measured in Hertz (cycles per second )
 is the angular frequency and is measured in radians per second
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ENGR201 AC Steady-State Analysis

3. ENGR201 AC Steady-State Analysis
Phase Shift - a sinusoid can be shifted right (left) by subtracting (adding) a phase angle.
x(t) = XMsin(t + )
sin(120t)
sin(120t- /4)
Note that one function lags (leads) the other function of time.
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ENGR201 AC Steady-State Analysis

4. ENGR201 AC Steady-State Analysis
sin(120t+ /4)
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ENGR201 AC Steady-State Analysis

5. ENGR201 AC Steady-State Analysis
Some useful trigonometric identities:
Multiple-angle formulas:
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ENGR201 AC Steady-State Analysis

6. Sinusoidal & Complex Forcing Functions
If we apply sinusoidal signals (voltages and/or currents) as inputs to linear electrical networks, all voltages and currents in the circuit will be sinusoidal also; only the amplitudes and phase angles will differ. The following example illustrates.
v(t)=VMcost R L
i(t)
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ENGR201 AC Steady-State Analysis

7. ENGR201 AC Steady-State Analysis
v(t)=VMcost R L
i(t)
Let us assume (this is a theorem from differential equations) that the solution to the differential equation, i(t), is also sinusoidal. That is, the forced response can be written
i(t) = Acos(t+)
The signal has the same frequency, but different magnitude and phase. However, this expression must satisfy the original differential equation.
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ENGR201 AC Steady-State Analysis

8. ENGR201 AC Steady-State Analysis
v(t)=VMcost R L
i(t)
i(t) = Acos(t+ )
Using the multiple angle formula:
i(t) = Acostcos - Asin tsin
i(t) = A1 cost + A2 sint
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ENGR201 AC Steady-State Analysis

9. ENGR201 AC Steady-State Analysis
Setting like terms on each side of the equation equal yields two simultaneous equations:
Solving these two simultaneous equations yields:
Since i(t) = A1 cost + A2 sint
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ENGR201 AC Steady-State Analysis

10. ENGR201 AC Steady-State Analysis
Using trig identities we get:
Conclusion: we do not want to use differential equations to analyze such circuits!
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ENGR201 AC Steady-State Analysis

11. ENGR201 AC Steady-State Analysis
Euler’s Identity
We can write our sinusoidal forcing functions in terms of the complex quantity ejt
v(t) = VMcost = Re[VMejt] = Re[VMcost + j VMsint ]
The currents and voltages produced can also be written in terms of ejt
i(t) = Imcos(t +) = Re[Imej(t + )]
i(t) = = Re[Imcos(t +) + j Imsin(t +) ]
Using complex forcing functions allows us to use (complex) algebra in place of differential equations.
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ENGR201 AC Steady-State Analysis

12. ENGR201 AC Steady-State Analysis
v(t)=VMejt R L
i(t)
i(t) = Imej(t +)
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ENGR201 AC Steady-State Analysis