Lecture%2010a

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Lecture 10a
Types of Circuit Excitation Why Sinusoidal
Excitation? Phasors
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Types of Circuit Excitation
Steady-State Excitation
OR
Transient Excitation
Sinusoidal (Single- Frequency) Excitation
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Why is Sinusoidal Single-Frequency Excitation
Important?

• Some circuits are driven by a single-frequency
• sinusoidal source.
• Example The electric power system at frequency
  of
• 60/-0.1 Hz in U. S. Voltage is a sinusoidal
  function of time because it is produced by huge
  rotating generators powered by mechanical energy
  source such as steam (produced by heat from
  natural gas, fuel oil, coal or nuclear fission)
  or by falling water from a dam (hydroelectric).
• Some circuits are driven by sinusoidal sources
  whose
• frequency changes slowly over time.
• Example Music reproduction system (different
  notes).

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Why (continued)

• You can express any periodic electrical signal as
  a
• sum of single-frequency sinusoids so you can
• analyze the response of the (linear,
  time-invariant)
• circuit to each individual frequency component
  and
• then sum the responses to get the total response.

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Representing a Square Wave as a Sum of Sinusoids

• Square wave with 1-second period. (b)
  Fundamental compo-
• nent (dotted) with 1-second period,
  third-harmonic (solid black)
• with1/3-second period, and their sum (blue). (c)
  Sum of first ten
• components. (d) Spectrum with 20 terms.

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PHASORS You
can solve AC circuit analysis problems that
involve Circuits with linear elements (R, C, L)
plus independent and dependent voltage
and/or current sources operating at a single
angular frequency w 2pf (radians/s) such as
v(t) V0cos(wt) or i(t) I0cos(wt) By using
any of Ohms Law, KVL and KCL equations,
doing superposition, nodal or mesh analysis,
and Using instead of the terms below on the
left, the terms below on the right
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Resistor I-V relationship vR iRR .VR
IRR where R is the resistance in ohms,  
VR phasor voltage, IR phasor current
(boldface indicates complex quantity) Capacitor
I-V relationship iC CdvC/dt
...............Phasor current IC phasor voltage
VC / capacitive impedance ZC ? IC VC/ZC
where ZC 1/jwC , j (-1)1/2 and boldface
indicates complex quantity Inductor I-V
relationship vL LdiL/dt
...............Phasor voltage VL phasor current
IL/ inductive impedance ZL ? VL ILZL
where ZL jwL, j (-1)1/2 and boldface
indicates complex quantity
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RULE Sinusoid in-- Same-frequency sinusoid
out is true for linear time-invariant circuits.
(The term sinusoid is intended to include both
sine and cosine functions of time.) Intuit
ion Think of sinusoidal excitation (vibration)
of a linear mechanical system every part
vibrates at the same frequency, even though
perhaps at different phases.




SAME
Given
Given
Given
Given
?
?
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Example 1 Well explain what phasor currents and
voltages are shortly, but first lets look at an
example of using them Heres a circuit
containing an ac voltage source with angular
frequency w, and a capacitor C. We represent the
voltage source and the current that flows (in
boldface print) as phasors VS and I -- whatever
they are!

V
S
I
C
-

We can obtain a formal solution for the unknown
current in this circuit by writing KVL
-VS ZCI 0 We can solve symbolically for I
I VS/ZC jwCVS

