EECS 40 Fall 2002 Lecture 4 W. G. Oldham and S. Ross Lecture 4 Last time –Gate delay: why it happens (Capacitance! Resistance!) –Voltage rise and fall.

EECS 40 Fall 2002 Lecture 4 W. G. Oldham and S. Ross

Lecture 4• Last time

– Gate delay: why it happens (Capacitance! Resistance!)– Voltage rise and fall through logic circuits is gradual – (delay between input and output at 50% of final value) found graphically– Used clock signal to prevent false output (due to signals delayed different

amounts)

• Now we will– Get to calculating delays for circuits (RC analysis)– Review fundamental knowledge of electric circuits


REVIEW OF ELECTRICAL QUANTITIES AND BASIC CIRCUIT ELEMENTS

• Solids in which outer-most atomic electrons are free to move around are called metals.– Metals typically have ~ 1 “free electron” per atom (~ 5 X1022/cm3)– Charge on a free electron is ”e” or “q”, where |e| = 1.6 x 10-19 C

Free ChargeMost matter is macroscopically electrically neutral most of the time. Exceptions: clouds in thunderstorm, people on carpets in dry weather, plates of a charged capacitor, etc.

Microscopically, of course, matter is full of charges. Consider solids:

Al or Cu – good metallic conductor – great for wires

Quartz – good insulator – great for dielectric

Si or GaAs – classic semiconductors

• Solids in which all charges are bound to atoms are called insulators.

• Semiconductors are insulators in which electrons are not tightly bound and thus can be easily “promoted” to a free state (by heat or even by “doping” with a foreign atom).


CHARGE (cont.)Charge flow Current Charge storage Energy

Definition of current i (or I)

Note: Current has sign

S

C

dt

dq (A) i

second per coulomb 1 offlow Amperes)(in i

where q is the charge in coulomb and t is the time in sec

Examples:

(a)

right) to (leftA 165106.1910

19106.1510i

nanosecondevery direction) this (call right toflow

19106.1 value of charges unit positive 510

A

Ce

right) to (leftmA 6.13106.1610

16106.11010-i

dmicroseconevery wirea in right toflow electrons 1010

(b)


CURRENT & CURRENT DENSITY

Define Current Density by Example:

Semiconductor with free electrons

2 c m

10 cm

1 cmC2

C1

X

Wire attached to ends

sec 1025.6

/106.1

sec/1 1819

electrons

electronC

C

•Electron flow (in negative X direction) consists of

Suppose we force a + 1A current from C1 to C2. Then

•Current of 1A flows in the semiconductor (in +X direction)

•Current Density = Current / (Cross-Sectional area current goes thru)

•Current density of above = 1 A / (2 cm • 1 cm) = 0.5 A/cm2


CURRENT & CURRENT DENSITY (cont.)

Note: Typical dimensions of integrated circuit components are in the range of . What is the current density in a component with [1m]² area carrying 5A?

)cm410(m1

• A wire carrying 1 amp has a cross-section, then the current density is

2cm 2102 1 mm

2 100

2101

cmA

2A/cm 5002cm)410(1

A6105

Answer


POSSIBLE CONCEPTUAL ISSUES

1 How does charge move through the wire?

Remember a wire has a huge number of free carriers moving very fast but randomly (because of thermal energy)

Drift concept: Now add even a modest electric field

Carriers “feel” an electric field along the wire and tend to drift with it (+ sign charge) or against it ( charge carrier). This drift is still small compared to the random motion.

2 Sign of the charge carriers: It is often negative (for metals); in silicon, it can be either negative or positive…we don’t care for circuit analysis.


POSSIBLE CONCEPTUAL ISSUES (con’t)

4. Field or Current can have positive or negative sign.

Examples: I = 0.2 mA same as I = 0.2 mA going left to right going right to left

3. Electric field is in the direction positive charge carriers move. Thus if we have an electrical field in the Z direction, positive charges (ions, positrons, whatever) will experience a force in the positive Z direction. Negatively charged particles will experience the force in the –Z direction.

Thus the free carriers we will be concerned with (electrons with negative charge and “holes” with positive charge) will move against and with the electric field respectively.


POSSIBLE CONCEPTUAL ISSUES (con’t)

5. When we have unknown quantities such as current or voltage, we of course do not know the sign. A question like “Find the current in the wire” is always accompanied by a definition of the direction:

6. To solve circuits, you may need to specify reference directions for currents. But there is no need to guess the reference direction so that the answer comes out positive….Your guess won’t affect what the charge carriers are doing! Of course you will find that your intuition and experience will often guide you to define a current direction so that the answer comes out positive.

In this example if the current turned out to be 1mA, but flowing to the left we would merely say I = -1mA.

(or)I I


THINKING ABOUT VOLTAGE

but potential difference may be present whether or not a conducting path is present…

Generalized circuit element with two terminals (wires) a and b, with a potential difference across the elementab

v

a

b

Vab

“Vab” means the potential at a minus the potential at b.

