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VAAGDEVI INSTITUTE OF TECHNOLOGY & SCIENCE
ANALOG CIRCUITS
UNIT- 1 Multistage and Differential Amplifiers
Syllabus:
Introduction – Recap of Small Signal Amplifiers
Multistage Amplifiers, Cascode amplifier
Darlington pair
the MOS Differential Pair
Small-Signal Operation of the MOS Differential Pair
The BJT Differential Pair
Nonideal Characteristics of the Differential Amplifier.
AMPLIFIER:
A circuit that increases amplitude of given signal is an amplifier. The amplifier is an electronic circuit, which amplifies or increases strength of weak signal.
A small AC signal fed into amplifier is obtained as large AC signal of same frequency. A amplifier is essential part in radios, TV’s and other communications A linear amplifier magnifies an input signal and produces an output signal whose magnitude is larger and directly proportional to the input signal.
The ac analysis, called a small-signal analysis, can be performed with the dc source set to zero.
Small-Signal Hybrid-π Equivalent Circuit of the Bipolar Transistor
All the transistor amplifiers are two port networks having two voltages and two currents. The positive directions of voltages and currents.Out of four variables two can be selected as are independent variables and two are dependent variables.
In h- parameters
(V 1 & I 2 ) – Dependent Variables (I 1 & V 2 ) – Independent Variables
Assuming the transistor as two port network. The iB vs vBE graph the small time varying signal is superimposed on Q-point. The slope of Q-point is constant with units of conductance.
Substitute equation 2 in 1
We can then write the small-signal collector current as 𝒊𝒄 = 𝒈𝒎𝑽𝒃𝒆
By using above equation we have to draw output port
Combine input port an output port, we will get small signal Hybrid-π Equivalent Circuit of the Bipolar Transistor
Fig: A simplified small-signal hybrid- π equivalent circuit
Alternative Form of Equivalent Circuit
𝑖𝐶 = (^) 𝜕 𝑖𝜕 𝑖𝐶𝐵 |𝑄−𝑝𝑡. 𝑖𝐵
𝑤ℎ𝑒𝑟𝑒 𝜕 𝑖 𝜕 𝑖𝐶𝐵 |𝑄−𝑝𝑡 ≡ 𝛽
We can then write as 𝒊𝒄 = 𝜷 𝒊𝒃
𝒓𝝅𝒈𝒎 = [𝜷𝑽 𝑰𝑪𝑸𝑻 ] [𝑰 𝑽𝑪𝑸𝑻 ] = 𝜷
Small-Signal Equivalent-Circuit Models:
Fig. MOSFET The FET behaves as a voltage-controlled current source. It accepts a signal vgs between gate and source and provides a current gmvgs at the drain terminal. The input resistance of this controlled source is very high and ideally infinite. The output resistance looking into the drain also is high, and we have assumed it to be infinite. The DC bias current is 𝐼𝐷 = 12 𝑘𝑛′ 𝑊𝐿 (𝑉𝑔𝑠 − 𝑉𝑡)^2 𝑘𝑛′^ = 𝜇𝑛𝐶𝑂𝑋 VOV = VGS - Vt is the overdrive voltage at which the MOSFET is biased to operate. 𝑔𝑚 = 𝑘𝑛′ 𝑊𝐿 (𝑉𝑔𝑠 − 𝑉𝑡) = 𝑘𝑛′ 𝑊𝐿 𝑉𝑂𝑉
𝑔𝑚 = (^) 𝑉𝑔𝑠2 𝐼−𝑉𝐷𝑡 = 2 𝐼 𝑉𝑂𝑉𝐷
The small-signal model of the MOSFET
𝒈𝒎𝑽𝝅
𝑅 1 || 𝑅 2 ||𝑟𝜋
𝑉𝑂 = −𝑔𝑚𝑉𝜋[𝑟𝑜||𝑅𝐶 ] ---
And the control voltage 𝑉𝜋 is found to be
𝑉𝜋 = (𝑅 1 𝑅||𝑅^1 ||𝑅 2 ||𝑟^2 ||𝑟𝜋)+𝑅𝜋 𝑠. 𝑉𝑆 ---
Substitute eq2 in eq
𝑉𝑂 = −𝑔𝑚 (𝑅 1 𝑅||𝑅^1 ||𝑅 2 ||𝑟^2 ||𝑟𝜋)+𝑅𝜋 𝑠. 𝑉𝑆 [𝑟𝑜||𝑅𝐶 ]
𝑨𝑽 = 𝑽 𝑽𝑶𝑺 = −𝒈𝒎 [𝒓𝒐||𝑹𝑪] [ (𝑹𝑹𝟏||𝑹𝟐||𝒓𝝅
𝟏||𝑹𝟐||𝒓𝝅)^ + 𝑹𝒔
]
Assume 𝑅 1 ||𝑅 2 ||𝑟𝜋 = 𝑅𝐵 and 𝑟𝑜||𝑅𝐶 = 𝑅𝐶
Multistage amplifiers:
In practical applications, the output of a single state amplifier is usually
insufficient, though it is a voltage or power amplifier. Hence they are replaced
by Multistage amplifiers.
