basic ac circuits part b - rc and l/r time constants - rlc circuits - complex numbers for ac...

Basic AC circuits part BBasic AC circuits part B

- RC and L/R time constants- RLC circuits- Complex numbers for AC circuits- Resonance- Filters

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

Homework Unit 2Homework Unit 2

Grob’s Basic Electronics

problems: 23-20, 23-21, 24-34, 24-35,

25-17, 25-25, 26-25, 26-39

Due September 26th 2011

Response of Resistance AloneResponse of Resistance Alone

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Fig. 22-1:

When the switch S is closed in Fig. 22-1 (a), the battery supplies 10 V across the 10-Ω R and the resultant I is 1 A. The graph in Fig. 22-1 (b) shows that I changes from 0 to 1 A instantly when the switch is closed.

L/RL/R Time Constant Time Constant


Fig. 22-2:

When S is closed, the current changes as I increases from zero. Eventually, I will reach the steady value of 1 A, equal to the battery voltage of 10 V divided by the circuit resistance of 10 Ω.

High Voltage Produced by High Voltage Produced by Opening an Opening an RLRL Circuit Circuit


Fig. 22-3:

In Fig. 22-3, the neon bulb requires 90 V for ionization, at which time it glows. The source is only 8 V, but when the switch is opened, the self-induced voltage is high enough to light the bulb for an instant.The sharp voltage pulse or spike is more than 90 V just after the switch is opened.

RCRC Time Constant Time Constant


Fig. 22-4:

T = 3 x 106 x 1 x 10−6

= 3 s

RCRC Charge and Charge and Discharge CurvesDischarge Curves


T in ms0 1 2 3 4 5

2

4

6

8

0v C

in V

olts

On discharge, C loses its charge at a declining rate. At the start of discharge, vC has its highest value and can produce maximum discharge current. As the discharge continues, vC goes down and there is less discharge current. The more C discharges, the more slowly it loses the remainder of its charge.

High Current Produced by High Current Produced by Short-Circuiting Short-Circuiting RCRC Circuit Circuit


Fig. 22-5:

With large capacitors, this can be dangerous!

The circuit of Fig. 22-5 illustrates the application of a battery-capacitor (BC) unit to fire a flashbulb for a camera. The flashbulb needs 5 A to ignite, but this is too much load current for the small 15-V battery. Instead of using the bulb as a load for the battery, the 100-μF capacitor is charged. The capacitor is then discharged through the bulb in Fig. 22-5 (b).

RCRC Waveshapes Waveshapes


Fig. 22-6:

Long and Short Time ConstantsLong and Short Time Constants

VA

R

CvOUT

v OU

T

Integrators use a relatively long time constant.

Differentiators use a relatively short time constant.

VA R

CvOUT

v OU

T


Integrators and Differentiators

Charge and Discharge with ShortCharge and Discharge with Short RC RC Time ConstantTime Constant

Fig. 22-7

Fig. 22-7 illustrates the charge and discharge of an RC circuit with a short time constant. Note that the waveshape of VR in (d) has sharp voltage peaks for the leading and trailing edges of the square-wave applied voltage.

Long Time Constant for Long Time Constant for RCRC Coupling CircuitCoupling Circuit


Fig. 22-8:

Fig. 22-8 illustrates the charge and discharge of an RC circuit with a long time constant. Note that the waveshape of VR in (d) has the same waveform as the applied voltage.

Advanced Time Advanced Time Constant AnalysisConstant Analysis

.


Fig. 22-9:

Comparison of Reactance and Comparison of Reactance and Time ConstantTime Constant

Table 22-2 Comparison of Reactance XC and RC Time Constant

Sine-Wave Voltage Nonsinusoidal Voltage

Examples are 60-Hz power line, af signal voltage, rf signal voltage

Examples are dc circuit turned on and off, square waves, rectangular pulses

Reactance XC = 1/(2πfC) Time constant T = RC

Larger C results in smaller reactance XC Larger C results in longer time constant

Higher frequency results in smaller XC Shorter pulse width corresponds to longer time constant

IC = VC/XC iC = C(dv/dt)

XC makes IC and VC 90° out of phase Waveshape changes between iC and vC

AC Circuits with Resistance but AC Circuits with Resistance but No ReactanceNo Reactance

In this figure, combinations of series and parallel resistances are shown. All voltages and currents throughout the resistive circuits are in phase. There is no reactance to cause a lead or lag in either current or voltage.

