Engr.Tehseen Ahsan

Lecturer, Electrical Engineering Department

EE-307 Electronic Systems Design

HITEC University Taxila Cantt, Pakistan

Amplifier Frequency Response

(Part 1)

10.1 Introduction Previously we neglected the effects of input frequency on an

amplifier’s operation due to capacitive elements in the circuit

in order to focus on other concepts (in EE-205)

The coupling and bypass capacitors were considered to be

ideal shorts and the internal transistor capacitances were

considered to be ideal opens. This treatment is valid when the

frequency is in an amplifier’s midrange.

Since capacitive reactance is inversely proportional to the

input frequency. When the frequency is low enough, the

coupling and bypass capacitors can no longer be considered

as shorts because their reactances are large enough to have a

significant effect.

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10.1 Introduction continue… Also, when the frequency is high enough, the internal

transistor capacitances can no longer be considered as opens

because their reactances become small enough to have a

significant effect on amplifier operation.

Frequency response of an amplifier is the change in gain

or phase shift over a specified range of input signal

frequencies.

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Effect of Coupling Capacitors Since capacitive reactance is inversely proportional to frequency. At

lower frequencies ( Audio Frequencies below 10 Hz)- capacitively

coupled amplifiers such as those in figure 10-1 have less voltage

gain than they have at higher frequencies. The reason is that at lower

frequencies more signal is dropped across C1 and C3 because their

reactances are higher. This higher signal voltage drop at lower

frequencies reduces the voltage gain. Also a phase shift is introduced

by the coupling capacitors because C1 forms a lead circuit with Rinof the amplifier and C3 forms a lead circuit with RL in parallel RCwith or RD

Recall an RC circuit ( Output voltage across R leads the input

voltage in phase)

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Effect of Coupling Capacitors continue..

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Effect of Bypass Capacitors At low frequencies, the reactance of the bypass capacitor C2 in

figure 10-1, becomes significant and the emitter ( or FET source

terminal) is no longer at ac ground. The capacitive reactance XC2 in

parallel with RE (or RS ) creates an impedance that reduces the gain.

For example when the frequency is sufficiently high, XC = OΩ and

the voltage gain of the CE amplifier is Av = RC/r'e. At Lower

frequencies, XC >> OΩ and the voltage gain is Av= RC/(r'e+ Ze)

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Effect of Internal Transistor Capacitors At high frequencies the coupling and bypass capacitors become

effective ac shorts and do not affect an amplifier’s response. Internal

transistor capacitances, however do come into play, reducing an

amplifier’s gain and introducing phase shift as the signal frequency

increases.

In figure 10-3 , in the case of BJT, Cbe is the base-emitter junction

capacitance and Cbc is the base-collector junction capacitance

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Effect of Internal Transistor Capacitors continue…

At lower frequencies, the internal capacitances have a very high

reactance therefore they look like opens and have no effect on the

transistor’s performance.

As the frequency goes up, the internal capacitive reactances go

down and at some point they begin to have a significant effect on

the transistor’s gain.

When the reactance of Cbe becomes small enough, a significant

amount of voltage drop is lost due to voltage divider effect of the

signal source resistance and the reactance Cbe as illustrated in figure

10-4(a)

When the reactance of Cbc becomes small enough, a significant

amount of voltage is fed back out of phase with input (-ve

feedback), thus effectively reducing the voltage gain as shown in

figure 10-4(b)8

Effect of Internal Transistor Capacitors continue…

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Miller’s Theorem

It is used to simplify the analysis of inverting amplifiers at high

frequencies where the internal capacitances are important.

The capacitance Cbc between the input (base) and output (collector)

is shown in figure 10-5(a) in a generalized form.

Av is the absolute voltage gain of the amplifier and C represents Cbc

Miller’s theorem states that C effectively appears as a capacitance

from input to ground as shown in figure 10-5(b) that can be

expressed as follows: Cin(Miller) = C (Av+1)…..( 1)

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Miller’s Theorem continue…

Equation 1 shows that Cbc has a much greater impact on input

capacitance than its actual value. If Cbc = 6 pF and amplifier gain is

50 then Cin(Miller) = 306 pF.

Figure 10.6 shows how this effective input capacitance appears in

the actual ac equivalent circuit in parallel with Cbe

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Miller’s Theorem continue…

Miller’s theorem also states that C effectively appears as a

capacitance from output to ground as shown in Figure 10.5(b), that

can be expressed as follows: Cout(Miller) = C (Av+1/Av)….(2)

Equation 2 indicates that if the voltage gain is 10 or greater

Cout(Miller)=C=Cbc because (Av+1/Av) is approximately equal to 1

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10.2 The Decibel

Decibels are a form of gain measurement and are commonly used

to express amplifier response.

The decibel is a logarithmic measurement of the ratio of one power

to another or one voltage to another.

