Voltage drop

**Voltage drop** is the decrease of electrical potential along the path of a current flowing in an electrical circuit.

Kirchhoff's circuit laws states that the sum of the voltage drops around any closed loop is equal to the supply voltage.

Electricity can be thought of as being like fluid flowing in a pipe. Voltage can be thought of as being like the pressure of the fluid in the pipe.

## Voltage drop in direct-current circuits: resistance

See also: Pull-up resistor
Ohm's Law can be used to calculate voltage drop.

# V
I
R

{\displaystyle V = IR}

.
Consider a direct-current circuit with a 300 volt supply voltage and two resistors of 50 ohms and 100 ohms connected in series. 2 amps of current will flow because

300
 Volts
=
2
 Amps
∗
150
 Ohms

{\displaystyle 300 \text{ Volts} = 2 \text{ Amps} \* 150 \text{ Ohms}}

The voltage drop across the first resister is 100 volts

100
 Volts
=
2
 Amps
∗
50
 Ohms

{\displaystyle 100 \text{ Volts} = 2 \text{ Amps} \* 50 \text{ Ohms}}

The voltage drop across the second resister is 200 volts

200
 Volts
=
2
 Amps
∗
100
 Ohms

{\displaystyle 200 \text{ Volts} = 2 \text{ Amps} \* 100 \text{ Ohms}}

The resistors use and dissipate supplied energy.
The energy dissipated by a resistor is given by

# E

I
2

R

{\displaystyle E = I^2 R}

## Voltage drop in alternating-current circuits: reactance

In alternating-current circuits, opposition to current flow occurs because of resistance, just as in direct-current circuits. However, alternating current circuits also include a second kind of opposition to current flow: reactance.

Electrical reactance is computed as the vector sum of capacitive reactance and inductive reactance.

The amount of reactance in an alternating-current circuit depends on the frequency of the alternating current.

### Capacitive reactance

Main articles: Capacitance
A capacitor consists of two conductors separated by an insulator, also known as a dielectric.

*Capacitive reactance* is an opposition to the change of voltage across an element. Capacitive reactance

X
C

{\displaystyle \scriptstyle{X\_C}}

 is inversely proportional to the signal frequency

f

{\displaystyle \scriptstyle{f}}

 (or angular frequency ?) and the capacitance

C

{\displaystyle \scriptstyle{C}}

.

X
C

=
−

1

ω
C

=
−

1

2
π
f
C

{\displaystyle X\_C = -\frac {1} {\omega C} = -\frac {1} {2\pi f C}}

Reactance is the imaginary part of impedance.

Z
c

=
−
j

X
c

{\displaystyle Z\_c=-jX\_c}

.
A given voltage applied across a capacitor causes a given amount of positive charge to accumulate on one side and negative charge to accumulate on the other side; the electric field due to the accumulated charge is the source of the opposition to the current. When the potential associated with the charge exactly balances the applied voltage, the current goes to zero.

* At zero frequency (DC) a capacitor is an open circuit so no current flows in the dielectric. Opposition to the current is infinite.

Driven by a given alternating voltage a capacitor will accumulate a given amount of charge each cycle. The higher the frequency, the faster the charge will accumulate and the greater the current and therefore the smaller the opposition to the current.

### Inductive reactance

Main articles: Inductance
Inductive reactance is a property exhibited by an inductor, and inductive reactance exists based on the fact that an electric current produces a magnetic field around it. In the context of an AC circuit (although this concept applies any time current is changing), this magnetic field is constantly changing as a result of current that oscillates back and forth. It is this change in magnetic field that induces another electric current to flow in the same wire (counter-EMF), in a direction such as to oppose the flow of the current originally responsible for producing the magnetic field (known as Lenz's Law). Hence, *inductive reactance* is an opposition to the change of current through an element.

For an ideal inductor in an AC circuit, the inhibitive effect on change in current flow results in a delay, or a phase shift, of the alternating current with respect to alternating voltage. Specifically, an ideal inductor (with no resistance) will cause the current to lag the voltage by a quarter cycle, or 90°.

Power is not completely transferred when voltage and current are out-of-phase because there will be points during which instantaneous current is positive while instantaneous voltage is negative, or vice versa, implying negative power transfer. Hence, real work is not performed when power transfer is "negative".

Inductive reactance

X
L

{\displaystyle \scriptstyle{X\_L}}

 is proportional to the sinusoidal signal frequency

f

{\displaystyle \scriptstyle{f}}

 and the inductance

L

{\displaystyle \scriptstyle{L}}

, which depends on the physical shape of the inductor.

X
L

=
ω
# L
2
π
f
L

{\displaystyle X\_L = \omega L = 2\pi f L}

The average current flowing through an inductance

L

{\displaystyle \scriptstyle{L}}

 in series with a sinusoidal AC voltage source of RMS amplitude

A

{\displaystyle \scriptstyle{A}}

 and frequency

f

{\displaystyle \scriptstyle{f}}

 is equal to:

I
L

=

A

ω
L

=

A

2
π
f
L

.

{\displaystyle I\_L = {A \over \omega L} = {A \over 2\pi f L}.}

Any conductor of finite dimensions has inductance; the inductance is made larger by the multiple turns in an electromagnetic coil. Faraday's law of electromagnetic induction gives the counter-emf

E

{\displaystyle \scriptstyle{\mathcal{E}}}

 (voltage opposing current) due to a rate-of-change of magnetic flux density

B

{\displaystyle \scriptstyle{B}}

 through a current loop.