If a conducting path exists between A and B, and there is a potential difference, charges will drift due to the electric field current flows

abV aVPotential is always referenced to some point ( in the example; is

meaningless without an understood reference point)


Suppose you have an unlabelled battery and you measure its voltage with a digital voltmeter. It will tell you magnitude and sign of the voltage.

SIGN CONVENTIONS

With this circuit, you are measuring

(or ). DVM indicates 1.4, so by 1.4 V. Which is the positive battery terminal? 1.401

DVM

a

b

bVaV

abV

bVaV

Answer: terminal b )aVbV(

DVM

b

a

Now you make a change. What would this circuit measure?

Answer: +1.401VNote that we have used “ground” symbol ( ) for reference node on DVM. Often it is labeled “C” or “common.”


SIGN CONVENTIONS (cont.)

How is VAD related to VAB, VBC etc ?

Lets put a bunch of batteries, say 1.5V and 9V in series to see what we already know about sign conventions:

Example 1

+ ++ 1.5V1.5V 9VA B C D

VAB = -1.5, VBC = -1.5, and VAC = -3 so clearly VAC =VAB+VBC etc.

What is VAD ? Answer: Clearly A is above D by 9V – 3V or 6V. That is consistent with VAB+VBC + VCD= -1.5 –1.5 + 9 = 6V

Example 2 How are single-subscript voltages related to double-subscript voltages?

+ ++ 1.5V1.5V 9VA B C D

V1

VX

+

+

-

-

Answer: Clearly V1 = 1.5 = - VAB = VBA

VX = -6 = VDA


SIGN CONVENTIONS (cont.)

Solve for

Just as in the physical world, we do not know a priori the sign (or magnitude) of voltages or currents; we don’t know them in the theoretical world until solving for them, so we just define voltages and currents and accept that half of the time they will be negative (quite analogously to probing with a DC meter).

Example 1 xi

Remembering Ohm’s law,

Here

mA1105.1

V5.1i

3x

mA1ix

Note that in this case, the current flows counterclockwise, i.e., opposite to arrow defining

Here

xi

mA1ix

+

1.5K1.5

+Batt

Vx

ix

Example 2 Solve for ix

+

ix

1.5K1.5

+

Vx

Batt


KEEPING THE VOLTAGE SIGNS STRAIGHT

Labeling Conventions• Indicate + and terminals clearly; or label terminals with letters• The + sign corresponds to the first subscript; the sign to the second

subscript. Therefore, Vab = - Vba

Note: The labeling convention has nothing to do with whether or not v > 0 or v < 0

+

vcd

+ vbd

d

+

1 V

a

b

V +c Using sign conventions:

1V12V

thus,2V ; 1V

cb

caab

Obviously Vcd +Vdb = Vcb too. Then if Vbd = 5V, what is Vcd?

Answer: Vcd -5 = -1 so Vcd = 4V


POWER IN ELECTRIC CIRCUITS

Power: Transfer of energy per unit time (Joules per second = Watts)

Concept: in falling through a positive potential drop V, a positive charge q gains energy• potential energy change = qV for each charge q• Rate is given by # charges/sec

Power = P = V (dq/dt) = VI

P = V I Volt Amps = Volts Coulombs/sec = Joules/sec = Watts

Circuit elements can absorb or release power (i.e., from or to the rest of the circuit); power can be a function of time.

How to keep the signs straight for absorbing and releasing power?

+ Power absorbed into element

Power delivered from element


“ASSOCIATED REFERENCE DIRECTIONS”

It is often convenient to define the current through a circuit element as positive when entering the terminal associated with the + reference for voltage

For positive current and positive voltage, positive charge “falls down” a potential “drop” in moving through the circuit element: it absorbs power

• P = VI > 0 corresponds to the element absorbing power if the definitions of I and V are associated.How can a circuit element absorb power?

By converting electrical energy into heat (resistors in toasters); light (light bulbs); acoustic energy (speakers); by storing energy (charging a battery).

Negative power releasing power to rest of circuit

VThis box representsthe re st of the c ircui t

i

Circuit element

+

-

Here, I and V are “associated”


A) For Resistor: P = i vR (because we are using “associated reference directions”)

“ASSOCIATED REFERENCE DIRECTIONS” (cont.)

1.5

+

+

vRvB

i

Hence, for the resistor, P = i vR

= +1.5mW (absorbed)

1.5K

B) For battery, current i is opposite to associated

mW 5.1i)v(P B So,

(delivered out of battery)

We know i = +1 mA


“ASSOCIATED REFERENCE DIRECTIONS” (cont.)

