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Section 5.6 Small Signal Model & Analysis
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Quiz No 1 DE 27 (CE) State the purpose and four steps (each) that are to be taken for carrying out DC Analysis (b) Small signal Analysis
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The operation of the transistor as an amplifier.
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Conceptual circuit with the signal source eliminated .
(vbe =0) sedr42021_0548a.jpg
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DC Analysis Signal source eliminated
Active Mode Verification VC>VB-0.4 V
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The collector Current & Trans-conductance
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The collector Current & Trans-conductance
For vbe<< VT, the transistor behaves as a voltage-controlled current device The trans-conductance of the controlled source is gm Output resistance is infinity
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Linear operation of the transistor under the small-signal condition:
sedr42021_0549.jpg
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Base Current & Input Resistance at the Base
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Emitter Current & Input Resistance @ Emitter
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Voltage Gain
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DC-AC Models
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Large Signal Model Small Signal Model
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The amplifier circuit sedr42021_0550.jpg
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Figure Two slightly different versions of the simplified hybrid-p model for the small-signal operation of the BJT. sedr42021_0551a.jpg
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Figure 5.52 Two slightly different versions of what is known as the T model of the BJT.
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Small Signal Analysis Coupling Capacitors
Couples the input signal vi to the emitter while blocks the DC signals Don’t let dc biasing established by VCC &VEE be disturbed. when vi is connected Capacitor is of very large value –infinite, acts as short circuit at signal frequency of interest.
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Application (Steps) : Small Signal Model
Suppress ac independent sources ac Voltage Sources be short circuited ac Current Sources be open circuited Capacitors be Open circuited Determine DC operating Point IC Suppress DC independent sources DC Voltage Sources be short circuited DC Current Sources be open circuited Capacitors be short circuited Replace BJT with small signal Model Analyze the resulting circuit of find voltage gain & input/output resistance Active Mode Verification VBE > 0.7 V VC> VB-0.4 V Small Signal Analysis
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Figure 5.53 Example 5.14: (a) circuit; (b) dc analysis;
sedr42021_0553a.jpg Î’ = 100 Find Voltage Gain
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Figure 5.53 Example 5.14: (a) circuit; (c) small-signal model.
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Figure 5.54 Signal waveforms in the circuit of Fig. 5.53.
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Figure 5.55 Example 5.16: (a) Common Base circuit; (b) dc analysis;
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Figure 5.55 Example 5.16: (a) circuit; (c) small-signal model;
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Figure 5.55 Example 5.16: (a) circuit; (d) small-signal analysis performed directly on the circuit.
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Figure 5. 56 Distortion in output signal due to transistor cutoff
Figure Distortion in output signal due to transistor cutoff. Note that it is assumed that no distortion due to the transistor nonlinear characteristics is occurring. sedr42021_0556.jpg
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Figure 5. 57 Input and output waveforms for the circuit of Fig. 5. 55
Figure Input and output waveforms for the circuit of Fig Observe that this amplifier is noninverting, a property of the common-base configuration. sedr42021_0557.jpg
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The Early Effect In real world
(a) Collector current does show some dependence on collector voltage (b) Characteristics are not perfectly horizontal line
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Figure (a) Conceptual circuit for measuring the iC –vCE characteristics of the BJT. (b) The iC –vCE characteristics of a practical BJT. sedr42021_0519.jpg
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Common Emitter Configuration
Emitter serves as a common terminal between input and output terminal Common Emitter Characteristics (ic-vCE) can be obtained at different value of vBE and varying vCE (dc), Collector current can be measured
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The Early Effect vCE < V CBJ become forward biased & BJT leaves active mode & enters saturation mode Characteristics is still a straight line but with a finite slope when extra-polated, the characteristics lines meet at a point on the negative vCE vCE = -VA Typical value of VA ranges v & called early voltage, after the name of english scientist JM Early
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The Early Effect At given vBE , increasing vCE increases reverse biased voltage on CBJ & thus depletion region increases, Resulting in a decrease in the effective base width W Is is inversely proportional to the base width Is increases , and Ic also increases proportionally called Early Effect
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The Early Effect
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Figure Large-signal equivalent-circuit models of an npn BJT operating in the active mode in the common-emitter configuration. sedr42021_0520a.jpg
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Figure 5.58 The hybrid-pi small-signal model, in its two versions, with the resistance ro included.
