Week 5a 1

EE42: Running Checklist of Electronics Terms 14.02.05 – Dick White

Terms are listed roughly in order of their introduction. Most definitions can be found in your text.

TERM

Charge, current, voltage, resistance , conductance, energy, power Coulomb, ampere, volt, ohm, siemen (mho), joule, watt

Reference directions Kirchhoff’s Current Law (KCL), Kirchhoff’s Voltage Law (KVL),

Ohm’s Law Series connection, parallel connection

DC (steady), AC (time-varying) Independent and dependent ideal voltage and current source

Voltage divider, current divider Analog (A/D), Digital (D/A)

Multimeter (DMM), Oscilloscope Prefixes (milli-, etc.)

Linear, nonlinear elements Superposition (analysis)

Nodal analysis (node, supernode) Loop analysis (mesh, branch)

Power delivery, dissipation, storage, maximum power transfer Equivalent circuits (Rs, Cs or Ls in series/parallel; Thevenin,

Norton) Frequency; angular frequency; period; phase (Hz; radian/s)

Capacitor, inductor, transformer Phasor, impedance, reactance

Amplifier, filter, transfer function Steady-state, transient, sinusoidal excitation

Terms2

Week 5a 2

Lecture 5a

Review: Types of Circuit Excitation

Why Sinusoidal Excitation?

Phasors

Week 5a 3

Types of Circuit Excitation

Linear Time- Invariant Circuit

Steady-State Excitation

Linear Time- Invariant Circuit

OR

Linear Time- Invariant Circuit

DigitalPulseSource

Transient Excitation

Linear Time- Invariant Circuit

Sinusoidal (Single-

Frequency) Excitation

Week 5a 4

Why is Sinusoidal Single-Frequency Excitation Important?

1. Some circuits are driven by a single-frequency sinusoidal source.

Example: The electric power system at frequency of60+/-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).

Week 5a 5

Bonneville Dam (Columbia River) Where Much of California’s Electric Power Comes From

Week 5a 6

Turbine-generator sets at Bonneville Dam

Week 5a 7

Voltage

Time

A B C

Time for which rotor position is shown

Coil A Plane of Coil C

Rotation of Rotor

Plane ofCoil A

Plane of Coil B

Where 3-Phase Electricity Comes From

Generatordriven byfallingwater has3 separatecoils

Output voltagesfrom the 3 coils(they leave thegenerating planton 3 separatecables)

Direct currentin the rotor(rotating coil)produces amagnetic fieldthat generatescurrents in stationary coilsA, B and C

Week 5a 8

2. Some circuits are driven by sinusoidal sources whose frequency changes slowly over time.

Example: Music reproduction system (different notes).

Why Sinusoidal Excitation? (continued)

3. And, you can express any periodic electrical signal as asum of single-frequency sinusoids – so you cananalyze the response of the (linear, time-invariant) circuit to each individual frequency component and then sum the responses to get the total response.

Week 5a 9

a b

c d Time (ms)

Frequency (Hz)

Sig

na

l (V

)

Re

lative

Am

plitu

de

Sig

na

l (V

)

Sig

na

l (V

)

Representing a Square Wave as a Sum of Sinusoids

(a) 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 tencomponents. (d) Spectrum with 20 terms.

Week 5a 10

Single-frequency sinusoidal-excitation AC circuit problems1. The technique we’ll show works on circuits composed of linear elements (R, C, L) that don’t change with time “linear time-invariant circuits”. 2. 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 thesource voltages or currents as functions of time as

v(t) = V0cos(t) or i(t) = I0cos(t),where f 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 = 377 radians/s and v(t) = 170cos(377t) V. More generally, we might have sources v(t) = V0sin(t) or i(t) = I0cos(t = ), where is a phase angle.

Week 5a 11

We could solve our circuit equations using such functions of time, but we’d 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.

Week 5a 12

SAME

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.)

Intuition: Think of sinusoidal excitation (vibration) of a linear mechanical system – every part vibrates at the same frequency, even though perhaps at different phases.