B) Battery: iB and vB are associated, therefore P= iB vB . Thus

1.5

+

RvB

iRiBAgain, iR = 1mA

therefore iB = -1mA

mW5.1)101(5.1P 3

A) Resistor: P = iB vB = +1.5mW


EXAMPLES OF CALCULATING POWER

Find the power absorbed by each element

Element :

Element : mW 6mA) 2V(3

Element : mW 0.5mA) 1V(0.5

mW 9mA) 33V(

Element : mW 2.5mA) 1V(2.5

+

+

V

1 V

+a

b

c

V

+

1 V2.5 mA

0.5 mA

3 mA

3V

3 mA

flip current direction:


BASIC CIRCUIT ELEMENTS

• Voltage Source• Current Source

• Resistor• Capacitor

• Inductor

(like ideal battery)(always supplies some constant given current)(Ohm’s law)(capacitor law – based on energy storage in electric field of a dielectric)

(inductor law – based on energy storage in magnetic field produced by a current)


DEFINITION OF IDEAL VOLTAGE SOURCE

Special cases:

upper case V constant voltage … called “DC”

lower case v general voltage, may vary with time

Symbol

V

Note: Reference direction for voltage source is unassociated (by convention)

Examples:

1) V = 3V

2) v = v(t) = 160 cos 377t

Current through voltage source can take on any value (positive or negative) but not infinite


IDEAL CURRENT SOURCE

Actual current source examples – hard to find except in electronics (transistors, etc.), as we will see

i

+ note unassociated

v direction

upper-case I DC (constant) value

lower-case implies current could be time-varying i(t)

“Complement” or “dual” of the voltage source: Current though branch is independent of the voltage across the branch


CURRENT-VOLTAGE CHARACTERISTICS OF VOLTAGE & CURRENT SOURCES

Describe a two-terminal circuit element by plotting current vs. voltage

V

i

Assume unassociated signs

Ideal voltage source

V

If V is positive and I is only positive

releasing power

absorbing power (charging)

But this is arbitrary; i might be negative so we extend into 2nd quadrant

But this is still arbitrary, V could be negative; all four quadrants are possible

absorbing power

releasing power


CURRENT-VOLTAGE CHARACTERISTICS OF VOLTAGE & CURRENT SOURCES (con’t)

V

iabsorbing power releasing power

Remember the voltage across the current source can be any finite value (not just zero)

Ideal current source

i

+

v

If i is positive then we are confined to quadrants 4 and 1:

And do not forget i can be positive or negative. Thus we can be in any quadrant.


RESISTOR

If we use associated current and voltage (i.e., i is defined as into + terminal), then v = iR (Ohm’s law)

+

v

i

R

Question: What is the I-V characteristic for a 1K resistor? Draw on axis below. I (mA)

V1 2 3

1

2

3

Slope = 1/R

Answer: V = 0 I = 0

V = 1V I = 1 mA V = 2V I = 2 mA

etc

Note that all wires and circuit connections have resistance, though we will most often approximate it to be zero. But we can (and do) deliberately construct circuit elements with some desired resistance, even very large values such as 10M.


CAPACITORAny two conductors a and b separated by an insulator with a difference in voltage Vab will have an equal and opposite charge on their surfaces whose value is given by Q = CVab, where C is the capacitance of the structure, and the + charge is on the more positive electrode.

We learned about the parallel-plate capacitor in physics. If the area of the plate is A, the separation d, and the dielectric constant of the insulator is , the capacitance equals C = A /d.

A d

Symbol or

But so

where we use the associated reference directions.dt

adQi

dtdv

Ci

i

+a

b

VabConstitutive relationship: Q = C (Va Vb).

(Q is positive on plate a if Va > Vb)


ENERGY STORED IN A CAPACITOR

You might think the energy (in Joules) is QV, which has the dimension of joules. But during charging the average voltage was only half the final value of V.

Thus, energy is .22

1

2

1CVQV


ENERGY STORED IN A CAPACITOR (cont.)

More rigorous derivation: During charging, the power flow is v i into the capacitor, where i is into + terminal. We integrate the power from t = 0 (v = 0) to t = end (v = V). The integrated power is the energy

i

v

+

Vv

0vdq vdt

endt

0

dtdqend

0vdt ivE

but dq = C dv. (We are using small q instead of Q to remind us that it is time varying . Most texts use Q.)

2CV21Vv

0vdv CvE


Capacitor Inductor

dt

dVCi

dt

diLv

2CV2

1E 2LI

2

1E

INDUCTORS

Inductors are the dual of capacitors – they store energy in magnetic fields that are proportional to current.

Switching properties: Just as capacitors demand v be continuous (no jumps in V), inductors demand i be continuous (no jumps in i ). Reason? In both cases the continuity follows from non-infinite, i.e., finite, power flow.

Capacitor Inductor v is continuous i is continuous i can jump V can jump Do not short circuit a charged capacitor (produces current)

Do not open an inductor with current flowing (produces voltage)

EECS 40 Fall 2002 Lecture 4 W. G. Oldham and S. Ross Lecture 4 Last time –Gate delay: why it happens (Capacitance! Resistance!) –Voltage rise and fall.

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