TEC 284Capacitors and Inductors

Capacitors

A capacitor is a device that stores an electrical charge

It is made of two metallic plates separated by an insulator or dielectric (plastic, ceramic, air)

Capacitors

Capacitors

A capacitor’s storage potential or capacitance is measured in Farads (f) – typically small values i.e. picofarads (pf 10-12) or nanofarads (nf 10-9) or microfarads (µf 10-6)

Capacitors in a DC circuit will be charged and current stops when the capacitor is at the same charge as the dc source

In an AC circuit, capacitors charge and discharge as current fluctuates making it appear that the ac current is flowing

Capacitors

Difference between capacitor and a battery A capacitor can dump its entire charge

in a fraction of a second

Uses of capacitors Store charge for high speed use (flash,

lasers) Eliminate ripples or spikes in DC voltage

– absorbs peaks and fills in valleys As a filter to block DC signals but pass

AC signals

Capacitor Calculations

Parallel Plate CapacitorC = k A / d

C - capacitance in Farads k – dielectric constant A –area in square meters d – distance between the electrodes in

meters

Capacitor Calculations

Parallel Plate CapacitorC = Q / V or Q = CV

C - capacitance in Farads (charge on plate) Q – charge in Coloumbs V – voltage in Volts

Q= CV Charge on the plates of the capacitor

E = ½ QV = ½ CV2

Energy stored on the plates of a capacitor Measured in Joules (J)

Capacitor Energy

Capacitors return stored energy to the circuit

They do not dissipate energy and convert it to heat like a resistor

Energy stored in a capacitor is much smaller than energy stored in a battery so they cannot be used as a practical energy source

Time Constant

τ = R x C τ - Time constant (Tau) R – Resistance in Ohms (Ω) C – Capacitance in farads (F)

This is the time response of a capacitive circuit when charging or discharging

Time Constant for Inductor circuits

τ = L / R τ - Time constant (Tau) R – Resistance in Ohms (Ω) L – Inductance in Henrys (H)

This is the time constant for circuits that contain a resistor and an inductor

Voltage across capacitor

The voltage across a capacitor at any given point in time is given by the equation above

Vc – Voltage across the capacitor Vs - Supply voltage t – Time elapsed since application of

supply voltage RC – Time constant of the RC charging

circuit

Time constant

Time taken for charging or discharging current to fall to 1/e its initial value

e = 2.71828So roughly the time constant is the

time taken for the current to fall to 1/3 its value

Charging Discharging

Time Constant

After 5 time constants (5RC) the current has fallen to less than 1% of its initial value

For all intents and purposes the capacitor can be considered to be fully charged / fully discharged after 5RC

A large time constant means a capacitor charges /discharges slowly

Charging a Capacitor

Charging current I = (Vs-Vc)/RAs Vc increases, I decreasesWhen charge builds up on the

capacitor, the voltage across it increases

This reduces the voltage across the resistor and reduces charging current

Rate of charging becomes slower

Charging a Capacitor

Discharging a Capacitor

At first current is large because voltage is large and charge is lost quickly

As charge/voltage decreases, current becomes smaller so rate of discharging becomes progressively slower

After 5RC voltage across capacitor is almost 0

Discharging a Capacitor

Charging and Discharging

Capacitors in Parallel

Ceq = C1 + C2Ceq = Q / V = (Q1 + Q2 ) / V = Q1 /

V + Q2 /V= C1 + C2

Capacitors in Series

1/Ceq = 1/C1 + 1/C21/Ceq = V /Q = (V1 + V2) / Q = V1 /

Q + V2 /Q= 1 /C1 + 1/C2

Capacitors in AC Circuits

Capacitors typically block DC signals but allow AC signals to pass

When the current in an AC circuit reverses direction, the capacitor discharges so it appears that current is flowing continuously

A capacitor will oppose the flow of an AC current

This opposition to current flow is called the reactance of the capacitor

Capacitor Reactance

Xc = 1 / 2πfCXc – Reactance of the capacitor in

ohms (Ω) f – Frequency of the input signal in

(Hz)C – Capacitance of the capacitor in

farads (F)

Capacitors and Resistors in Series

The output sine wave has the same frequency as the input sine wave

Because of the reactance (opposition to the flow of ac current) of the capacitor the amplitude of the output is smaller than the input

Voltage Divider

A capacitor and resistor in series as above functions as a voltage divider

Total Resistance for a Series – RC Circuit

For a normal series circuit Rt = R1 + R2

Total opposition to the flow of an electric current in a circuit containing a capacitor and a resistor in series is called the impedance