E

=
−

d

Φ
B

d
t

{\displaystyle \mathcal{E} = -{{d\Phi\_B} \over dt}}

For an inductor consisting of a coil with

N

{\displaystyle \scriptstyle N}

 loops this gives.

E

=
−
N

d

Φ
B

d
t

{\displaystyle \mathcal{E} = -N{d\Phi\_B \over dt}}

The counter-emf is the source of the opposition to current flow. A constant direct current has a zero rate-of-change, and sees an inductor as a short-circuit (it is typically made from a material with a low resistivity). An alternating current has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

## Impedance

Main articles: Electrical impedance
The sum of oppositions to current flow from both resistance and reactance is called impedance.

Electrical impedance is commonly represented by the variable *Z* and measured in ohms at a specific frequency. Electrical impedance is computed as the vector sum of electrical resistance, capacitive reactance, and inductive reactance.

The amount of impedance in an alternating-current circuit depends on the frequency of the alternating current and the magnetic permeability of electrical conductors and electrically isolated elements (including surrounding elements), which varies with their size and spacing.

Analogous to Ohm's law for direct-current circuits, electrical impedance may be expressed by the formula

# E
I
Z

{\displaystyle E = I Z}

. So, the voltage drop in an AC circuit is the product of the current and the impedance of the circuit.

Both reactance

X

{\displaystyle \scriptstyle{X}}

 and resistance

R

{\displaystyle \scriptstyle{R}}

 are components of impedance

Z

{\displaystyle \scriptstyle{Z}}

.

# Z
R
j
X

{\displaystyle Z = R + jX}

where:

* Z

{\displaystyle Z}

 is the impedance, measured in ohms;
* R

{\displaystyle R}

 is the resistance, measured in ohms. It is the real part of the impedance:

# R
ℜ

(
Z
)

{\displaystyle {R=\Re{(Z)}}}
* X

{\displaystyle X}

 is the reactance, measured in ohms. It is the imaginary part of the impedance:

# X
ℑ

(
Z
)

{\displaystyle {X=\Im{(Z)}}}
* j

{\displaystyle j}

 is the square root of minus one, usually represented by

i

{\displaystyle i}

 in non-electrical formulas (

j

{\displaystyle j}

 is used so as not to confuse the imaginary unit with current, commonly represented by

i

{\displaystyle i}

).

When both a capacitor and an inductor are placed in series in a circuit, their contributions to the total circuit impedance are opposite. Capacitive reactance

X
C

{\displaystyle \scriptstyle{X\_C}}

 and inductive reactance

X
L

{\displaystyle \scriptstyle{X\_L}}

 contribute to the total reactance

X

{\displaystyle \scriptstyle{X}}

 as follows.

# X

X
L

X
C

=
ω
L
−

1

ω
C

{\displaystyle {X = X\_L + X\_C = \omega L -\frac {1} {\omega C}}}

where:

* X
L

{\displaystyle \scriptstyle{X\_L}}

 is the inductive reactance, measured in ohms;
* X
C

{\displaystyle \scriptstyle{X\_C}}

 is the capacitive reactance, measured in ohms;
* ω

{\displaystyle \omega}

 is the angular frequency,

2
π

{\displaystyle 2\pi}

 times the frequency in Hz.

Hence:

* if

X
>
0

{\displaystyle \scriptstyle X > 0}

, the total reactance is said to be inductive;
* if

# X
0

{\displaystyle \scriptstyle X = 0}

, then the impedance is purely resistive;
* if

X
<
0

{\displaystyle \scriptstyle X < 0}

, the total reactance is said to be capacitive.

Note however that if

X
L

{\displaystyle \scriptstyle{X\_L}}

 and

X
C

{\displaystyle \scriptstyle{X\_C}}

 are assumed both positive by definition, then the intermediary formula changes to a difference:

# X

X
L

−

X
C

=
ω
L
−

1

ω
C

{\displaystyle {X = X\_L - X\_C = \omega L -\frac {1} {\omega C}}}

but the ultimate value is the same.

### Phase relationship

The phase of the voltage across a purely reactive device (a capacitor with an infinite resistance or an inductor with a resistance of zero) *lags* the current by

π

/

2

{\displaystyle \scriptstyle{\pi/2}}

 radians for a capacitive reactance and *leads* the current by

π

/

2

{\displaystyle \scriptstyle{\pi/2}}

 radians for an inductive reactance. Without knowledge of both the resistance and reactance the relationship between voltage and current cannot be determined.

The origin of the different signs for capacitive and inductive reactance is the phase factor

e

±
j

π
2

{\displaystyle e^{\pm j{\pi \over 2}}}

 in the impedance.

Z
~

C

=

1

ω
C

e

j
(
−

π
2

)

=
j

(

−

1

ω
C

)

=
j

X
C

Z
~

L

=
ω
L

e

j

π
2

=
j
ω
# L
j

X
L

{\displaystyle \begin{align}
 \tilde{Z}\_C &= {1 \over \omega C}e^{j(-{\pi \over 2})} = j\left({ -\frac{1}{\omega C}}\right) = jX\_C \\
 \tilde{Z}\_L &= \omega Le^{j{\pi \over 2}} = j\omega L = jX\_L\quad
\end{align}}

For a reactive component the sinusoidal voltage across the component is in quadrature (a

π

/

2

{\displaystyle \scriptstyle{\pi/2}}

 phase difference) with the sinusoidal current through the component. The component alternately absorbs energy from the circuit and then returns energy to the circuit, thus a pure reactance does not dissipate power.