Electrical Engineering Units and Formulas

Unit of Measurement

Symbol Formulas NotesDC AC

Ohm’s Law E=IR= PI=√PR E=IZ ; I= E

Z; Z= E

T

Current A - Amp I I=VR

I= PV

I=√PR

I= EZT

IS (RMS) = VS / ZT (A)IC (RMS) )¿V S BC , AIL (RMS)¿V SY L , A

IR + IC + IL = IS (RMS), AIS = VS * YT

Voltage V - Volt V,E V=IR

V= PI V=√PR

E=IZV R=I ZR

V C=I ZC

V R=ZR E

Z R+Z L+ZCV C=

ZCEZR+ZL+ZC

Resistance Ω - Ohm R R=VI

R= PI 2

R=V 2

P

Series: RT=R1+R2+R3+…+Rn+…

Parallel: RT=R1 R2R1+R2

Power W - Watt P PR=V R I SP=I 2RP=V 2

RPS=V S I S cosθ ° PT=EI cosθ

Capacitance F – Farad CSeries:

1C1

+ 1C2

+…+ 1Cn

+…

Parallel: CT=C1+C2+C 3+…+Cn+…Inductance H – Henry L Series: LT=L1+L2+L3+…+Ln+…

Parallel: LT=L1L2L1+L2

Impedance Ω - Ohm Z Total Circuit Impedance: ZT=R+ j(X ¿¿ L−XC)¿

ZT=1Y T

Inductive ZL=RL + j XL

Impedance (Z) is the total measure of opposition to electric current and is the complex (vector) sum of (“real”) resistance and (“imaginary”) reactance.

Admittance S - Siemens Y Y= 1Z=G+ jBTotal Circuit Admittance: Y T=

1R

+Y L+BC Admittance (Y), the ease at which a circuit composed of resistances and reactances allows current to flow when voltage is applied, is the reciprocal of impedance (Z).

Conductance S - Siemens G G= 1R

G= RR2+X2

Conductance (G), the ease at which a resistor allows current to flow, is the reciprocal of resistance (R).

Susceptance S - Siemens B BL=1X L

, BC=1XC

Susceptance (B), the ease at which a reactance allows current to flow, is the reciprocal of reactance (X).

Angular Velocity

Radians per second ω ω=2πf

Period s – seconds T T=1f

Frequency Hz - Hertz f f= 1T

f (Frequency) = 1/T (period)

Inductive Reactance

Ω - Ohm XL X L=ωL=2πfL

Capacitive Reactance

Ω - Ohm XC XC=1ωC

= 12πfC

Gain dB - Decibel dB=10 log10P1P2

Define levels of audio, voltage gain, energy, field strength, etc.

v (t )=A sin (2π ft+θ )

v ( t ) is the value of the voltage at any point in time.

A is the signal voltage amplitude in volts peak (VPK)

f is the signal frequency in Hz, also known as cycles per second.

t is for time in seconds.

θ is the angular phase shift with units of degrees or radians.

Decibels

Unit of Measurement


If N=(b)x , then x=logbN

log e x=2.3 log10 x Base conversion

Value=10x×10d1d2

Logarithmic level at a particular point between known levels using a ruler or simply estimating the distances.

log10 1=0log10 ab=log10 a+ log10 b

log10ab=log10 a− log10b

log10 an=n log10a

Properties of Logarithms

Power Gain 1 bel = 10 decibels B, dB

B=log10P1P2

dB=10 log10P1P2

dBm=10 log10P

1mW

Two power levels can be compared using a unit of measure calle3d the bel. However, to provide a unit of measure of less magnitude, a decibel is defined.

The resulting equation compares power levels P2 and P1

in decibels.

For some applications, a reference level is established to permit a comparison of decibel levels from one situation to another. For communications systems, a commonly applied reference level is: Pref=1mW (across a 600 Ω load)

Voltage Gain 1 bel = 10 decibels B, dBdBv=20 log10

V 2

V 1

Decibels are also used to provide a comparison between voltage levels. This is done by substituting in the basic power equations.

Sound Pressure Level (SPL)

1 bel = 10 decibels B, dB dB s=20 log10P

0.0002µbarTo establish a basis for comparison between audio levels, a reference level of 0.0002 microbar (µbar) was chosen, where 1 µbar is equal to the sound pressure of 1 dyne per square centimeter, or about 1 millionth of the normal atmospheric pressure at sea level. The 0.0002 µbar level is the threshold level of hearing.