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The hybrid- small-signal model, with the resistance ro included.
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Figure E5.40 sedr42021_e0540.jpg
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Table 5.4 sedr42021_tb0504a.jpg
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Figure Basic structure of the circuit used to realize single-stage, discrete-circuit BJT amplifier configurations. sedr42021_0559.jpg
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Exercise 5.41 Consider the circuit shown for the case VCC = VEE = 10V, I = 1 mA, RB = 100 kΩ , RC = 8 kΩ and β = 100. Find all DC currents and voltages. What are the allowable signal swings at the collector in both directions? How do these values change as β is changed to 50? To 200? Find the value of the BJT small-signal parameters at the bias point (with β = 100). The Early Voltage VA = 100 V.
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Figure E5.41 sedr42021_e0541a.jpg
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Solution Signal swing: for β = 100, +8V, -3.4V; for β= 50, +8V, -4.4V;
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Problem 5.112 The transistor amplifier in fig. P5.112 is biased with a current source I and has a very high B. find the dc voltage at the collector, Vc. Also, find the value of Replace the transistor with the simplified hybrid- model of Fig 5.51(a)(note that the dc current source I should be replaced with an open circuit). Hence find the voltage gain
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Figure P5.112 sedr42021_p05112.jpg
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Problem 5.115 For the circuit shows in Fig P5.115, draw a complete small-signal equivalent circuit utilizing an appropriate T model for the BJT (use = 0.99). Your circuit should show the value of all components, including the model parameters. What is the input resistance Rin? Calculate the overall voltage gain
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Figure P5.115 sedr42021_p05115.jpg
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Problem 5.116 In the circuit show in the Fig P5.116, the transistor has a of 200. What is the dc voltage at the collector? Find the input resistance Rib and Rin and the overall voltage gain For an output signal of , what values of are required?
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Figure P5.116 sedr42021_p05116.jpg
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Problem 5.124 The transistor in the circuit shown in Fig. P5.124 is biased to operate in the active mode. Assuming that β is very large, find the collector bias current Ic. Replace the transistor with the small-signal equivalent circuit model of Fig 5.52(b) (remember to replace the dc power supply with a short circuit). Analyze the resulting amplifier equivalent circuit to show that Find the value jof these voltage gain (for α = 1). Now, if the terminal labeled vo1 is connected to ground, what does the voltage gain become?
