Fundamental Physics/Electricity/Electricity formulas

Charged particle

Charged Particle	Symbol	Mass	Charge
Electron	$e^{-}$	$m_{e}$	Negative (-)
Proton	$p^{+}$	$m_{p}$	Positive (+)
Neutron	$n^{o}$	$m_{n}$	Zero (0)

Electric charge

Electric charge process

Charge type	Charge acquired process
Negative charge	Neutral object receives electron	$O+e$
Positive charge	Neutral object gives electron	$O-e$

Electric charge characterisitics

Charge type	Charge quantity	Electric field	Magnetic field
Negative charge	-Q		B ↓
positive charge	+Q		B ↑

Electric charge interaction force

Electrostatic force	Coulomb force is used in calculating the force of negative charge to attract positive charge	$F_{Q}={\frac {\|Q_{+}\|\|Q_{-}\|}{r^{2}}}$
Electrodynamic force	Ampere force is used in calculating the force of electricity to make electric charge to move in a straight line path	$F_{E}=QE$
Electromagnetomotive force	Lorentz force is used in calculating the force of magnet to make electric charge to move in a straight line perpendicular to the original path	$F_{B}=\pm QvB$
Electromagnetic force	is the sum of the 2 electrodynamic and electromagnetomotive force	$F_{EB}=F_{E}+F_{B}=Q(E\pm vB)$

Electricity

Electricity source

Electricity can be generated from the following sources below

Electricity souce	Picture	Types of electricity
Electrolysis		DC electricity
Electrovoltaic		DC electricity
Photonvoltaic		DC electricity
Electromagnetic induction		AC electricity

Electricity types

Electricity types	definition	Mathematical formula
DC electricity	provides constant voltage over time	$v(t)=V$
AC electricity	provides time varying voltage over time	$v(t)=V\sin(\omega t+\theta )$

Electricity and matter

Matter that interacts with electricity are divided into three types Conductor, Semi Conductor, and Non Conductor

Conductor	referes to all matter allow current to flow easily like Metal for example Ferrite, Copper	Conductors are used in making passive electric components like Resistor, Capacitor, Inductor
Semi Conductor	referes to all matter does not allow current to flow easily like Semi Conductor for example Silicon, Germanium	Semi - Conductors are used in making active electric components like Diode, Transistor, FET
Non Conductor	referes to all matter does not allow current to flow like Rubber

DC and AC response of electric conductor

Resistor

Characteristics	DC	AC
Voltage	$V=IR$	$v=iR$
Current	$I={\frac {V}{R}}$	$i={\frac {v}{R}}$
Resistance	$R={\frac {V}{I}}$	$R={\frac {v}{i}}$
Power provided	$P_{V}=IV$	$P_{V}=iv$
Power Loss	$P_{R}=I^{2}R(T)={\frac {V^{2}}{R(T)}}$	$P_{R}=i^{2}{\frac {v^{2}}{R(T)}}$
Power delivered	$P=P_{V}-P_{R}$
Reactance		$X_{R}=0$
Impedance		$Z_{R}=X_{R}+R=R$
Phase		$0$

Capacitor

Characteristics	DC	AC
Voltage	$V=QC={\frac {W}{Q}}$	$v={\frac {1}{C}}\int idt$
Charge	$Q={\frac {V}{C}}$
Capacitance	$C={\frac {V}{Q}}$
Current	$I={\frac {Q}{t}}$	$i=C{\frac {dv}{dt}}$
Power provided	$P_{V}=IV=({\frac {Q}{t}})({\frac {W}{Q}})={\frac {W}{t}}$	$p={\frac {1}{2}}Cv^{2}$
Reactance		$X_{C}(t)={\frac {v}{i}}$ $X_{C}(j\omega )={\frac {1}{j\omega C}}$ $X_{C}(\omega \theta )={\frac {1}{\omega C}}\angle -90$
Impedance		$Z_{C}(t)=X_{C}+R_{C}$ $Z_{C}(j\omega )={\frac {1}{j\omega C}}+R_{C}$ $Z_{C}(\omega \theta )={\frac {1}{\omega C}}\angle -90+R\angle 0$
Phase		$Tan\theta ={\frac {1}{\omega T}}$
Time Constant		$T=CR_{C}$