Fig. 23-1

Circuits with Circuits with XXLL Alone Alone

Fig. 23-2

A series inductive circuit is shown in Fig. 23-2.The ohms of XL are just as effective as ohms of R in limiting the current or producing a voltage drop. XL has a phasor quantity with a 90° phase angle.

Circuits with Circuits with XXLL Alone Alone

Fig. 23-3

A parallel inductive circuit is shown in Fig. 23-3. The ohms of XL are just as effective as ohms of R in limiting the current or producing a voltage drop. XL has a phasor quantity with a 90° phase angle.

Circuits with Circuits with XXCC Alone Alone

Fig. 23-4

Series XC

Capacitive reactances are shown in Fig. 23-4 Since there is no R or XL, the series ohms of XC can be combined directly.


Fig. 23-5

Parallel XC

Capacitive reactances are shown in Fig. 23-5. Since there is no R or XL, parallel IC currents can be added.


XC and XL are phasor opposites.

R

XLR

XC

When analyzing series circuits:Opposite reactances in series must be subtracted. If XL is larger, the net reactance is inductive. If XC is larger, the net reactance is capacitive.


Opposite Reactances CancelOpposite Reactances Cancel


Fig. 23-6:

Opposite Reactances CancelOpposite Reactances Cancel


Fig. 23-7:

Series Reactance Series Reactance and Resistanceand Resistance

The resistive and reactive effects of series reactance and resistance must be combined by phasors.

For series circuits, all the ohms of opposition are added to find ZT.

First, add all series resistances for one total R. Combine all series reactances, adding all XLs and all

XCs and finding X by subtraction.

The total R and net X can be added by phasors to find the total ohms of opposition in the entire series circuit.

Series Reactance Series Reactance and Resistanceand Resistance

Magnitude of ZT

After the total R and net reactance X are found, they can be combined by the formula

ZT = R2 + X2

Parallel Reactance Parallel Reactance and Resistanceand Resistance

In parallel circuits, the branch currents for resistance and reactance are added by phasors.

Then the total line current is found by

IT = IR2 + IX

2


IR

IC IR

IL

Parallel IC and IL are phasor opposites.

Opposite currents in parallel branches are subtracted. If IL is larger, the circuit is inductive. If IC is larger, the circuit is capacitive.


Parallel RCL Circuit Analysis

IT = 5 A

VA = 120 R = 30 Ω XC = 60 Ω XL = 24 Ω

4 A

3 A IT = 5 A

2 A

4 A

5 A

The circuitis inductive.

IT = IR2 + IX

2 = 42 + 32 = 5A


Parallel RCL Circuit Impedance

VA = 120 R = 30 XC = 60 XL = 24

IT = 5 A

ZEQ = = = 24 ΩVA

IT

120

53 A IT = 5 A

4 A


Parallel RCL Circuit Phase Angle

IT = 5 A

VA = 120 R = 30 XC = 60 XL = 24

Θ = Tan-1 −IX

IR

3

4= Tan−1 − = −37°

−37°4 A

3 A IT = 5 A

Series-Parallel Reactance and Series-Parallel Reactance and ResistanceResistance


Fig. 23-12:

Figure 23-12 shows how a series-parallel circuit can be reduced to a series circuit with just one reactance and one resistance.

The triangle diagram in (d) shows total impedance Z (141 Ω).


Waveforms and Phasors for a Series RCL Circuit

VR

VC

VL

VR

Θ = 0I

Θ = −90I

VC

Θ = 90I

VL

R

CI

L


Amy Hill

Author: For this & the three slides following, I am not sure where to place them. Please advise.


Series RCL Circuit Analysis


The net reactance is 3 , capacitive.