Power gain is expressed in decibels (dB) as Ap(dB) = 10 log Ap where

Ap is the actual power gain, Pout/Pin

Voltage gain is expressed in decibels (dB) as Av(dB) = 20 log Av

If Av is greater than 1, the dB gain is positive and if Av is less than 1,

the dB gain is negative and is usually called attenuation.

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O dB Reference

It is often convenient in amplifiers to assign a certain value of gain

as the O dB reference.

This does not mean that actual voltage gain is 1 ( 0 dB); it means

that the reference gain, no matter what its actual value , is used as a

reference with which to compare other values of gain and is there

assigned a 0 dB value.

Many amplifiers exhibit a maximum gain over a certain range of

frequencies and a reduced gain at frequencies below and above this

range.

The maximum gain occurs for the range of frequencies between the

upper and lower critical frequencies and is called midrange gain,

which is assigned a 0 dB value.

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O dB Reference continue…

Any value of gain below this range can be referenced to 0 dB and

expressed as a negative dB value.

Figure 10-7 illustrates a normalized gain-versus-fequency curve

showing several dB points. The term normalized means that the

midrange voltage gain is assigned a value of 1 or 0 dB.

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Mid range voltage

gain is 100

Gain at a certain frequency

below mid range is 50 thus

the reduced voltage gain

can be expressed as 20

log(50/100)= 20 log (0.5) =

- 6 dB

O dB Reference continue…

Table 10-1 shows how doubling or halving voltage gains translates

into decibel values. Notice that every time the voltage gain is

doubled, the decibel value increases by 6 dB and every time the gain

is halved, the dB value decreases by 6 dB.

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The Critical Frequency

Also called cutoff or corner frequency is the frequency at which the

output power drops to one-half of its midrange value. This

corresponds to a 3 dB reduction in power gain, as expressed in dB

by Ap(dB) = 10 log(0.5) = -3 dB

The output voltage is 70.7% of its midrange value at critical

frequency and expressed in dB as Av(dB) = 20 log(0.707) = -3 dB

The voltage gain is down to 3 dB or is 70.7% of its midrange value

and at the same frequency, the power is one-half of its midrange

value.

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Power Measurement in dBm

The dBm is a unit for measuring power levels referenced to 1mW.

Positive dBm values represent power levels above 1mW and

negative dBm values represent power levels below 1mW.

Each 3 dBm increase corresponds to a doubling of the power, and a

3 dBm decrease corresponds to a halving the power. See table 10-2

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10-3 Low Frequency Amplifier Response

“We will examine how the voltage gain and phase shift

of a capacitively coupled amplifier are affected by

frequencies below ( below midrange) which the

reactance of the coupling capacitors become too large

to neglect.”

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BJT Amplifiers

A typical capacitively coupled CE amplifier is shown in Figure 10-8.

Assuming that coupling and bypass capacitors are ideal shorts at the

midrange signal frequency, we can determine the midrange

voltage gain using equation (1), where Rc= RC ∥ RD Av(mid) = Rc/r'e …… (1)

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BJT Amplifiers Continue…

The BJT amplifier in fig 10-8 has three high-pass RC circuits that

affect its gain as the frequency is reduced below midrange.

The three high-pass RC circuits in low-frequency ac equivalent

circuit is shown in Fig 10-9.

Unlike the ac equivalent circuit used previously which represented

midrange response(XC=OΩ ), the low-frequency equivalent circuit

retains the coupling and bypass capacitors because XC is not small

enough to neglect when the signal frequency is sufficiently low.

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The Input RC Circuit

The input RC circuit is formed by C1 and the amplifier’s input

resistance as shown in fig 10-10

As the signal frequency decreases increases. This causes less voltage

across Rin because more voltage is dropped across C1 and

consequently the overall voltage gain of the amplifier is reduced.

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The Input RC Circuit Continue…

The Vbase in fig 10-10( neglecting the internal resistance of the input

source) can be stated as

As mentioned previously, a critical point in the amplifier’s response

occurs when the output voltage is 70.7% of its midrange value. This

condition occurs in the input RC circuit when XC1=Rin

23 Attenuation / Attenuation factor

Lower Critical Frequency

The condition where the gain is down 3 dB is logically called the

-3 dB point of the amplifier response; the overall gain is 3 dB less

than at midrange frequencies because of the attenuation of the input

RC circuit.

The frequency , fc , at which this condition occurs is called lower

critical frequency ( lower cutoff frequency, lower corner frequency) and can

be calculated as follows:

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25

Voltage Gain Roll-off at low frequencies

The input RC circuit reduces the overall voltage gain of an

amplifier by 3 dB when the frequency is reduced to the critical

value fc.

As the frequency continues to decrease below fc , the overall

voltage gain also continues to decrease.

