Review of Core Material Relevant to EN123

(1) Ohm's Law
We would like to start with Ohm's Law, Voltage = Current X Resistance, V = I * R.
But first let's get comfortable with voltage, current and resistance.

For voltage we start with Coulomb's Law, from E&M class: "opposites attract, likes repel."


In the case we're describing, the charges are stationary, not allowed to move.

Voltage between point P and G is any path integral from G to P over the conservative E field.

A fluid analogy for voltage is water pressure.

Define current as charge flow, Q/sec. Now charge can be stationary. But given a "pressure" in the form of voltage, and a pathway of resistance R, Ohm's Law says

I=Q/t, and implies charges moving at a "constant" terminal velocity. Corn syrup demo. contrast with acceleration of charge is a vacuum or plasma... F=m*a? not for Ohm's Law...
The charge on an electron is negative, so a positive current arrow implies electron flow in the oppositie direction.
Corn syrup demo. contrast with acceleration of charge is a vacuum or plasma... F=m*a? not for Ohm's Law...

Ohm's Law is about resistance.
Formula for Resistance R = rho*Length/Area
* Measuring resistance; tolerance; resistor color code

Voltage divider:

Resistors in series, parallel. G = 1/R ; conductance.

and what about 3 R's in parallel?

Current flow through by ions through membrane channels

Voltage divider, and a formula for Vout

Potentiometer (show and tell aircraft angle sensor; servo motor component... )

Power = Voltage * Current = V^2/R = I^2*R

Resistors dissipate energy, heat up. Demo: 1/4 watt resistor trying to dissipate 1W of power...

Power transfer by impedance matching...

Wheatstone bridge

Ground vs common vs reference

(2) The Leaky Integrator, and capacitance C.
Next, a hydraulic analogy, the leaky bucket, where voltage is like water pressure, current is like water flow and electrical resistance is like resistance to water flow through a pipe. See below, bucket with area A, flow in F cc/sec, water height h, and leak coefficient G cm^2/sec
(dwg by Jo Schulak)

* Resistance through pipe:

http://www.emedicine.com/radio/byname/basic-fluid-dynamics-principles---application-to-percutaneous-intervention.htm

Capacitance C is analogous to area A of the bucket. Remembering your exposure to differential equations, and assuming an initial condition of no water in the bucket, solve the leaky integrator equation above to find

C = Q/V; Q = C*V ; I = C dV/dt

where epsilon is the permittivity of the dielectric medium between the capacitor plates.

 ; look forward to defibrillator design...

Capacitors as sensors.

formulas capacitors in parallel (add), in series 1/C = 1/C1 + 1/C2.

First order natural HP and LP filters, with capacitors.

Capacitors are short circuits at high frequency and open circuits at low freq...impedance...

Laplace transform; s =j*w, frequency

How impedance is different from resistance: "complex resistance"

(3) Laplace transforms & Kirchoff's Current Law:

As you may recall from a diff eq course (AM33), the Laplace transform takes you from the time to the "frequency" domain
and allows you to solve a differential equation "algebraically".

A property of the Laplace transform, give you know F(s), the Laplace transform of f(t):

The Laplace integration property: ; see

http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/LaplaceZTable/LaplacePropTable.html

Taking the Laplace transform of the relationship between current and voltage for a capacitor,

The low pass VCVS second order filter: output = K*V+; (See Horowitz and Hill, chpt 5, p. 273...)

define KCL: Algebraic sum of branch currents into a node equals zero.

my convention at a node: current outflow is positive...

Two nodes to write KCL for, end up with

which can with a little practice be written down by inspection...

(4) Fourier Series, spectrum

good for a periodic function...
meaning of fundamental frequency, 1/f = period (omega in radians...)
how many unknowns are there? 20, no counting a0, the mean of the periodic formula
calculus book formula for solving...

solving another integral...

what about 20 equations in 20 unknowns? samples spaced how far apart?
another matrix inversion problem...

(5) Diodes, rectifiers, power supply

Current-voltage characteristic of a diode

where k is Boltzmann's constant, q is the charge on the electron and T is temperature in Kelvin.
Note the implication, corrected below, that resistance increases with temperature...

the Reverse saturation current Is (leakage current in the "wrong direction"):

which implies that for every 5 degree increase in temperature the current doubles.

ON voltages of p-n junction diode, green LED, Schottky diode
0.6, 1.7. 0.2v
Zener diodes as voltage references (H&H p. 44ff).

Effect of temperature on resistance: metals vs insulators

full wave rectifier: Wheatstone bridge again?

Power supply = transformer + rectifier + LP filter + regulator
Demo of E3631A limiting current and voltage with 4.7 ohm resistor
on 25v supply. resistance of jumpers! 2 ohms...

Transistor as "back to back diodes",
BJT thickness 1 micron of base region;
common emitter inverter; Ic/Ib current gain.
emitter follower circuit:
phototransistor and photons releasing electrons in base.

(6) Logic, E&M

ref: Digital Design from Zero to One, John Wiley & Sons, 1996.
Logic Noise Margins
for example the TTL (bipolar junction transistor logic family 7400 series)
Vcc the supply voltage is guaranteed between 4.75 and 5.25 v; below 4.75 chips may not work as advertised; above Vcc-max 7.0 volts chips may be permanently damaged.
Logic LO = 0 = FALSE = inactive = open switch is between 0v and 0.8v
Logic HI= 1= TRUE = active = closed switch is above 2.2v up to 5.0v
Region in TTL between 0.8 and 2.2 is "not allowed" Noise Margin represents the advantage of digital or analog representation; a digital recording can be faithfully reproduced, whereas reproduction of an analog "master tape" will always be degraded...

To change from LO to HI the voltage of the TTL logic gate must pass through the noise margin region--as quickly as possible--nanoseconds. An voltage between 0.8 and 2.2 must be "converted" to 0 or 1, from its metastable region. TTL chip mfgrs make chips that have stable outputs only for logic 0 or logic 1... same for any other chip family...

Base 2 numbers, hexidecimal representation: 0 1 2 3 4 5 6 7 8 9 A B C D E F
in base 2, C = 1 1 0 0
in C++ code for IAR editor hex code is 0xCF = decimal 12*16 + 15 = 207

Combinatorial Logic: logic gates "without memory" circuits without feedback.

7400 series hex inverters: input hysteresis 7414,
open collector output 7406...interface between voltage levels, drive LEDs...

C B A OUT
0 0 0    0
0 0 1    0
0 1 0    0
0 1 1    1
1 0 0    0
1 0 1    0
1 1 0    1
1 1 1    0

Realize as "Sum of Products" AND to OR two-level circuit

SEQUENTIAL: logic circuits with memory, with feedback.
SR latch
S/ R/   Q
0 0     not allowed
0 1    1 (active LO SET)
1 0    0 (active LO RESET)
1 1    no change: the memory state

----------- ---------------------- ----------------

Lorentz force Law

Faraday's Law

The circular magnetic field around a current-carrying wire:
Biot-Savart Law;Ampere's Law

How a transformer works: primary, secondary windings
Vout =(num_sec_wind/num_pri_wind)*Vin...
inductors, I = L*dv/dt
solenoid: moving a iron piston with electro-magnetic B field

Electromagnetic radiation, ultraviolet, x-rays, infrared, microwaves...the spectrum

References for BMEs:
Horowitz and Hill, The Art of Electronics
G. Bendek & F. Villars, Physics with Illustrative Examples from Medicine and Biology:
vol 3, Electricity and Magnetism