Review of Core Material Relevant to EN123

(1) Resistors & 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: Electrostatics.

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

Ground vs common vs reference

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

Ohm's Law is about resistance.
Calculating resistance R = rho*Length/Area, where rho is resistivity, a material property. What is effect of increasing temperature on resistance? For metals, increasing resistance, for semiconductors, decreasing.

* Measuring resistance; tolerance; resistor color code

Voltage divider:
Vout = (R2/(R2+R1)) * Vcc

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

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

and what about 3 R's in parallel? add conductances.

Current flow through by ions through membrane channels
ions as charge carriers in aqueous solution. electrochemistry.

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

Thevenin - Norton equivalents

Resistivity rho (a material property):

Current-voltage characteristic of a diode: temperature dependence?!

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.

demo: 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).

full wave rectifier: Wheatstone bridge again...

* Resistance through pipe:

node admittance matrix: KCL: current flow into a node is negative.
V = inv(nY)*Is
which can with a little practice be written down by inspection...
example with all-resistor network: node voltages, branch currents.

---------------- ------------------- --------------------- end of Resistors

(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)

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 for 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"

Laplace transforms--

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, given you know F(s), the Laplace transform of f(t):

The Laplace integration property: ; see

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 handout, p. 273...)

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

repeat node-admittance-matrix convention: current flowing out of a node is positive...

Filter with two capacitors: Two nodes to write KCL for, end up with

solving with a dependent source.

Fourier Series, spectrum

good for a periodic function...
meaning of fundamental frequency in radians, omeg0 = 2*pi/T where T is "period"
how many unknowns are there? 20, no counting a0, the mean of the periodic formula
calculus book formula for solving...

solving another integral...

but if we are recording/sampling waveform h(t) we don't know it analytically.
what about 20 equations in 20 unknowns? samples spaced how far apart?
another matrix inversion problem...

LP and HP natural first order filters.

---------------- ------------------- --------------------- end of Capacitors

(3) Inductors and Faraday's Law

Faraday's Law

V = I * R
Q = C * V
φ = L * I

flux = phi = L * I ==> V = L dI/dt ==> V(s) = s*L*I(s)

The unit of inductance is the Henry. self inductance L, and mutual inductance M (between 2 coils).

Griffiths p. 295: calculate inductance of a solenoid radius R, length l, turns N/l: L = mu * pi * R^2*l*N...

what about minus sign in Faraday's Law? from Griffiths:
"Inductance (like capacitance) is an intrinsically positive quantity. Lenz's Law, which is enforced by the minus sign, dictates that the EMF (voltage) is in such a direction as to oppose any change in current. For this reason, it is called a back EMF. Whenever you try to alter the current in a wire you must fight againt this back EMF. Thus inductance plays somewhat the same role in electric circuits that mass plays in mechanical systems: The greater L is, the harder it is to change the current, just as the larger the mass, the harder it is to change an object's velocity."

Inductors in series, in parallel.

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...

demo solenoid: moving a iron piston with electrically-induced magnetic field B

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...

How a switching power supply works to have no loss at the regulator.

The role of an inductor in a defibrillator. Second order system, overdamped please.

How transformers work:

Lorentz force Law

Transistor as "back to back diodes",
Braitten, Bardeen and Shockley, Bell Labs, 1949
doping silicon crystal: column 5: phosphorus for n-type, column 3: boron for p-type

BJT thickness 1 micron of base region; emitter more doped than collector;
AZ: electrons moving across E-B junction can diffuse across base to collector and then move up; a few holes go the opposite direction. See energy vs distance curve for another POV.
What happens if base is too thick? recombination...
Wikipedia: To minimize the percentage of carriers that recombine before reaching the collector–base junction, the transistor's base region must be thin enough that carriers can diffuse across it in much less time than the semiconductor's minority carrier lifetime. In particular, the thickness of the base must be much less than the diffusion length of the electrons.
common emitter inverter; Ic/Ib current gain = 100
emitter follower circuit: a GFI solenoid in the follower.

phototransistor and photons releasing electrons in base.
2N4401 and heat sink.

FETs: better saturation voltage, better freq response. High input impedance.

Horowitz and Hill, The Art of Electronics

---------------- ------------------- --------------------- end of Inductors.

(4) Logic, EM spectrum, electrochemistry half cells

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...

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

: logic circuits with memory, with feedback.
Set Reset Latch with output Q
S R   Q
0 0    no change: the memory state! 
0 1    0 (RESET)
1 0    1 (SET Q to logical 1)
1 1    not allowed!

toggle flip flops


states, and Finite state machines.

instructions: source, function, destination.

Coding, functions, interrupts.

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Electromagnetic radiation spectrum:
power lines, radio, microwave/radar, infrared, visible, ultraviolet, x-rays, gamma rays, cosmic rays.

half cells of electrochemistry

Reference for BMEs:
G. Bendek & F. Villars, Physics with Illustrative Examples from Medicine and Biology:
vol 3, Electricity and Magnetism