electronics

electrostatics

electrostatics

electric force

F = Q1Q2 / 4 x pi x r² x permittivity of free space

electric field strength

Q1 / 4 x pi x r² x permittivity of free space

electric potential integrated

Q1 / 4 x pi x rA x permittivity of free space

breakdown voltage

Vb = Eb x d

what is L in amperes law

• depends on shape of magnetic field

• for straight wire or toroid L = 2πr

derive magnetic flux density using amperes law

H x L = Ienc = NI

H x (2πr) = Ienc. B = UoUrH

B = Ienc x Uo x Ur / 2πr

assumptions with air gaps

• no leakage flux

• no fringing flux (assume all flux goes straight through air gap)

• flux follows a mean path through the core

leakage flux

• flux that doesn’t go through windings bu goes through surrounding air

fringing flux

• flux that doesn’t go straight through air gap, but bends around sides

reluctance of air gap

Rg = lg / Uo x A

reluctance of iron

Ri = 2πr - lg / Uo x Ur x A

magnetomotive force equation

F = NI = Rt x flux

faradays law

EMF induced = rate of change of total flux (depends on how quickly you move the magnet)
- e= dψ/dt = dφ/dt

lenz’s law

the direction of an induced current will always oppose the change in magnetic flux that produced it

important to note (electromagnetics)

• no EMF induced if you dont move conductor or magnet

• an EMF is induced NOT a current

• current will prodcue its own magnetic field

force acting on a charge (within magnetic and electric field)

F=BIL

self flux linkage

ψ=Nφ

self flux linkage in air or linear magnetic material

ψ=Li
- L is self inductance

self inductance (L)

the self inductance of a coil measures the coils ability to store energy in its magnetic field
- unit Henry(H)

self inductance in faradays law

e= -L di/dt

leakage inductance

L_leak=ψ_leak/I

total self inductance

L = L + L_leak

iron core is infintely permeable with air gap, whats reluctance

• if iron infinitely permeable - Rcore = 0

• Rt = Rair

MMF equation

MMF = NI = R_tφ

• MMF does not depend on reluctance

• MMF stays fixed, if reluctance increases, flux drops

• like V=IR

MMF

• about producing a magnetic field in a core

• for finding how much magnetic flux produced

capactiance between parallel plate

C = e_re₀ A / d

B

mag flux density

φ

magnetic flux

Ψ

magnetic flux linkage

Ψ equation

Ψ=Nφ

φ (flux) equation

φ=BAcostheta (theta is angle to normal of plane)
- those r the things that can change flux inducing emf

emf through changing area

e = -NBAωcosωt

stator

• usually outside machine

• holds the field

rotor

• inside

• holds the armature (windings)

commutator

• In generators emf induced=AC (sine wave)

• commutator reverses current direction every half turn (electrically)

  • 180-360 coil end on the positive brush moves to negative brush

• so output is DC (on voltage-time always positive)

back emf

• voltage generated in a motor due to the rotating coil, that opposes applied voltage

magnetic saturation

• when mag flux becomes constant even if increase field, as magnetic material is saturated

motor torque production

• both sides of coil experience opposite forces

• one push up one push down (turns)

• commutator

motor energy conversion

electricity to mechanical work
- current into the machine (positive)

generator energy conversion

mechanical work to electricity
- current out of the machine (negative)

separately excited machine

armature winding (coils) and field winding (acting magnet) not connected to the same source

separately excited machine

• defined like motor (pos current)

armature power, P_a (separately excited circuit) IF YOU IGNORE LOSSES

E_aI_a = electrical power = mechanical power

voltage equation for all machines

motor: E_a = V - I_aR_a (V > E_a) (I is positive though)

generator: E_a = V + I_aR_a (E_a > V) (I is negative)

“internal EMF” and “torque” equations with constant flux (mag field) ****

E_a = k_ww (speed relates to voltage)
T=k_wI_a (torque relates to current)

armature power WITH LOSSES

• P_a = P_e - P_arm = E_aI_a

• converted power = electrical input power - losses in the armature winding

electrical input power (P_e)

• P_e=VI_a

losses in armature winding (P_arm)

• P_arm = I_a²R_a

• due to resistance (so no Parm: R_a = 0)

mechanical power

• P_m=P_a - P_fw = Tw

• P_fw = rotational losses in the system (friction and windage of machine)

rpm to rad/s

N x 2pi/60

Methods of Speed Control in machines

• varying supply voltage (reduce)

• field weakening - reduce strength of mag field

varying supply voltage

• usually to reduce speed

• because cannot exceed max armature voltage

• add too much current causes overheating in windings

field weakening

• reduce mag. field

• exceeds the rated speed on armature wihtout changing voltage

• controlling field current I_f (cant be done on perm. magnet machine)

• only works on separately excited cause need diff sources

field weaking eq. (neglecting R_a (armature loss))

w = V / k₁I_f

starting current

• rotor starts stationary so E_a = 0 as voltage relates to speed

• from voltage equation I_a = V / R_a

• as R_a is very small, leads to high starting current

starting resistor

• required to reduce starting current

• variable

• turns to 0 once started

• find from I = V / R with new I

series-excited machine

field winding is in series with the armature winding

• same current in both windings

• I = I_a = I_f (apply for finding w and T equations)

• can get high torque and low speed

• cant do field weakening

series-excited equations

• E_a = V - I(R_a + R_f)

• T = k₁I²

• speed is same just I_f Is I

shunt-excited machine

field winding is in parallel with the armature winding

• same voltage in each parallel branch

• is a way to do field weakening or armature voltage control

no-load meaning

T=0

• so assume Pfw = 0 so Ia = 0

permanent magnet

• assume constant field

node

• a region of wires where the voltage is the same (represented by a point)

• think if you can move across a wire without crossing a component then its the same node

branch

starts at one node and ends at another connects them

• same current flow through one branch

parallel connection

components share same voltage

series connection

components share same current

KCL

sum of current in a node = 0

KVL

sum of voltage in a loop = 0

combining resistors in parallel

R1R2/R1 + R2

potential divider equation (for voltage across branch)

V_{above R2}= VR₂ / (R₁ + R₂)
V_{above R1}= VR₁ / (R₁ + R₂)

current divider equation (for current across node)

I₁= VR₂ / (R₁ + R₂)

voltmeter ideal resistance

Rin very large

ammeter ideal resistance

Rin very low

photodiode equation

I_p=RL

• I_p = photocurrent

• R = responsivity

• L = ligth intensity

Resistance complicated equation

R = l/A ρ

strain gauge resistance

ΔR/R = G(epsilon)

strain gauge voltage

V_out = -V ΔR/R

Vin for OP amps

Vin = V⁺ - V^-

• V⁺ = non-inverting input

• V^- = inverting input

Vout for OP amps

Vo = GVin

• G = gain

capacitor voltage (RC circuit)

V_c (t) = V₁ (1 - e^{-t / RC} )

• changes exponentially when theres a step change in input (not continuous like AC circuit)

KVL for RC circuit

V_in = V_r + V_c
input voltage = resistor voltage + capacitor voltage

RC

RC is defined as time constant

• controlled by selection of components

• determines how fast voltage/current reaches final value

small t

• exponential decays quicker

• final value reached faster

CIVIL

• in a capacitor I leads V by pi/2 (voltage down)

• in an inductor V leads I by pi/2 (voltage up)

• current in phase with voltage for resistors