#electrical eng

Tonight’s math set up! It’s time to finally revise the dreaded inner products 😅

- a time function always uses peak V or A values - in polar form however it is always rms - Vavg=Vmean=Vdc - real and imaginary power exists because of real and imaginary I and V - S= P + Qj - S is total power, Q is imaginary power - instantaneous power is P(t) = V(t) i(t) - average power is given by P = VI cosθ

- angular freq: W = 2pi*f  = 1/RC

- convert electromagnetic energy into kenetic energy - v(t)=N(dΦ/dt)=di/dt L - can be modelled as an RL circuit - p=LI^2/2 - as current increases more energy is stored in magnetic field in inductor - if L changes then conservation of energy states that the change in electrical energy stored equals work done moving the actuator rod in/out the coil - this is basis of electric solenoid - if the rod is high reluctance, it effects inductance greatly by its presence - if rod is attached to spring, the work done on it keeps it in circuit, else with circuit off it is pushed out - F=1/2 i^2 dL/dx - F is force, x is distance moved by actuator rod - B(g)= µ0 Ni/2g - air gap is g - Fx=Fspring at equilibrium - torque T=1/2 i^2 dL/dθ where θ is angle - DC generators work by moving a coil through magnetic flux lines - there are 2 main components, the stator and the rotor - stator is stationary and holds magnetic field in place - rotor spins, holding coils which pass through field lines - e=Blv - generated emf using flux density, length of coil and velocity - emf is only induced as coils pass through flux lines - thus as coils pass parallel to flux lines nothing is induced - time taken to turn from position 1 to 2 of rotation = θ2-θ1/ω =θ2-θ1/2πf - where f is revolutions per second - peak generated = epk= NΦmax*n/15 *α - where a is pole arc/pole pitch ~0.7 - e=CnΦ in general, where C is machine constant - electromagnetic torque T=BILR - where R is radius of armature - T = K(m)ΦI(a) where K(a) is machine constant - the reversibility of electric machines is the principle that a generator can be a motor and vice versa - In a motor, n motor speed = V-IR/CΦ - e= K(e)n for back emf - If is approximately constant determined by supply voltage - R(a) is always very small - because resistance is s small current surge is possible, so starts with low controlled current - increasing R(a) reduces speed - greater resistance in field windings increases

- there are two forms of transformers, shell and core type

- a phenomenon of charge in motion, ie current creating a magnetic field - simplest form of magnetic circuit a magnetic ring around which a current carrying conductor is wrapped - magnetic flux in mag circuits relate to resistance - F=BIL -F=Bev where v is velocity - a magnetic field circuit is usually a means to an end - Φ=µ(A/l)Ni - flux is proportional to magnetomotive force -V/I=R like F/Φ=Reluctance - permeability is 1/reluctance - it is said that the coil is the supply of flux - field strength is H=NI/L - flux density =B = flux/area - µ is permiability=µ0*µr - µ0 is permeability of free space - µr is relative perm - ferromagnetic materials have perm. Of 100 to10000 - paramagnetic materials are mid to low range - diamagnetic materials have very low perm. - a solenoid is a long coil containing a large number of coils - the ohms law equivalent is Hopkinson's law - MMF=F*Reluctance - magnetic circuits can be analysed in electrical circuit form - flux is equal in series and mmf is constant in parallel, like I and V - fringing is when magnetic flux lines pass in loops outwards from the mag circuit - B=µH -

- A filter removes certain bands of frequencies from a signal and allowing others through - pass band allows certain frequencies though - stop band stops certain frequencies - RLC circuits are used to set the pass and stop bandwidths - filters are often used for removing noise from signals - active rc filters made from op amps are used for v low frequencies - digital filters are most commonly used today - low pass filters allow frequencies from 0 to a corner frequency - after the corner frequency, almost all frequencies are attenuated - where nothing passes is a stop band -  the corner frequency is given by ωc=1/√LC - this is also the bandwidth - for damping factor:  ζ= (1/2ωc)*√L/C - high pass does the same but opposite - is orientated opposite to low pass - band pass allows certain defined frequencies through - gives a bell curve - al frequencies above 50% output power are considered part of the pass band - the center of the bandwidth is 1/√LC - the bandwidth is Δω=Rl/L - band stop do the opposite - it's bandwidth stretches beyond either end of its stop band - can be narrow or wide - 50% point and below is bandwidth of stop pass - band pass has RLC in series - band stop have R in parallel with LC - the attenuation rate or order is the gradient of the slope beyond the -3dB gain point - passive and active filters exist - passive filters however have a tendency to affect frequency -

- In many cases it is desirable to express a phasor  with respect to another phasor. - use the shortest time interval when finding phase and use smallest phase angle - values must be calculated as rms - if angle v1-angle v2 • = < 0, v1 is leading • = >0, v2 is leading - found by phasor V1/V2 - adding the two phasors gives the resultant magnitude - an rlc circuit is a resistor inductor capacitor circuit - X=Xl-Xc - remember, reactance is purely capacitive or inductive, thus has phase of ±90 deg for one component - Xl = 2pi*fL - Xc = 1/2pi*fC - impedance Z = ZL+ ZC - in parallel use product over sum - ZRLC=R-j(XL-XC) - if total phase is positive, circuit is more inductive - more negative means more capacitive - angle 0 is resonance, and thus circuit is purely resistive - this means current is in phase with voltage