Lecture 5a

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Lecture 5a
Review Types of Circuit Excitation Why
Sinusoidal Excitation? Phasors
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Types of Circuit Excitation
Steady-State Excitation
OR
Sinusoidal (Single- Frequency) Excitation
Transient 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).

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Bonneville Dam (Columbia River) Where Much
of Californias Electric Power Comes From
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Turbine-generator sets at Bonneville Dam
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Where 3-Phase Electricity Comes From
Generator driven by falling water has 3
separate coils
Direct current in the rotor (rotating
coil) produces a magnetic field that
generates currents in stationary coils A, B and C


Output voltages from the 3 coils (they leave
the generating plant on 3 separate cables)
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Why Sinusoidal Excitation? (continued)

• Some circuits are driven by sinusoidal sources
  whose frequency changes slowly over time.
• Example Music reproduction system (different
  notes).

• And, 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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Single-frequency sinusoidal-excitation AC circuit
problems

• The technique well show works on circuits
  composed of linear
• elements (R, C, L) that dont change with time ?
  linear
• time-invariant circuits.
• The circuit is driven with independent voltage
  and/or current
• sources whose voltages or currents vary at a
  single frequency, f,
• measured in Hertz (abbreviated Hz) this is the
  number of cycles the
• voltages or currents execute per second. We can
  represent the
• source voltages or currents as functions of time
  as
• v(t) V0cos(wt) or i(t) I0cos(wt),
• where w 2pf is the angular frequency in radians
  per second.
• Example In the U. S. the AC power
  frequency, f, is 60 Hz and the peak voltage V0
  is 170 V, so w 377 radians/s and
• v(t) 170cos(377t) V. More generally,
  we might have sources
• v(t) V0sin(wt) or i(t) I0cos(wt f),
  where f is a phase angle.

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We could solve our circuit equations using such
functions of time, but wed have to do a lot of
tedious trigonometric transformations. Instead
we use a mathematical trick to eliminate time
dependence from our equations! The trick is
based on a fundamental fact about linear,
time-invariant circuits excited with sinusoidal
sources the frequencies of all the voltages and
currents in the circuit are identical.
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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.) Intuiti
on Think of sinusoidal excitation (vibration)
of a linear mechanical system every part
vibrates at the same frequency, even though
perhaps at different phases.




SAME
SAME w
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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
analysis, nodal analysis or mesh
analysis,
AND Using instead of the terms
below on the left (general
excitation), the terms below on the right
(sinusoidal excitation)
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Resistor I-V relationship General excitation
Sinusoidal excitation vR
iRR VR IRR where R is the
resistance in ohms,   VR phasor voltage
across the resistor,
IR phasor current through
the resistor, and boldface indicates
complex quantity. Capacitor I-V relationship
General excitation Sinusoidal
excitation iC CdvC/dt IC
VC / ZC where IC phasor current
through the
capacitor, VC phasor voltage
across the capacitor,
the capacitive
impedance ZC in ohms is ZC 1/jwC
, j (-1)1/2, and boldface
capital letters are
complex quantities.
(Note EEs use j for (-1)1/2
instead of i,
since i might suggest current)
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Inductor I-V relationship General excitation
Sinusoidal excitation vL
LdiL/dt VL IL ZL where VL is the
phasor voltage
across the inductor, IL is the
phasor
current through the inductor, the
inductive
impedance in ohms ZL is
ZL jwL , j (-1)1/2 and
boldface capital
letters are complex quantities.

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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 IZC 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 phasor voltages and currents, and
impedances! 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 ejf cos(f)
jsin(f), 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
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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).Note Often
(especially in this class) we may not care about
the phase angle, and will focus just on the
amplitude of the voltage or current that we
obtain. This will be particularly true of
filters and amplifiers.
Finishing Example 1

vS(t) 4 cos(wt)
i(t)
C
-
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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
Note We obtained the current as a function of
time (the current waveform) without ever having
to work with trigonometric identities!
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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?


R
V
C
V
out
in
I
-
-

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 for ZC, we have 1
R/ ZC 1 jwRC, whose magnitude is
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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Notice that weve obtained a lot of information
about how this particular RC circuit performs
just by looking at the magnitude of the ratio of
phasor output voltage to phasor input
voltage (i.e., we havent had to study the phase
angles associted with those phasor voltages. (In
more detailed studies the phase angles can be
important but not in this course.) Does this
behavior make sense from what we know
about capacitors? YES! At low frequency a
capacitor is like an open circuit
so the output voltage would equal the input
voltage At high frequency a capacitor
is like a short circuit so the
output voltage would be very small.
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Here is a useful web site to explore http//www.
phys.unsw.edu.au/jw/AC.html Youll find some
demonstrations dealing with phasors and
impedances there.
And Appendix A in Hambley is a review of complex
numbers.
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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. Skim over this
   LIGHTLY.

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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 which some of you may have taken.