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Passive components and circuits - CCPLecture 5

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Content

Capacity Properties DC behavior AC behavior Transient regime behavior

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Web addresses

http://en.wikipedia.org/wiki/RC_circuit http://www.solarbotics.net/bftgu/starting_elect_pass_cap.html http://www.st-andrews.ac.uk/~jcgl/Scots_Guide/info/comp/passive/

capacit/capacit.htm http://www.phy.ntnu.edu.tw/oldjava/rc/rc.htm http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/DC-Current/

RCSeries.html http://academic.evergreen.edu/projects/biophysics/technotes/electron/

rc.htm#time_const http://www.sciences.univ-nantes.fr/physique/perso/charrier/tp/rcrlrlc/

index.html http://zone.ni.com/devzone/cda/ph/p/id/217

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Electric capacitance - circuit element The electrical property of a

circuit element to store electric charge when a voltage is applied.

Measurement unit: Farad [F]

F, nF, pF

QC

V

C

V

+ -

n eg a tiv e

ch a rg e

p o sitiv e

ch a rg e

c

dt

dVCI

dtdVI

dtdVdtdQ

C CC

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Electric capacitance - circuit element Electronic component -

capacitor.

It is characterized by its capacity.

For a capacitor, the capacitance depends on the geometrical size and the dielectric properties

C a p a c itor p la te w ith

sur fa ce A

d

C a p a c itor p la te w ith

sur fa ce A

D ie le c tr ic c h a ra c te r ize d

b y re la t iv e p e rm it iv ity

T erm in a l

r

pF/m85419,80

0

d

AC r

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Stored energy The capacitor doesn’t dissipate power, but it is storing

energy when it is charging and is releasing the energy when it is discharging:

2

000 2

1CVdt

dt

dVVCVIdtPdtW

TTT

e

We is energy stored when the capacitance is charging to VC voltage or is discharged from VC to 0.

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Parallel connection The equivalent capacitance is equal

with the sum of capacitances

n

iiech

ii

echi

ii

CC

VVQQ

V

QC

V

QC

1

;

;

B

C1 C2 Cn Cech

A

B

A

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Series connection

The equivalent capacitance is given by the following formula:

n

i iech

ii

echi

ii

CC

QQVV

V

QC

V

QC

1

11

;

;

C1 C2 Cn

Cech

A

A

B

B

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DC regime behavior In DC, the capacitor corresponds to an open-circuit.

21 VVVAB 0CC

dt

dVC

dt

dvCi AB

C

A B

V1 V2

VAB

A B

V1 V2

VAB

DC

R1

iC

R2 R1 R2

iC =0

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AC regime behavior In AC regime, the capacitance is equivalent with ZC

impedance. The sinusoidal voltage applied is considered through the phasorial representation..

CZX

CjZ

i

vvCj

eeVCjdt

eeVdCi

eeVvdt

dvCi

CC

CC

CC

jtjjtj

C

jtjC

CC

1

1

;

Capacitance’s reactance

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AC regime behavior Impedance (reactance) is frequency dependent. In alternative current, the imittances of the circuits with

capacitors are dependent of the signals frequency. The property of a circuit to pass or reject some

frequencies is called filtering. Low-pass filter High-pass filter Band-pass filter Band reject filter

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RC low-pass filter;

1

1

( ) 1( )

( ) 1

oo i C C

C

Co i i

C

o

i

vv v R i i

Z

Z j Cv v v

R Z Rj C

v jH j

v j j RC

For R=1,6 K and C=100pF :

67 101

1

106,121

1)(

fjfj

jfH

vi is a sinusoidal voltage with frequency f = ω/2π (or ω=2π f)

C

R

vi

voZ

C

iC

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RC low-pass filter - frequency characteristics

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RC low-pass filter - frequency characteristics Answer to the following questions:

What kind of representation is used for modulus-frequency and phase-frequency characteristics?

What is the modulus-frequency characteristic slope for 100Hz – 100KHz domain? And in10MHz – 100MHz domain?

What is the phase difference between input and output voltages at the 1MHz?

Represent, at a 1MHz frequency, the input voltage phasor and the output voltage phasor. How does this representation look at 100MHz?

At what frequency is the phase difference between input and output φ=30?

How are the previous characteristics modified if R=16KΩ and C=100pF? How are the previous characteristics modified if R=1KΩ and C=1nF?

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RC high-pass filter

RCj

RCj

jv

jvjH

v

CjR

Rv

ZR

Rv

Z

vv

Z

viiRv

i

o

iiC

o

C

oi

C

CCCo

1)(

)()(

1

;

For R=1,6 K and C=100pF:

6

6

7

7

101

10

106,121

106,12)(

fj

fj

fj

fjjfH

vi is a sinusoidal voltage with frequency f = ω/2π (or ω=2π f)

vi

voR

C

ZC

iC

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High-pass RC filter - frequency characteristics

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High-pass RC filter - frequency characteristics Answer to the following questions:

What kind of representation is used for modulus-frequency and phase-frequency characteristics?

What is the modulus-frequency characteristic slope for 100Hz – 100KHz domain? And in 10MHz – 100MHz domain?

What is the phase difference between input and output voltages at the 1MHz?

Represent, at a 1MHz frequency, the input voltage phasor and the output voltage phasor. How does this representation look at 100MHz?

At what frequency is the phase difference between input and output φ= =30?

How are the previous characteristics modified if R=16KΩ and C=100pF?

How are the previous characteristics modified if R=1KΩ and C=2.2nF?

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The high frequencies behavior At high frequencies, the capacitive reactance is much lower

than resistances from the previous circuits. The capacitance is equivalent with a short-circuit.