The rate of decrease in voltage gain with frequency is called roll-

off.

For each ten times reduction in frequency below fc , there is a 20 dB

reduction in the voltage gain.

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Voltage Gain Roll-off at low frequencies Continue…

Let’s consider a frequency that is one-tenth of the critical frequency

(f=0.1fc ). Since XC1= Rin at fc ,then XC1= 10Rin at 0.1fc because of

the inverse relationship of XC1 and fc. The attenuation of the input

RC circuit is, therefore,

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dB/decade

A ten-times change in frequency is called a decade.

For input RC circuit, the attenuation is reduced by 20 dB for each

decade that the frequency decreases below the critical frequency.

This causes the overall voltage gain to drop 20 dB per decade.

Figure 10-11 shows a graph of dB voltage versus frequency.

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Frequency is reduced to

one-hundredth of fc ( a

two-decade decrease) i.

e, 20 log( 0.01) = -40 dB

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Phase Shift in the Input RC Circuit

In addition to reducing voltage gain, the input RC circuit also

causes an increasing phase shift through an amplifier as the

frequency decreases.

At midrange frequencies, the phase shift through the input RC

circuit is approximately zero because XC1= 0Ω.

At lower frequencies higher values of XC1 causes a phase shift to be

introduced and the output voltage (base voltage),Vb of the RC

circuit leads the input voltage Vin .

The phase angle in an input RC circuit is expressed as ( Recall ac

circuit theory)

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Phase Shift in the Input RC Circuit Continue…

A continuation of this analysis reveals that the phase shift through

the input RC circuit approaches 90˚ as the frequency approaches

zero.

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The Output RC Circuit

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Fig 10-8 is formed

by C3, the resistance

looking in at the

collector and the

load resistance RL

For Rout , looking in

at the collector, the

transistor is treated

as an ideal current

source( with infinite

internal resistance)

Thevenize the circuit to the left

of capacitor C3 produces an

equivalent voltage source equal

to the collector voltage and a

series resistance equal to RC

The Output RC Circuit Continue…

The critical frequency of the output RC circuit is given by

The effect of the output RC circuit on the amplifier voltage gain is

similar to that of the input RC circuit.

As the signal frequency decreases, XC3 increases. This causes less

voltage across the load resistance because more voltage is dropped

across C3.

Phase shift in the Output RC Circuit The phase angle in the

output RC circuit is

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The Bypass RC Circuit

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Amplifier voltage

gain Av = Rc / r'e

Amplifier voltage gain

Av = Rc / (r'e + Ze)

The Bypass RC Circuit Continue…

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The bypass RC

circuit is formed by

C2 and the

resistance looking

in at the emitter

Rin(emitter)

Rin(emitter) is derived by first applying Thevenin’s

theorem looking from the base of the transistor

towards the input source Vin.This results in an

equivalent resistance (Rth) and an equivalent

voltage source(Vth(1)) in series with the base as

shown in fig 10-16 (c) next slide.

The Bypass RC Circuit Continue…

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Rin(emitter) is determined

with equivalent input

source shorted and is

expressed as

The Bypass RC Circuit Continue…

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Looking from capacitor C2,

Rth/βac + r'e is in parallel

with RE

Thevenizing again, we get the equivalent RC

circuit. The critical frequency for this equivalent

bypass RC circuit is

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FET Amplifiers

A zero-biased D-MOSFET amplifier with capacitive coupling on

the input and output shown in figure 10-18, you have learned

previously that the midrange voltage gain of a zero-biased amplifier

is

This is the gain at frequencies high enough so that the capacitive

reactances are approximately zero

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FET Amplifiers Continue…

The amplifier in fig 10-18 has only two high-pass RC circuits that

influence its low-frequency response.

One RC circuit is formed by the input coupling capacitor C1 and

the input resistance as shown in fig-10-19.

The second RC circuit is formed by the output coupling capacitor

C2 and the output resistance looking in at the drain

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The Input RC Circuit

Just like BJT amplifier, the reactance of the input coupling capacitor

increases as the frequency decreases when XC1= Rin, the gain is

down 3 dB below its midrange.The lowest critical frequency is

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43

The Output RC Circuit

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2nd RC circuit is formed by a

coupling capacitor C2 and the

output resistance looking in at

the drain

Just like BJT, the FET is

also treated as a current

source

The Thevenin equivalent of

the circuit to the left of C2

The Output RC Circuit Continue…

The critical frequency for this RC circuit is

The phase angle in the low-frequency output RC circuit is

Again at the critical frequency, the phase angle is 45˚ and

approaches 90˚ as the frequency approaches zero.

The effect of the output RC circuit on the amplifier’s voltage gain

below the midrange is similar to that of input RC circuit.

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Reference(s)

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Chapter 1O: “Electronic Devices”, by Thomas L .Floyd, 7th Edition.