A generalized three-stage amplifier is shown in below figure.
Fig. A generalized three-stage amplifier
In Multi-stage amplifiers, the output of first stage is coupled to the input of next
stage using a coupling device. These coupling devices can usually be a
capacitor or a transformer. This process of joining two amplifier stages using a
coupling device can be called as Cascading.
In general, the input impedance of the first stage must be high and the output
impedance of the last must be low.
The overall gain is the product of the individual gains.
Cascode Amplifier :
Cascode amplifier is a type of multistage amplifier. The CE amplifier followed by CB amplifier is called cascode amplifier. The high bandwidth and high gain are major advantages of cascode amplifier. The input is into a common-emitter amplifier (Q 1 ), which drives a common-base amplifier (Q 2 ). The output signal current of Q 1 is the input signal of Q 2.
Fig. Cascode amplifier
F
i g
. Small-signal equivalent circuit of the cascode amplifier
The bias resistance of the amplifier is 𝑹𝑩𝟏 = 𝑹𝟐 ∥ 𝑹𝟑
The input resistance of the amplifier is 𝑹𝒊 = 𝑹𝑩𝟏 ∥ 𝒓𝝅𝟏 The out resistance of the amplifier is 𝑹𝒐 = 𝑹𝑪 Vπ1 = Vs since we are assuming an ideal signal voltage source.
The Voltage gain of the amplifier is
The output voltage is 𝑉𝑜 = −𝑔𝑚2𝑉𝜋2[𝑅𝑐 ∥ 𝑹𝑳]^ --- Writing a KCL equation at E 2 , we have 𝑔𝑚1𝑉𝜋1 = 𝑉 𝑟𝜋2𝜋2 + 𝑔𝑚2𝑉𝜋 The control voltage V π2 (noting that Vπ1 = Vs ), we find 𝑉𝜋2 = 𝑔𝑚1𝑉𝑠𝑟𝜋2 − 𝑔𝑚2𝑉𝜋2 𝑟𝜋 𝑉𝜋2 + 𝑔𝑚2𝑉𝜋2 𝑟𝜋2 = 𝑔𝑚1𝑉𝑠𝑟𝜋 𝑉𝜋2(1 + 𝑔𝑚2 𝑟𝜋2) = 𝑔𝑚1𝑉𝑠𝑟𝜋 𝑉𝜋2 = (^) (1+𝑔𝑔𝑚1𝑚2𝑉𝑠𝑟 𝜋2𝑟𝜋2) ∴ 𝛽 2 = 𝑔𝑚2 𝑟𝜋
𝑉𝜋2 = ( (^) 1+𝛽𝑟𝜋2 2 ) 𝑔𝑚1𝑉𝑠 --- Substitute eq. (3) in eq. (2), we get output voltage as 𝑉𝑜 = −𝑔𝑚1𝑔𝑚2 ( (^) 1+𝛽𝑟𝜋2 2 ) 𝑔𝑚1𝑉𝑠[𝑅𝑐 ∥ 𝑹𝑳] Therefore, the small-signal voltage gain is
𝑨𝒗𝒔 = 𝑽 𝑽𝒐𝒔 = −𝒈𝒎𝟏𝒈𝒎𝟐 ( 𝟏+𝜷𝒓𝝅𝟐𝟐 ) 𝒈𝒎𝟏 [𝑹𝒄 ∥ 𝑹𝑳]
We know that 𝑔𝑚2 ( (^) 1+𝛽𝑟𝜋2 2 ) = (^) 1+𝛽𝛽^22 ≅ 1 The gain of the cascode amplifier is then approximately
𝑨𝒗𝒔 ≅ = −𝒈𝒎𝟏 [𝑹𝒄 ∥ 𝑹𝑳]
Which is the same as for a single-stage common-emitter amplifier.
The current gain of the amplifier is 𝑨𝑰𝑺 = 𝑰 𝑰𝑶𝒔
The overall current gain of darrlington pair is multiplication of individual current gains. If matched transistors then β1=β2 then βd=𝛽1^2
Generally darlington pair available in package contain 3 terminals i.e. base ,emitter ,and collector Darlington pair can be used as emitter follower its equivalent is cascading of two emitter followers.
Fig: A Darlington pair amplifier or configuration The Darlington pair basically consisting of two bipolar transistors with the emitter of one transistor connected to the base of the other, such that the current amplified by the first transistor is amplified further by the second one.
The collectors of both transistors are connected together. It is often packaged as a single transistor. The high current gain and high input impedance are the advantages of Darlington pair. One drawback is doubling of the base–emitter voltage.