4 A

L

XL = 9

R = 4

XC = 12 20 V

4 R

3

XNETZ = 5

I = = =4AV

Z

20

5

Z= R2 + X2 = 42 + 32 = 5Ω


I

Series RCL Circuit Phase Angle

ΘZ= Tan-1 −X

R

3

4= Tan−1 − = −37

The net reactance is 3 , capacitive.

L

XL = 9

R = 4

XC = 12 20 V

Z = 5Ω

ΘZ= Tan-1 ± X / R

4 R

3

XNET5

−37° Note: Since the circuit is capacitive, the source voltage lags the source current by 37 degrees.




Series RCL Voltage Drops

VC and VL are phasor opposites, so the net reactivevoltage is the difference between the two or 12 V.

VT = 162 + 122 = 20 V

16 V

12 V

R

XNET

R = 4 Ω

XC = 12 ΩL

XL = 9 Ω

20 V4 A VR = IR = 4 × 4 = 16 V

VC = IXC = 4 × 12 = 48 V

VL = IXL = 4 × 9 = 36 V

Real PowerReal Power

In an ac circuit with reactance, the current I supplied by the generator either leads or lags the generator voltage V.

The product VI is not the real power produced by the generator, since the instantaneous voltage may have a high value while at the same time the current is near zero, or vice versa.


The real power in watts can always be calculated as I2R, where R is the total resistive component of the circuit.

To find the corresponding value of power as VI, this product must be multiplied by the cosine of the phase angle Θ. Then

Real power = P = I2R

or

Real power = P = VI cos Θ


Series RCL Circuit Power Dissipation

R = 4

XC = 12 L

XL = 9

20 V 4 ANote: the power dissipation is zero in ideal capacitors and ideal inductors. All of the dissipation takes place in the circuit’s resistance.

4 A

20 V

−37 The source voltage and source current are not in phase and the true power is not equal to VI. It is equal to VI × power factor.

P = V × I × Cos Θ = 20 × 4 × 0.8 = 64 W

P = I2R = 42× 4 = 64 W


Parallel RCL Circuit Power Dissipation

IT = 5 A

VA = 120 R = 30 XC = 60 XL = 24

The source voltage and source current are not in phase and the true power is not equal to VI. It is equal to VI × power factor.

−37°4 A

3 A IT = 5 A

P = V × I × Cos Θ = 120 × 5 × 0.8 = 480 W

P = = = 480 WV2

R

1202

30

Summary of Types of Summary of Types of Ohms in AC CircuitsOhms in AC Circuits

Ohms of opposition limit the amount of current in dc circuits or ac circuits.

Resistance is the same for either case. Ac circuits can have ohms of reactance because of

the variations in alternating current or voltage. Reactance XL is the reactance of an inductor with

sine-wave changes in current. Reactance XC is the reactance of a capacitor with

sine-wave changes in voltage.

Summary of Types of Summary of Types of Phasors in AC CircuitsPhasors in AC Circuits


Fig. 23-15

TheThe j j OperatorOperator

The operator of a number can be any angle between 0° and 360°. Since the angle of 90° is important in ac circuits, the factor j is used to indicate 90°. In Fig. 24-2, the number 5 means 5 units at 0°, the number −5 is at 180°, and j5 indicates the number 5 at the 90° angle.

Fig. 24-2

TheThe j j OperatorOperator


Fig. 24-3:

The angle of 180° corresponds to the j operation of 90° repeated twice. This angular rotation is indicated by the factor j2. Note that the j operation multiplies itself, instead of adding.

Definition of a Definition of a Complex NumberComplex Number

The combination of a real and an imaginary term is called a complex number.

Usually, the real number is written first. As an example, 3 + j4 is a complex number including

3 units on the real axis added to 4 units 90° out of phase on the j axis.

Complex numbers must be added as phasors.

Definition of a Definition of a Complex NumberComplex Number


Fig. 24-4

Phasors for complex numbers are shown in Fig. 24-4 The phasors are shown with the end of one joined to the start of the next, to indicate addition.Graphically, the sum is the hypotenuse of the right triangle formed by the two phasors.