SAME

Circuit of linear elements (R, L, C)

Excitation: Output:

vS(t) = VScos(t + ) Iout(t) = I0cost(t + )

Given Given

Given

? ?

Week 5a 13

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 = 2f (radians/s) such as v(t) = V0cos(t) or i(t) = I0cos(t).

By using any of Ohm’s 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):

Week 5a 14

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/jC ,

j = (-1)1/2, and boldface capital letters are complex quantities. (Note: EE’s use j for (-1)1/2 instead of i, since i might suggest current)

Week 5a 15

Inductor I-V relationshipGeneral 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 = jL , j = (-1)1/2 and boldface capital

letters are complex quantities.

Week 5a 16

Example 1

We’ll explain what phasor currents and voltages are shortly, but first let’s look at an example of using them:

Here’s a circuit containing an AC voltage source with angular frequency , 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!

VS 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 = jCVS

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Note that so far we haven’t had to include the variable of time in our equations -- no sin(t), no cos(t), etc. -- so our algebraic task has been almost trivial. This is the reasonfor 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 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: ej = cos() + jsin(), 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 ejt, and

2. Take the real part of the product For example, if I = 3 amps, a real quantity, then i(t) = Re[Iejt] = Re[3ejt] = 3cos(t) amps where Re means “take the real part of”

imaginary

real

j

1

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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 ejt and take the real part of the product.

For example, if V = V0ej, then v(t) = Re[Vejt] = Re[V0ejejt] = Re[V0ej(t + )] =

V0cos(t + )

Week 5a 20

Apply this approach to the capacitor circuit above, where the voltage source has the value vS(t) = 4 cos(t) volts.The phasor voltage VS is then purely real: VS = 4. The phasor current is I = VS/ZC = jCVS = (C)VSej/2, wherewe use the fact that j = (-1)1/2 = ej/2; thus, the current in a capacitor leads the capacitor voltage by /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.

+

- Ci(t)vS(t) = 4 cos(t)

Finishing Example 1

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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 ejt, and taking the real part of the product. i(t) = Re[j (C) x 4ejt] = Re[4(C)ej(t + /2)]

i(t) = 4(C)cos(t + /2) amperes

Note: We obtained the current as a function of time (thecurrent waveform) without ever having to work with trigonometric identities!

Week 5a 22

Analysis of an RC Filter

+

-

VoutVin

+

-

RC

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(t) and representit by the phasor Vin. A phasor current I flows clockwise in thecircuit.

I

Week 5a 23

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 = Vin[ZC /(R + ZC)]If we are only interested in the dependence upon frequencyof 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 + jRC,whose magnitude is Thus,

Vout

Vin---------

1

RC 21+

---------------------------------·

=

1)( 2 RC

Week 5a 24

Explore the ResultIf RC << 1 (low frequency) then | Vout / Vin | = 1If RC >> 1 (high frequency) then | Vout / Vin | ~ 1/RC

If we plot | Vout / Vin | vs. RC we obtain roughly the plot below, which was plotted on a log-log plot:

1

|Vo ut /Vi n |

=RC

The plot shows that this is a low-pass filter. Its cutoff frequency is at the frequency for which RC = 1.

Vout

Vin---------

1

RC 21+

---------------------------------·

=

Week 5a 25

Notice that we’ve obtained a lot of information about how thisparticular RC circuit performs just by looking at the magnitudeof the ratio of phasor output voltage to phasor input voltage(i.e., we haven’t had to study the phase angles associted withthose phasor voltages. (In more detailed studies the phase anglescan be important – but not in this course.)

Does this behavior make sense from what we know aboutcapacitors? 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.

Week 5a 26

Here is a useful web site to explore:http://www.phys.unsw.edu.au/~jw/AC.html

You’ll find some demonstrations dealing with phasors and impedances there.

And Appendix A in Hambley is a review of complex numbers.

Week 5a 27

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.

Week 5a 28

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.