Inductor

Characteristics	DC	AC
Magnetic Field Strength	$B=LI$
Current	$I={\frac {B}{L}}$
Inductance	$C={\frac {B}{I}}$
Current	$I={\frac {Q}{t}}$
Power provided	$P_{V}=IV=({\frac {B}{l}})({\frac {W}{Q}})={\frac {W}{t}}$
Reactance		$X_{L}(t)={\frac {v}{i}}$ $X_{L}(j\omega )=j\omega L$ $X_{L}(\omega \theta )=\omega L\angle 90$
Impedance		$Z_{C}(t)=X_{C}+R_{C}$ $Z_{C}(j\omega )=j\omega L+R_{L}$ $Z_{C}(\omega \theta )=\omega L\angle 90+R\angle 0$
Phase		$Tan\theta =\omega T$
Time Constant		$T={\frac {L}{R_{L}}}$

Diode

A device made from joining one negative typed semi conductor with one positive typed semi conductor

- o--[N|P]--o +

Diode's circuit sysmbol

The IV chracteristic of diode is specified in IV Graph

As seen in the graph above the diode actually works in both the forward region and the backward region. In the forward region the value of I and V are positive and in the backward region I and V are negative.

Diode does not conduct current or turned OFF at

V=V_{d}

of

I=0

Diode starts to conduct current or turned ON at

V=V_{d}

of

I=1mA

Diode conducts current at

V>V_{d}

at

I=I_{o}[e^{({\frac {V_{d}}{nV_{T}}})t}-1]

Transistor

Transistor	Symbol	Construction
NPN transistor		o-- [N P N] --o
PNP transistor		o-- [P N P] --o

Manufacturer's IV Graph specification tell transistor operation

V<V_{d}.I=0

. Transistor is OFF

V=V_{d}.I=1mA

. Transistor is ON

V>V_{d}.I=I_{o}e^{\frac {V}{V_{d}}}

V=V_{S}.I=I_{s}

. Transistor is saturated

With the right connection, by connecting transistor with resistor(s) Transistor can acts as

An electronics amplifier

i_{b}>0

i_{e}>\alpha i_{b}

i_{c}>\beta i_{b}

Provided that, transistor should be turned ON with biased voltage at the base must be greater than diode's break over voltage

v_{b}>v_{be}

.

v_{be}=v_{d}

An electronics switch

Transistor is switch On when

V_{b}>V_{be}

Transistor is switch Off when

V_{b}<V_{be}

Electric device

Electric device referes to an electric circuit of many components connected in a closed loop to perform a specific task

Electric exponential decay

Electric exponential decay is an electric device that can generate current or voltage decay in the form of an exponential decay

Electric current decay circuit	Configuration	Formula
RL series		$V_{L}+V_{R}=0$ $L{\frac {di}{dt}}+iR=0$ ${\frac {di}{dt}}=-{\frac {1}{T}}i$ $T={\frac {L}{R}}$ $\int {\frac {di}{i}}=-{\frac {1}{T}}\int dt$ $Lni=-{\frac {1}{T}}+c$ $i=e^{-{\frac {1}{T}}t+c}=Ae^{-{\frac {1}{T}}t}$ $A=e^{c}$
/RC series/		$V_{C}+V_{R}=0$ $C{\frac {dv}{dt}}+{\frac {v}{R}}=0$ ${\frac {dv}{dt}}=-{\frac {1}{T}}v$ $T=RC$ $\int {\frac {dv}{v}}=-{\frac {1}{T}}\int dt$ $Lnv=-{\frac {1}{T}}+c$ $v=e^{-{\frac {1}{T}}t+c}=Ae^{-{\frac {1}{T}}t}$ $A=e^{c}$

Electric oscillation wave

Electric oscillation wave is an electric device that can generate sinusoidal wave