How Complex Numbers Are How Complex Numbers Are Applied to AC CircuitsApplied to AC Circuits

Circuit Values Expressed in Rectangular Form

6+j0

6+j6

3−j3

0+j6 XL

0−j6 XC

6 6

3 3


Impedance in Complex FormImpedance in Complex Form

The rectangular form of complex numbers is a convenient way to state the impedance of series resistance and reactance.

The general form of stating impedance is Z = R ± jX. If one term is zero, substitute 0 for this term to keep Z

in its general form. This procedure is not required, but there is usually

less confusion when the same form is used for all types of Z.

Operations with Complex Operations with Complex NumbersNumbers

Real numbers and j terms cannot be combined directly because they are 90° out of phase.

For addition or subtraction, add or subtract the real and j terms separately.

To multiply or divide a j term by a real number, multiply or divide the numbers. The answer is still a j term.

To multiply or divide a real number by a real number, just multiply or divide the real numbers, as in arithmetic.

Operations with Complex Operations with Complex NumbersNumbers

To divide a j term by a j term, divide the j coefficients to produce a real number; the j factors cancel.

To multiply complex numbers, follow the rules of algebra for multiplying two factors, each having two terms.

To divide complex numbers, the denominator must first be converted to a real number without any j term.

Converting the denominator to a real number without any j term is called rationalization.

Magnitude and Angle Magnitude and Angle of a Complex Numberof a Complex Number

In electrical terms, the complex impedance (4 + j3) means 4 Ω of resistance and 3 Ω of inductive reactance with a leading phase angle of 90°.

The magnitude of Z is the resultant of 5 Ω. Finding the square root of the sum of the squares is

vector or phasor addition of two terms in quadrature, 90° out of phase.

The phase angle of the resultant is the angle whose tangent is 0.75. This angle equals 37°

Magnitude and Angle Magnitude and Angle of a Complex Numberof a Complex Number


Fig. 24-8:

Polar Form for Polar Form for Complex NumbersComplex Numbers

Calculating the magnitude and phase angle of a complex number is actually converting to an angular form in polar coordinates.

The rectangular form 4 + j3 is equal to 5 in polar form. In polar coordinates, the distance from the center is the

magnitude of the phasor Z. Its phase angle Θ is counterclockwise from the 0° axis. To convert any complex number to polar form,

Find the magnitude by phasor addition of the j term and real term.

Find the angle whose tangent is the j term divided by the real term.

37

Polar Form for Polar Form for Complex NumbersComplex Numbers

Phasors Expressed in Polar Form

Magnitude is followed by the angle. 0 means no rotation. Positive angles provide CCW rotation. Negative angles provide CW rotation.

6

6

6

8.496

64.24

Converting Polar to Rectangular Converting Polar to Rectangular FormForm

Complex numbers in polar form are convenient for multiplication and division, but cannot be added or subtracted if their angles are different because the real and imaginary parts that make up the magnitude are different.

When complex numbers in polar form are to be added or subtracted, they must be converted into rectangular form.

Converting Polar to Rectangular Converting Polar to Rectangular FormForm


Fig. 24-9:

Consider the impedance Z in polar form. Its value is the hypotenuse of a right triangle with sides formed by the real and j terms. In Fig. 24-9, note the polar form can be converted to rectangular form by finding the horizontal and vertical sides of the right triangle.

Real term for R = Z cos Θj term for X = Z sin Θ

Complex Numbers in Complex Numbers in Series AC CircuitsSeries AC Circuits

Refer to Fig. 24-10 (next slide). Although a circuit like this with only series resistances

and reactances can be solved graphically with phasor arrows, the complex numbers show more details of the phase angles.

The total ZT in Fig. 24-10 (a) is the sum of the impedances:

ZT = 2 + j4 − j12

= 6 − j8

Convert ZT to polar and divide into VT to determine I.

Complex Numbers in Complex Numbers in Series AC CircuitsSeries AC Circuits


Fig. 24-10:

Complex Numbers in Complex Numbers in Parallel AC CircuitsParallel AC Circuits

A useful application is converting a parallel circuit to an equivalent series circuit.

See Fig. 24-11 (next slide), with a 10-Ω XL in parallel with a 10-Ω R.

In complex notation, R is 10 + j0 and Xl is 0 + j10.