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Note that so far we havent had to include the
variable of time in our equations -- no sin(wt),
no cos(wt), etc. -- so our algebraic task has
been almost trivial. This is the reason for
introducing phasors! In order to reconstitute
our phasor currents and voltages to see what
functions of time they represent, we use the
rules below. Note that often (for example, when
dealing with the gain of amplifiers or the
frequency characteristics of filters), we may not
even need to go back from the phasor domain to
the time domain just finding how the magnitudes
of voltages and currents vary with frequency w
may be the only information we want.
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Rules for reconstituting phasors (returning to
the time
domain) Rule 1 Use the Euler relation for
complex numbers ejx cos(x)
jsin(x), where j (-1)1/2 Rule 2 To obtain
the actual current or voltage i(t) or v(t) as
a function of time 1. Multiply the phasor I
or V by ejwt, and 2. Take the real part of the
product For example, if I 3 amps, a
real quantity, then i(t)
ReIejwt Re3ejwt 3cos(wt) amps where Re
means take the real part of Rule
3 If a phasor current or voltage I or V is not
purely real but is complex, then multiply it by
ejwt and take the real part of the product.
For example, if V V0ejf, then v(t)
ReVejwt ReV0ejfejwt ReV0ej(wt f)
V0cos(wt f)
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Apply this approach to the
capacitor circuit above, where the voltage source
has the value
vS(t) 4 cos(wt) volts.The phasor voltage VS is
then purely real VS 4. The phasor current is
I VS/ZC jwCVS (wC)VSejp/2, wherewe use
the fact that j (-1)1/2 ejp/2 thus, the
current in a capacitor leads the capacitor
voltage by p/2 radians (90o). The actual
current that flows as a function of time, i(t),
is obtained by substituting VS 4 into the
equation for I above, multiplying by ejwt, and
taking the real part of the product. i(t)
Rej (wC) x 4ejwt Re4(wC)ej(wt p/2)
i(t) 4(wC)cos(wt
p/2) amperes
Finishing Example 1

vS(t) 4 cos(wt)
i(t)
C
-
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Analysis of an RC Filter
Consider the circuit shown below. We want to use
phasors and complex impedances to find how the
ratio Vout/Vin varies as the frequency of the
input sinusoidal source changes. This circuit is
a filter how does it treat the low frequencies
and the high frequencies?
Assume the input voltage is vin(t) Vincos(wt)
and represent It by the phasor Vin. A phasor
current I flows clockwise in the circuit.
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Write KVL -Vin IR IZC 0 -Vin I(R
ZC) The phasor current is thus I Vin/(R
ZC) The phasor output voltage is Vout I
ZC. Thus Vout VinZC /(R ZC) If we are
only interested in the dependence upon
frequency of the magnitude of (Vout / Vin) we can
write Vout / Vin ZC/(R ZC)
1/1 R/ ZC Substituting ZC 1/jwRC, we have
1 R/ ZC 1 jwRC, whose magnitude is the
square root of (wRC)2 1. Thus,
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Explore the Result
If wRC ltlt 1 (low frequency) then Vout / Vin
1 If wRC gtgt 1 (high frequency) then Vout / Vin
1/wRC If we plot Vout / Vin vs. wRC we
obtain roughly the plot below, which was
plotted on a log-log plot
The plot shows that this is a low-pass filter.
Its cutoff frequency is at the frequency w for
which wRC 1.
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Why Does the Phasor Approach Work?

1. Phasors are discussed at length in your text
   (Hambley 3rd Ed., pp. 195-201) with an
   interpretation that sinusoids can be visualized
   as the real axis projection of vectors rotating
   in the complex plane, as in Fig. 5.4. This is
   the most basic connection between sinusoids and
   phasors.
2. We present phasors as a convenient tool for
   analysis of linear time-invariant circuits with a
   sinusoidal excitation. The basic reason for
   using them is that they eliminate the time
   dependence in such circuits, greatly simplifying
   the analysis.
3. Your text discusses complex impedances in Sec.
   5.3, and circuit analysis with phasors and
   complex impedances in Sec. 5.4.

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Motivations for Including Phasors in EECS 40

1. It enables us to include a lab where you measure
   the behavior of RC filters as a function of
   frequency, and use LabVIEW to automate that
   measurement.
2. It enables us to (probably) include a nice
   operational amplifier lab project near the end of
   the course to make an active filter (the RC
   filter is passive).
3. It enables you to find out what impedances are
   and use them as real EEs do.
4. The subject was also supposedly included (in a
   way) in EECS 20.