Z = √X2C + R2 = (X2

C + R2)1/2

Z – the impedance of the circuit in ohms Xc – the reactance of the capacitor in

ohms R – the resistance of the resistor in ohms

Exercises

1. C = 530 µF, R = 12 ohms, Vin = 26 Vpp, f=60 Hz. Calculate the impedance and the current in the circuit.

2. C = 1.77 µF, R = 12 ohms, Vin = 150 Vpp, f=10 kHz. Calculate the impedance and the current in the circuit.

Exercises

10 Vpp f = 1kHz

C = 0.32 µF

R = 1 k ohm

• Find Xc• Find Z• Find Vout

Attenuation

In the previous examples, the output signal is attenuated

The output amplitude is smaller than the input amplitude

The frequency of the input and output is the same

RMS value of a sine wave

RMS is the root mean square valueFor a sign wave S of amplitude aSRMS = a / √2

a 2a

Measurements for ac waveforms

RMS – Root Mean Square

AVG – Practical Average

P-P – Peak to Peak

High Pass Filter

As the frequency increases, the reactance decreases and the voltage drop becomes larger over the resistor

Low Pass Filter

As the frequency increases, the reactance decreases and the voltage drop across the capacitor becomes smaller

This circuit is used in many electronic devices

Cutoff Frequency for RC Circuits

fcutoff = 1/2 π RC fcutoff – cutoff frequency in Hertz (Hz) R – resistance of resistor in ohms (Ω) C – capacitance of the capacitor in

farads (F)The cutoff or corner frequency is the

frequency above which the output voltage falls to 70.7% (3 DB) (√ 1/2) of the source voltage

This is an important in the design of filters

Cutoff Frequency for RL Circuits

fcutoff = R/2 π L fcutoff – cutoff frequency in Hertz (Hz) R – resistance of resistor in ohms (Ω) L – inductance of the inductor in Henrys

(H)

Phase Shift of an RC Circuit In an RC circuit, the current and the

voltage across the resistor is in phase. They have no phase difference

The voltage across the capacitor lags the current through the capacitor by 90 degrees (they are 90 degrees out of phase)

The input voltage and the output voltage across a component of an RC circuit have the same frequency but the output is attenuated and is out of phase with the input

Phase Shift of an RC Circuit

Output voltage of the LP lags the input voltage

Output voltage of HP leads the input voltage

Low Pass Filter High Pass Filter

RC Phase Shift

HP Filter LP Filter

Output leads Input Voltage Output lags Input Voltage

Phase Angle

tan θ = Vc / VR

= 1 / 2 πfRC

= Xc / R

Inductors

Essentially a coil of wireWhen current flows, a magnetic field

is created and the inductor will store the magnetic energy until released

A capacitor stores voltage as electrical energy while a conductor stores current as magnetic energy

The strength of an inductor is it’s inductance which is measured in Henrys (H)

Capcitors vs Inductors

Capacitors block DC current and let AC pass

Inductors block AC and let DC passCapacitors oppose the change of

voltage in a circuit while an inductor opposes the change in current

Inductors in AC Circuits

Like capacitors, inductors cannot change the frequency of the sine wave but can reduce the amplitude of the output voltage

The opposition to the current flow is called the impedance

In many cases the DC resistance of the inductor is very low

Reactance of an Inductor

XL = 2 πfL XL – reactance of the inductor in ohms

(Ω) f – frequency of the signal in Hertz (Hz) L – inductance of the inductor in Henrys

(H)

Low Pass filter with an Inductor

1. Find DC Ouput Voltage if resistance of inductor is 0

2. Find the reactance of the inductor

3. Find the AC impedance

4. Find the AC output voltage

5. Draw actual output

Answers

1. DC Vout = 10 V (full 10 V DC is dropped across Vout)

2. XL = 2 πfL= 2 X π x 1000 x 0.163 = 1 K Ω approx

3. Z2 = XL 2+ R2 = 10002 + 10002

1. Z = 1.414 k Ω 4. AC out = 1 k Ω / 1.414 k Ω X 2Vpp

= 1.414 Vpp

Phase Shift for an RL Circuit

The current through an inductor lags the voltage across the inductor by 90 degrees

LP Filter HP Filter

Output lags Input Voltage Output leads Input Voltage

Phase Shift for RL Circuit

tan θ = VL / VR

= XL / R

=2 πfL/ R