Their combined parallel impedance ZT equals the product divided by the sum.

ZT in polar form is 7.04 45

Complex Numbers in Complex Numbers in Parallel AC CircuitsParallel AC Circuits


Fig. 24-11:

The rectangular form of ZT means that a 5-Ω R in series with a 5-Ω XL is the equivalent of a 10-Ω R in parallel with a 10-Ω XL, as shown in Fig. 24-11.

Recall the product over sum method of combining parallel resistors:

The product over sum approach can be used to combine branch impedances:

Combining Two Complex Branch Combining Two Complex Branch ImpedancesImpedances


Fig. 24-12:

REQ = R1 × R2

R1 + R2

ZEQ = Z1 × Z2

Z1 + Z2



ZT = Z1 × Z2

Z1 + Z2

Z1 = 6+j0 + 0+j8 = 6+j8 = 1053.1°

Z2 = 4+j0 + 0-j4 = 4-j4 = 5.6645°

Z1 + Z2 = 6+j8 + 4-j4 = 10+j4 = 10.821.8

Z1 × Z2 = 1053.1° x 5.6645° = 56.6

56.610.821.8ZT = = 5.24

Fig. 24-12:



Fig. 24-12:

4.5813.7A

56.68.110.821.8ZT = = 5.2413.7

24

5.2413.7IT = = 4.5813.7A

ZT = Z1 × Z2

Z1 + Z2

Note: The circuit is capacitive since the current is leading by 13.7°.

VA = 24 V

Combining Complex Combining Complex Branch CurrentsBranch Currents


Fig. 24-13:

4.5813.7A

Adding the branch currents,

IT = I1 + I2

= (6 + j6) + (3 − j4) = 9 + j2 A

In polar form, the IT of 9 + j2 is calculated as the phasor sum of the branch currents.

2 2T

I

T

I = 9 2 = 85

= 9.22 A

2tan = = 0.222

9 = arctan (0.22)

= 12.53

Therefore, I is 9 j2 A in rectangular form

or 9.22 12.53 in polar form.

Parallel Circuit with Three Parallel Circuit with Three Complex BranchesComplex Branches

Because the circuit in Fig. 24-14 (next slide) has more than two complex impedances in parallel, use the method of branch currents.

Convert each branch impedance to polar form. Convert the individual branch currents from polar to

rectangular form so they can be added for IT.

Convert IT from rectangular to polar form.

ZT can remain in polar form with its magnitude and phase angle or can be converted to rectangular form for its resistive and reactive components.

Parallel Circuit with Three Parallel Circuit with Three Complex BranchesComplex Branches


Fig. 24-14:

The Resonance EffectThe Resonance Effect

Inductive reactance increases as the frequency is increased, but capacitive reactance decreases with higher frequencies.

Because of these opposite characteristics, for any LC combination, there must be a frequency at which the XL equals the XC; one increases while the other decreases.

This case of equal and opposite reactances is called resonance, and the ac circuit is then a resonant circuit.

The frequency at which XL = XC is the resonant frequency.

The Resonance EffectThe Resonance Effect


Fig. 25-1:

The most common application of resonance in rf circuits is called tuning. In Fig. 25-1, the LC circuit is resonant at 1000 kHz. The result is maximum output at 1000 kHz, compared with lower or higher frequencies.

Series ResonanceSeries Resonance

At the resonant frequency, the inductive reactance and capacitive reactance are equal.

In a series ac circuit, inductive reactance leads by 90°, compared with the zero reference angle of the resistance, and capacitive reactance lags by 90°.

XL and XC are 180° out of phase.

The opposite reactances cancel each other completely when they are equal.



Series Resonant Circuit

L C

where:fr = resonant frequency in HzL = inductance in henrysC = capacitance in farads



Fig. 25-2:

Fig. 25-2 (b) shows XL and XC equal, resulting in a net reactance of zero ohms. The only opposition to current is the coil resistance rs, which limits how low the series resistance in the circuit can be.