Fig: Small-signal equivalent circuit of Darlington pair 𝑖𝐸1 = 𝑖𝑏1 + 𝛽 1 𝑖𝑏1 = 𝑖𝑏1[1 + 𝛽 1 ] = 𝑖𝑏2 --- 𝑖𝐸2 = 𝑖𝑏2 + 𝛽 1 𝑖𝑏2 = 𝑖𝑏2[1 + 𝛽 2 ]^ --- Substitute equation 1 in 2 𝑖𝐸2 = 𝑖𝑏2[1 + 𝛽 2 ] = 𝑖𝑏1[1 + 𝛽 1 ][1 + 𝛽 2 ]^ --- Current gain
AI = 𝑖 𝑖𝐸2𝑏1 = 𝑖𝑏1[1+𝛽𝑖^1 𝑏1][1+𝛽^2 ]
AI = [1 + 𝛽 1 ][1 + 𝛽 2 ]^ ∴ 𝛽 1 ≫ 1, 𝛽 2 ≫ 1 AI = 𝛽 1 𝛽 2 ∴ identical transistors 𝛽 1 = 𝛽 2
𝐀𝐈 = 𝜷𝟏𝟐^ = 𝜷𝟐𝟐^ = 𝜷𝑫
Voltage gain
𝐴𝑉 = 𝑉 𝑉𝑂𝑆 = 𝑖𝐸2 𝑉𝑅𝑆𝐸 = 𝑖𝑏1[1+𝛽^1 ][1+𝛽 𝑉𝑆^2 ] 𝑅𝐸 ---
Apply KVL for input loop
𝑉𝑆 = 𝑖𝑏1 𝑟𝜋1 + 𝑖𝑏2 𝑟𝜋2 + 𝑖𝐸2𝑅𝐸 ---
Equation 1 convert to equivalent circuit
For calculating output impedance 𝑉𝑆 = 0, the above circuit simplified to
𝑍𝑂 = (^) 𝑔^1 𝑚 ||𝑅𝐸 ∴Low resistance || high resistance ≅ Low resistance ∴ (^) 𝑔^1 𝑚 ≪ 𝑅𝐸
𝒎
Differential pair or Amplifier:
The differential-pair or differential-amplifier is the most widely used building block in analog integrated-circuit design. For instance, the input stage of every op-amp is a differential amplifier.
The BJT differential amplifier is the basis of a very-high-speed logic circuit family like emitter- coupled logic (ECL).
Differential Amplifiers is an amplifier in which the output voltage is directly proportional to the difference of input signal. This amplifier, also called a diff-amp There are two input terminals and one output terminal 𝑽𝒐 = 𝑨𝑽𝑶(𝑽𝟏 − 𝑽𝟐) Where A vo is called the open-loop voltage gain.
The differential-mode input voltage 𝑽𝒅 = 𝑽𝟏 − 𝑽𝟐
The common-mode input voltage 𝑽𝒄𝒎 = 𝑽𝟏+𝑽 𝟐 𝟐
The BJT Differential Pair:
The basic BJT differential pair is shown in fig.
Fig. The basic BJT differential pair
Fig: Differential pair with a small differential input signal
From Fig, It can be seen that we are able to steer the entire bias current from one side of the pair to the other with small difference voltages. This current steering property of the differential pair allows it to be used in logic circuits. Apply a very small differential signal, which will result in one of the transistors conducting a
current of (^) 𝟐𝑰 + 𝚫𝑰 ; the current in the other transistor will be 𝑰𝟐 − 𝚫𝑰..
𝑽𝒐 = 𝑽𝑪𝟐 − 𝑽𝑪𝟏 = [𝑽𝑪𝑪 − (𝑰 𝟐 𝑸− ∆𝑰) 𝑹𝑪] − [𝑽𝑪𝑪 − (𝑰 𝟐 𝑸+ ∆𝑰) 𝑹𝑪] = 𝟐∆𝑰𝑹𝑪
A voltage difference is created between vC 2 and vC 1 when a differential-mode input voltage is applied.
The magnitude of the small-signal collector current in each transistor is then ( gmvd ) / 2. 𝑉𝑜 = 𝑉𝐶2 − 𝑉𝐶 𝑉𝑜 = [𝑉𝐶𝐶 − 𝐼𝐶2𝑅𝐶] − [𝑉𝐶𝐶 − 𝐼𝐶1𝑅𝐶] = (𝐼𝐶1 − 𝐼𝐶2)𝑅𝐶 𝑉𝑜 = [(𝐼 2 𝑄 + 𝑔𝑚 2 𝑉 𝑑) − (𝐼 2 𝑄 − 𝑔𝑚 2 𝑉 𝑑)] 𝑅𝐶
Differential mode gain: The ratio of the output signal voltage to the differential-mode input signal is called the differential-mode gain, Ad , which is 𝑉𝑜 = 𝑔𝑚𝑉𝑑𝑅𝐶
𝑽𝒅^ = 𝒈𝒎𝑹𝒄
𝐼𝑄 2 𝑉𝑇^ =^
𝐼𝑄 2 𝑉𝑇 In many cases output is taken at one collect terminal which is called a one-sided output
𝑉𝑜 = 𝑔𝑚𝑉 2 𝑑𝑅𝐶
𝑨𝒅 = 𝒈𝒎 𝟐𝑹 𝑪 ∴ For Unbalanced output /Single output
Small signal analysis of BJT differential pair