Filters

Unit of Measurement


Magnitude

Phase Angle

Cutoff Frequency

Magnitude of the ratio

V o/V i=A v=V o

V i=

XC

√R2+X2C= 1

√( RX C )

2

+1

Phase Angle is determined by θ=−90 °+tan−1 XC

R=− tan−1 R

X C

Frequency at which XC = R is determined by f c=1

2πRC

Frequency at which XL = R is determined by f c=R2πL

R-C LOW-PASS FILTERS

f c=1

2πRC

For f <f c ,V o>0.707V i

whereas for f >f c ,V o<0.707V i

At f c ,V o lagsV iby 45 °

Magnitude

Phase Angle

Cutoff Frequency

Magnitude of the ratio

V o/V i=A v=V o

V i= R

√R2+X2C= 1

√1+( XC

R )2

Phase Angle is determined by θ=tan−1 XC

R

Frequency at which XC = R is determined by f c=1

2πRC

Frequency at which XL = R is determined by f c=R2πL

R-C HIGH-PASS FILTERS

f c=1

2πRC

For f <f c ,V o<0.707V i

whereas for f >f c ,V o>0.707V i

At f c ,V o leadsV iby 45 °

PASS-BAND FILTERS

BAND-REJECT FILTERS

Octave Frequency Hz Octave above f=2 f c Two frequencies separated by a 2 : 1 ratio are said to be an octave apart. For Bode plots, a change in frequency by one octave will result in a 6 dB change in gain.

Octave below f=f c

2

Decade Frequency Hz Decade above f=10 f c

Decade below f=f c10

Two frequencies separated by a 10 : 1 ratio are said to be a decade apart. For Bode plots, a change in frequency by one decade will result in a 20 dB change in gain.

Transformers

Unit of Measurement

Symbol Formulas Notes

Primary and Secondary Coil Voltage

V- Volts e p

es

e p=N p

dϕ p

dt ; e p=Lp

dipdt

; e p=Mdisdt

es=N s

dϕm

dt ; es=N s

dϕ p

dt ; es=M

di pdt

The coil to which the source is applied is called the primary, and the coil to which the load is applied is called the secondary.

Coefficient of Coupling

ratio kk=

ϕm

ϕ p

Since the maximum level of ϕm isϕ p, the coefficient of coupling between two coils can never be greater than 1.

Mutual Inductance

H – Henry MM=N s

dϕm

dip

M=N p

dϕ p

disM=k √Lp Ls

Mutual inductance between two coils is proportional to the instantaneous change in flux linking one coil due to an instantaneous change in current through the other coil.

N s is the number of turns in the secondary winding and ϕm is the portion of the primary flux ϕ p that links the secondary winding.

Ep

E s=N p

N s

The ratio of the magnitudes of the induced voltages is the same as the ratio of the corresponding turns.

V g

V L=N p

N s

The instantaneous values of e1 and e2 are related by a constant determined by the runs ratio.

Transformation Ratio

aa=

N p

N s

The ratio of N p / N s usually represented by the lowercase letter a, is referred to as the transformation ratio.

Primary and Secondary Currents

Amps / Turns I/N I p

I s=

N s

N p

The primary and secondary currents of the transformer are related by the inverse ratio of the turns.

Reflected Impedance and Power

Zp=a2Z L

Bipolar Junction Transistors

Unit of Measurement


In the active region of a transistor, the base-emitter junction is forward-biased, whereas the collector-base junction is reverse-biased.

In the cutoff region the base-emitter and collector-base junctions of a transistor are both reverse-biased. In the saturation region the base-emitter and collector-base junctions are forward-biased.

V BE=0.7V Transistor in ON state the Base to Emitter Voltage will be assumed to be V BE=0.7V

α dc=ICI E

α= ββ+1

The quantity alpha (a) relates the collector and emitter currents and is always close to one.

βdc=ICI B

β= α1−α

The quantity beta (β) provides an important relationship between the base and collector currents, and is usually between 50 and 400.For linear amplification purposes, cutoff for the common-emitter configuration will be defined by IC = ICEO

Transistors

Unit of Measurement


Current A – Amps I I E=IC+ IB Applying Kirchoff’s Current Law

IC=α I E+ ICBO BJTIC≅ I E Common Base configuration

ICEO=ICBO1−α

¿I β=0µA

ICEO=β ICBOIC=β I B

I E=(β+1) IB

Common Emitter Configuration

I i=V i

Ri

Transistor Amplifying Action

Base-Emitter

Collector-Emitter

A - Amps IB

IC

IB=V CC−V BE

RB

IC=β I B

Fixed Bias Configuration

Collector-Emitter V - Volts V CE V CE=V CC−ICRC

V CE=V C−V E

V CE=V C

V BE=V B−V E

V BE=V B

IC sat=V CC

Rc

Collector-Emitter Loop

Bipolar Junction Transistors

DC Biasing-BJTs

gm0=2 IDSS|V P|

=2∗3mA|−2V| =3mS

gm=gm0(1−V GSQ

V P)

gm=3mS(1− −2V

)

Operational Amplifiers

V o=R f

R1V 1

Operational Amplifiers

V o=AdV d+A cV c

Electrical Engineering Units and Formulas

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