Resonant Rise in VL and VC

5 A R = 4

XC = 31

XL = 31

20 V5 kHz Ir = 20/4 = 5 A

VL = I × XL = 155 V

VC = I × XC = 155 V

Note: The reactive voltages are phasor opposites and they cancel (VXL+VXC

= 0).


Resonant Rise in VL and VC

4

0.25 F4 mH

Q = 32

5 A20 V5 kHz

VL = I × XL = 640 V

VC = I × XC = 640 V

32 × 20 V = 640 V

VL = I × XL = 155 V

VC = I × XC = 155 V

7.8 × 20 V = 155 V

R = 4

L

20 V5 kHz

5 A

Q = 7.8

1 F

1 mH

QVS = VX


LCπ2

1f r

Frequency Response

20 V

f

4 Ω

1 μF1 mH

1 2 3 4 5 6 7 8 9 10Frequency in kHz

5

0

3

4

2

1

Cur

rent

in A

LCπ2

1f r =

1

2 π 1× 10−3 × 1× 10−6= 5.03 kHz

Parallel ResonanceParallel Resonance

When L and C are in parallel and XL equals XC, the reactive branch currents are equal and opposite at resonance.

Then they cancel each other to produce minimum current in the main line.

Since the line current is minimum, the impedance is maximum.


where:fr = resonant frequency in HzL = inductance in henrysC = capacitance in farads

L

C

Parallel Resonant Circuit

LCf

r π2

1=

[Ideal; no resistance]Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.


Fig. 25-6


20 V R = 1 k C = 1 F L = 1 mH

Frequency Response

Frequency in kHz1 2 3 4 5 6 7 8 9 10

0

1

2

3

I T in

A

Inductive Capacitive


Resonant FrequencyResonant Frequency

The formula for the resonant frequency is derived from XL = XC.

LCπ2

1f r

For any series or parallel LC circuit, the fr equal to

is the resonant frequency that makes the inductive and capacitive reactances equal.

QQ Magnification Factor of Magnification Factor of Resonant CircuitResonant Circuit

The quality, or figure of merit, of the resonant circuit, in sharpness of resonance, is indicated by the factor Q.

The higher the ratio of the reactance at resonance to the series resistance, the higher the Q and the sharper the resonance effect.

The Q of the resonant circuit can be considered a magnification factor that determines how much the voltage across L or C is increased by the resonant rise of current in a series circuit.



Q is often established by coil resistance.

31.61

31.6==

rS

XLQ =

20 V5.03 kHz

C = 1 F L = 1 mH

rS = 1


4 20 V

1 F1 mH

4 20 V

0.25 F4 mH

5

1 2 3 4 5 6 7 8 9 10Frequency in kHz

0

3

4

2

1

Cur

rent

in A Half-power

point

Q = 7.8 Q = 32

Increasing the L/C Ratio Raises the Q


Bandwidth of Bandwidth of Resonant CircuitResonant Circuit

When we say that an LC circuit is resonant at one frequency, this is true for the maximum resonance effect.

Other frequencies close to fr also are effective.

The width of the resonant band of frequencies centered around fr is called the bandwidth of the tuned circuit.

Bandwidth ofBandwidth of Resonant Circuit Resonant Circuit


Fig. 25-10:

TuningTuning

Fig. 25-12

Tuning means obtaining resonance at different frequencies by varying either L or C. As illustrated in Fig. 25-12, the variable capacitance C can be adjusted to tune the series LC circuit to resonance at any one of five different frequencies.

TuningTuning

Fig. 25-13

Fig. 25-13 illustrates a typical application of resonant circuits in tuning a receiver to the carrier frequency of a desired radio station. The tuning is done by the air capacitor C, which can be varied from 360 pF to 40 pF.

MistuningMistuning

When the frequency of the input voltage and the resonant frequency of a series LC circuit are not the same, the mistuned circuit has very little output compared with the Q rise in voltage at resonance.

Similarly, when a parallel circuit is mistuned, it does not have a high value of impedance

The net reactance off-resonance makes the LC circuit either inductive or capacitive.

Analysis of Parallel Analysis of Parallel Resonant CircuitsResonant Circuits

Fig. 25-14

Parallel resonance is more complex than series resonance because the reactive branch currents are not exactly equal when XL equals XC. The coil has its series resistance rs in the XL branch, whereas the capacitor has only XC in its branch. For high-Q circuits, we consider rs negligible.