Electric current oscillation wave		$L{\frac {di}{dt}}+{\frac {1}{C}}\int vdt=0$ ${\frac {d^{2}i}{dt}}+{\frac {1}{T}}=0$ ${\frac {d^{2}i}{dt}}=-{\frac {1}{T}}$ $i(t)=e^{\pm j{\sqrt {\frac {1}{T}}}t}=e^{\pm j\omega t}=ASin\omega t$ $\omega ={\sqrt {\frac {1}{T}}}$ $T=LC$
Electric current oscillation standing wave		$Z_{L}-Z_{C}=0$ $Z_{C}=-Z_{L}$ ${\frac {1}{\omega C}}=-\omega L$ $\omega =\pm j{\sqrt {\frac {1}{LC}}}$ $V_{C}=-V_{L}$ $V(\theta )=ASin(\omega _{o}t+2\pi )-ASin(\omega _{o}t-2\pi )$
Electric current decay oscillation wave		$L{\frac {di}{dt}}+{\frac {1}{C}}\int idt+iR=0$ ${\frac {d^{2}i}{dt}}+{\frac {R}{L}}{\frac {di}{dt}}+{\frac {1}{LC}}i=0$ ${\frac {d^{2}i}{dt}}=-2\alpha {\frac {di}{dt}}-\beta i$ $\beta ={\frac {1}{T}}={\frac {1}{LC}}$ $\alpha =\beta \gamma ={\frac {R}{2L}}$ $T=LC$ $\gamma =RC$ Root of equation 1 real root . $\alpha =\beta$ $i=Ae^{-\alpha t}=A(\alpha )$ 2 real roots . $\alpha >\beta$ $i=Ae^{-\alpha \pm {\sqrt {\alpha -\beta }}t}$ 2 complex roots . $\alpha <\beta$ $i=Ae^{-\alpha \pm j{\sqrt {\beta -\alpha }}t}$ $i=Ae^{-\alpha t}e^{\pm j{\sqrt {\beta -\alpha }}t}$ $i=A(\alpha )Sin\omega t$ $A(\alpha )=Ae^{-\alpha t}$ $\omega ={\sqrt {\beta -\alpha }}$
Peak electric current oscillation wave		$Z_{L}=-Z_{C}$ $\omega _{o}={\sqrt {\frac {1}{T}}}$ $T=LC$ $Z_{t}=R$ $i={\frac {v}{R}}$ $i(\omega =0)=0$ $i(\omega =\omega _{o})={\frac {v}{R}}$ $i(\omega =00)=0$

Voltage stabilizer

Voltage stabilizer provides table voltage over frequency of time

Low frequency filter	${\frac {v_{o}}{v_{2}}}={\frac {\frac {1}{j\omega C}}{R+{\frac {1}{j\omega C}}}}={\frac {1}{1+j\omega T}}$	$T=RC$	$\omega _{o}={\frac {1}{T}}={\frac {1}{RC}}$
Low frequency filter	${\frac {v_{o}}{v_{2}}}={\frac {R}{R=j\omega L}}={\frac {1}{1+j\omega T}}$	$T={\frac {L}{R}}$	$\omega _{o}={\frac {1}{T}}={\frac {R}{L}}$
High frequency filter	${\frac {v_{o}}{v_{2}}}={\frac {j\omega T}{1+j\omega T}}$	$T=RC$	$\omega _{o}={\frac {1}{T}}={\frac {1}{RC}}$
High frequency filter	${\frac {v_{o}}{v_{2}}}={\frac {j\omega T}{1+j\omega T}}$	$T={\frac {L}{R}}$	$\omega _{o}={\frac {1}{T}}={\frac {R}{L}}$
Band pass filter	${\frac {v_{o}}{v_{i}}}=({\frac {1}{1+j\omega T_{L}}})({\frac {j\omega T_{H}}{1+j\omega _{H}}})$ $T_{L}={\frac {L}{R}}$	$T_{H}=RC$ $\omega _{L}-\omega _{H}={\frac {R}{L}}-{\frac {1}{RC}}$
Band pass filter	${\frac {v_{o}}{v_{i}}}=({\frac {1}{1+j\omega T_{L}}})({\frac {j\omega T_{H}}{1+j\omega _{H}}})$ $T_{L}=RC$ $T_{H}={\frac {L}{R}}$	$\omega _{L}-\omega _{H}={\frac {1}{RC}}-{\frac {R}{L}}$
Resonance tuned selected band pass filter	${\frac {v_{o}}{v_{i}}}={\frac {R}{R+j\omega L+{\frac {1}{j\omega C}}}}$ $\omega =\omega _{1}-\omega _{2}$ $v_{o}(\omega =0)=0$ $v_{o}(\omega =\omega _{o})=v_{i}$ $v_{o}(\omega =00)=0$

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