C

R

vi

vo

vi

voR

C

R

vi

vo

vi

voR

=0

vi

=

V H F

V H F

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Separation and pass capacitances In some circuits, the capacitances are used to

separate the DC components (DC - open-circuit) between two circuits without affecting the signal variation (AC short-circuit)

In these situations, they are called separation capacitances (separation of DC components). In other situations, realizing the same functions, they are called passing capacitances (for high frequency signals).

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Capacitance behavior in transient regime In this case, the transient regime consist in the

modification of a DC circuit steady state in a new DC steady state.

During these modifications, the capacitance cannot be considered open-circuit or short-circuit.

The transient regime analysis presumes the determination of the way of charging and discharging of the capacitance.

In transient regime, the circuit operations are described by differential equations.

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The capacitance charging from a constant voltage source

Considering the K switch on position 1, the capacitance will be discharged.

At the time t=t0, the switch is moved on position 2.

After enough time, t, the capacitance will be charged at the E voltage.

The transient regime is taking place between these two DC steady states.

C

R

E vC

1

2K

iC

vR

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The capacitance charging from a constant voltage source

CC

CC

CCCC

CR

vdt

dvE

RCvdt

dvRCE

dt

dvCiviRE

vvE

;

;

:TKV

t

CCCC evvvtv

)]()0([)()(

)()(

)()0( 0

tvv

ttvv

C

CC

The solution of the differential equation

RC Circuit time constant

C

R

E vC

1

K

iC

2

vR

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Voltage variation across the capacitance, vC

t

CR

t

C

CC

eEtvEtv

eEtv

Evv

)()(

)1()(

)(;0)0(

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Current variation through the capacitance, iC

t

CC e

R

E

R

tvEti

)()(

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Significance of the time constant If the transient process has the same slope like in origins

(initial moment), the final values of voltages and currents will be obtain after a time equal with circuit time constant.

As can be seen in the previous figures, the charging process continues to infinite.

Practically, the transient regime is considered to be finished after 3 (95% from the final values) or 5 (99% from the final values).

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Example (E=1V, R=1KΩ, C=1nF)

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The capacitance discharging regime on a resistance

At the initial time, the switch is considered on position 2. The capacitance is charged to E voltage.

At a reference time moment t=t0, The K switch is moved on position 1.

After enough time, t, the capacitance is totally discharged.

The transient regime is taking place between these two DC steady states.

C

R

E vC

1

K

iC

2

vR

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The capacitance discharging regime on a resistance

CC

CC

CCCC

CR

vdt

dv

RCvdt

dvRC

dt

dvCiviR

vv

0

;0

;0

0:TKV

t

CCCC evvvtv

)]()0([)()(

0)()(

)()0( 0

tvv

Ettvv

C

CC

The solution of the differential equation

tR

C

CR

t

C

eR

E

R

tvti

tvtveEtv

)()(

)()(;)(

C

R

E vC

1

2K

iC

vR

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Example (E=1V, R=1KΩ, C=1nF)

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Charging the capacitance from a constant current source The K switch is considered in

position 1. The capacitance is considered initially charged to the voltage vC(0).

At reference time t=t0, the K is switched in position 2.

The constant current source will charge the capacitance with the current I.

C

R

I

vC

1

2K

iC

vR

R 1

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Charging the capacitance from a constant current source

)()()(1

)()(1

)()(

000

0

0

0

tvttC

ItvdtI

C

tvdttiC

tvdt

tdvCi

CC

t

t

C

t

t

CCC

C

vC

vC

t

0

(t)

(0)

slope = I/C

C

R

I

vC

1

2K

iC

vR

R 1

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Observations

The voltage across the capacitance will raise linearly.

The charging slope (or discharging) is independent by the value of resistance R (so, the resistor can misform the circuit).

Theoretically, the voltage across the capacitance can raise infinitely . In these situations, we must take some measures to limit the voltage on the capacitance.

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The RC circuits behavior when pulses are applied

Consider a pulses signal source applied to a series RC circuit.

In analyzing the circuit behavior, we consider both voltages: the voltage across the capacitor, vC(t), and the voltage across the, vR(t).

Applying this signal source, the phenomena of charging and discharging described to transient regime are repetitive.

C

R

vC

iC

vR

vI

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Case A – the time constant is much lower than pulses duration

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Case B – the time constant is much greater than pulses duration

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Integrating circuit

If the output voltage is the voltage across the capacitor, the effect under the input signal is attenuation of edges, similarly with the integration operation.

In this situation,(when vO(t)=vC(t)), the circuit is called integration circuit.

The integration effect is higher in case B , when the time constant is greater then the pulse duration.

The integration function in transient regime corresponds to low-pass filtering in AC regime.

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Derivative circuit If the output voltage is the voltage across the resistor,

the circuit effect under the input signal is an accentuation of edges, similarly with derivative mathematical operation.

In this situation,(when vO(t)= vR(t)), the circuit is called derivative circuit.

The derivative effect is higher in case A , when the time constant is lower then pulse duration.

The derivative function in transient regime corresponds to high-pass filtering in AC regime.

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Answer to the following questions

How are the waveforms modified if the input source for the RC series circuit is a pulses current source?

Can the current source pulses be asymmetrical? (from 0 on the current axis)?

How does the voltage across the capacitor (or resistor) vary if the pulses have the same duration with the circuit time constant? Make the analysis starting with the initial time moment, when the capacitor is completed discharged.

For the homework problem from the end of lecture 2, consider that the resistor R is replaced by a capacitance C=10nF. Draw the waveform of the voltage across the capacitance.