Analysis of Parallel Analysis of Parallel Resonant CircuitsResonant Circuits

In low-Q circuits, the inductive branch must be analyzed as a complex impedance with XL and rs in series.

This impedance is in parallel with XC, as shown in Fig. 25-14.

The total impedance ZEQ can then be calculated by using complex numbers.

Fig. 25-14

Damping of Parallel Resonant Damping of Parallel Resonant CircuitsCircuits

Fig. 25-15

In Fig. 25-15 (a), the shunt RP across L and C is a damping resistance because it lowers the Q of the tuned circuit. The RP may represent the resistance of the external source driving the parallel resonant circuit, or Rp can be an actual resistor. Using the parallel RP to reduce Q is better than increasing rs.

Choosing Choosing LL and and C C for a Resonant for a Resonant CircuitCircuit

A known value for either L or C is needed to calculate the other.

In some cases, particularly at very high frequencies, C must be the minimum possible value.

At medium frequencies, we can choose L for the general case when an XLof 1000 Ω is desirable and can be obtained.

For resonance at 159 kHz with a 1-mH L, the required C is 0.001 μF.

This value of C can be calculated for an XC of 1000 Ω, equal to XL at the fr of 159 kHz.

Examples of FilteringExamples of Filtering

Electronic circuits often have currents of different frequencies corresponding to voltages of different frequencies because a source produces current with the same frequency as the applied voltage.

Examples: The ac signal input to an audio circuit can have high

and low audio frequencies. An rf circuit can have a wide range of radio frequencies

in its input.

Examples of FilteringExamples of Filtering

Filter examples (continued): The audio detector in a radio has both radio frequencies

and audio frequencies in the output. The rectifier in a power supply produces dc output with

an ac ripple superimposed on the average dc level. In such applications where the current has different

frequency components, it is usually necessary either to favor or to reject one frequency or a band of frequencies.

Direct Current Combined with Direct Current Combined with Alternating CurrentAlternating Current

Fig. 26-2

Current that varies in amplitude but does not reverse in polarity is considered pulsating or fluctuating direct current. Fig. 26-2 illustrates how a circuit can have pulsating direct current or voltage. The steady dc voltage of the battery VB is in series with the ac voltage VA. Since the two series generators add, the voltage across RL is the sum, as shown in (b).

Transformer CouplingTransformer Coupling

Fig. 26-5

A transformer produces induced secondary voltage just for variations in primary current. With pulsating direct current in the primary, the secondary has a voltage only for ac variations. The steady dc component in the primary has no effect in the secondary.

Capacitive CouplingCapacitive Coupling

Capacitive coupling is probably the most common type of coupling in amplifier circuits.

The coupling connects the output of one circuit to the input of the next.

The requirements are to include all frequencies in the desired signal, while rejecting undesired components.

Usually, the dc component must be blocked from the input to the ac amplifiers.

The purpose is to maintain a specific dc level for the amplifier operation.

Capacitive CouplingCapacitive Coupling


Fig. 26-6

Fig. 26-6 illustrates that the RC coupling blocks the dc component. With fluctuating dc voltage applied, only the ac component produces charge and discharge for the output voltage across R.

Bypass CapacitorsBypass Capacitors

A bypass is a path around a component. In circuits, the bypass is a parallel or shunt path. Capacitors are often used in parallel with resistance to

bypass the ac component of a pulsating dc voltage. The result is steady dc voltage across the RC parallel

combination.

Bypass CapacitorsBypass Capacitors


Fig. 26-7:

Fig. 26-7 illustrates that the low reactance of bypass capacitor C1 short-circuits R1 for an ac component of fluctuating dc input voltage.

Filter CircuitsFilter Circuits

In terms of their function, filters can be classified as either low-pass or high-pass.

A low-pass filter allows the lower frequency components of the applied voltage to develop output voltage across the load resistance.

A high-pass filter allows the higher frequency components of the applied voltage to develop voltage across the output load resistance.

The most common type filters using L and C are the L, T, and π.

The ability of a filter to reduce the amplitude of undesired frequencies is called attenuation.

Low-Pass FiltersLow-Pass Filters

With an applied input voltage having different frequency components, the low-pass filter action results in maximum low-frequency voltage across RL, while most of the high-frequency voltage is developed across the series choke or resistance.



Fig. 26-9: Using the series resistance R in Fig. 26-9 (f) instead of a choke provides a more economical π filter in less space.



Fig. 26-10:

Fig. 26-10 illustrates the response of a low-pass filter with cutoff at 15 kHz. The filter passes the audio signal but attenuates radio frequencies.

High-Pass FiltersHigh-Pass Filters

The high-pass filter passes to the load all frequencies higher than the cutoff frequency fc, whereas lower frequencies cannot develop appreciable voltage across the load.

High-Pass FiltersHigh-Pass Filters


Fig. 26-11:

Analyzing Filter CircuitsAnalyzing Filter Circuits

Any low-pass or high-pass filter can be thought of as a frequency-dependent voltage divider, since the amount of output voltage is a function of frequency.

Special formulas can be used to calculate the output voltage for any frequency of the applied voltage.

There is a mathematical approach in analyzing the operation of the most basic low-pass and high-pass circuits.

Analyzing Filter CircuitsAnalyzing Filter Circuits

Filter types RC and RL low-pass – pass low frequencies and

attenuate high frequencies. RC and RL high-pass – pass high frequencies and

attenuate low frequencies. RC band-pass – a filter designed to pass only a specific

band of frequencies from its input to its output. RC band-stop – a filter designed to block or severely

attenuate only a specific band of frequencies.

Decibels and Frequency Decibels and Frequency Response CurvesResponse Curves

In analyzing filters, the decibel (dB) unit is often used to describe the amount of attenuation offered by the filter.

In basic terms, the decibel is a logarithmic expression that compares two power levels.

Expressed mathematically,

NdB = 10log (Pout/Pin) If the ratio Pout/Pin is greater than one, the NdB value is

positive, indicating and increase in power. If the ratio Pout/Pin is less than one, the NdB value is

negative, indicating a loss and referred to as an attenuation.



Fig. 26-22:

Fig. 26-22 shows a log-log graph paper. Notice that each octave corresponds to a 2-to-1 range of values and each decade corresponds to a 10-to-1 range of values.



Fig. 26-23 (a)

Figure 26-23 illustrates an RC low-pass filter frequency response curve. The RC circuit is shown in (a) and the response curve is shown in (b). (Next slide)



Fig. 26-23 (b)

Resonant FiltersResonant Filters

Tuned circuits provide a convenient method of filtering a band of radio frequencies because relatively small values of L and C are necessary for resonance.

A tuned circuit provides filtering action by means of its maximum response at the resonant frequency.

The width of the band of frequencies affected by resonance depends on the Q of the tuned circuit; a higher Q provides a narrower bandwidth.

Resonant filters are often called band-stop or band-pass filters.

The band-stop filter is also referred to as a wavetrap.



Fig. 26-24

A series resonant circuit has maximum current and minimum impedance at the resonant frequency. Connected is series with RL, as in Fig. 26-24 (a), the series-tuned LC circuit allows frequencies at and near resonance to produce maximum output across RL. This is band-pass filtering.



Fig. 26-25

A parallel resonant circuit has maximum impedance at the resonant frequency. Connected in series with RL, as in Fig. 26-25 (a), the parallel-tuned LC circuit provides maximum impedance in series with RL at and near the resonant frequency. This is a band-stop filter for the bandwidth of the tuned circuit.

Interference FiltersInterference Filters

Voltage or current not at the desired frequency represents interference.

Usually, such interference can be eliminated by a filter.

Some typical applications are Low-pass filter to eliminate rf interference from the 60-

Hz power-line input to a receiver High-pass filter to eliminate rf interference from the

signal picked up by a television receiving antenna Resonant filter to eliminate an interfering radio

frequency from the desired rf signal

basic ac circuits part b - rc and l/r time constants - rlc circuits - complex numbers for ac...

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