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Figure P5.124 sedr42021_p05124.jpg
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sedr42021_p05126.jpg Figure P5.126
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Problem 5.130 Find the common-emitter amplifier shown in Fig. P5.130, Let VCC =9V, R1 = 27kΩ, R2 = 15kΩ, RE = 1.2kΩ, and Rc = 2.2kΩ. The transistor has β = 100 and VA = 100 V. Calculate the dc bias current IE. If the amplifier operates between a source for which Rsig = 10 kΩ and a load of 2kΩ replace the transistor with its hybrid-Πmodel, and find the value of Rm, the voltage gain and the current gain
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Figure P5.130 sedr42021_p05130.jpg
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Figure P5.130 DC Analysis Suppress the AC (independent Sources)
Short Circuit Voltage Sources Open Circuit the Capacitors Calculate DC Node Voltages & Loop Currents sedr42021_p05130.jpg
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Figure P5.130 DC Analysis β = 100 , α = 0.99 VA = 100V
IE = ?, Rin = ?, overall gain vo/vsig, io/i1 sedr42021_p05130.jpg
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Solution P5.130 DC Values 1.92 mA 9.64 KΩ 3.21 V 1.94 mA
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Solution P5.130 Check for Mode 3.21 V ACTIVE MODE VCB > - 0.4 V
1.94 mA 1.92 mA ACTIVE MODE VCB > V
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Solution P5.130 Small Signal Model IC = 1.92 mA VT = 25 mV
β = 100 , α = 0.99 VA = 100 V
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Solution P5.130
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CE with pi Model sedr42021_p05130.jpg
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CE with ‘T’ Model sedr42021_p05130.jpg
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Comparison ‘pi’ Vs ‘T’ Model
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sedr42021_p05134.jpg Figure P5.134
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Single Stage BJT Amplifier
Three Configurations Common Emitter (CE) Common Emitter (CE) with Emitter Resistance Common Base (CB) Common Collector (CC)
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Figure 5. 60 (a) A common-emitter amplifier using the structure of Fig
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Amplifiers Configurations
Common Emitter sedr42021_tb0506a.jpg
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Amplifiers Configurations Common Emitter
DC Analysis Suppress Independent ac Source Voltage source Short Cct Current Sources --- Open Capacitors ---- Open Cct Redraw the Circuit Analysis VC VB IE=I IC=αIE IB=(β+1)IE sedr42021_tb0506a.jpg VE VC=VCC-ICRC VB=-IBRB VE=VB-VBE gm=Ic/VT rл=β/gm re=α/gm
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Amplifiers Configurations Common Emitter
Small Signal Analysis Suppress Independent DC Source Voltage source Short Cct Current Sources --- Open Capacitors ---- Short Cct Redraw the Circuit by replacing BJT With pi Model sedr42021_tb0506a.jpg Analysis gm=Ic/VT rл=β/gm re=α/gm Find Rin, Rout, Voltage Gain vo/vi
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Common Emitter Rin=RB||rл Rout=RC|RL vbe + -
Short Circuit Current Gain Ais Ais = ios/ii ios=-gmvbe vbe=vi=iiRin Ais=-gmRin Rin=RB||rл Rout=RC|RL
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Summary : CE Input Resistance Output Resistance
Low to moderate typically a few kilohms Output Resistance Output Resistance is relatively low Open Circuit Voltage Gain Voltage gain of a few hundred Short Circuit Current Gain Current gain equal to β
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Figure 5.61 (a) A common-emitter amplifier with an emitter resistance Re.
sedr42021_0561a.jpg
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Quiz No 2 (DE 27 CE) Redraw the circuit for Small Signal pi model analysis Redraw the circuit for DC analysis
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Figure 5.61 (a) A common-emitter amplifier with an emitter resistance Re.
sedr42021_0561a.jpg
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Figure 5.61 (a) A common-emitter amplifier with an emitter resistance Re.
sedr42021_0561a.jpg
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Small Signal Analysis : CE with Emitter Resistance Input Resistance
Multiplication by a factor (1+β) is known as the Resistance Reflection Rule. Analysis
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Small Signal Analysis : CE with Emitter Resistance Voltage Gain
Voltage gain is lower than that of CE because of the additional term (1+β)Re
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Small Signal Analysis : CE with Emitter Resistance Current Gain
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Small Signal Analysis : CE with Emitter Resistance
Summary Re introduces negative feedback gives it the name emitter degenerative resistance
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Comparison ‘T’ Vs ‘pi’ Model
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A common-base amplifier
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A common-base amplifier with its T model.
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Small Signal Analysis : CB
CB has low input resistance CB is non-inverting amplifier sedr42021_0562a.jpg
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Summary : CB Very Low input resistance Rin=re
Short Circuit Current Gain is nearly unity Open circuit Voltage Gain is equal to CE and is positive gm RC Relatively high output resistance (Rc) same as CE Excellent high frequency performance As short circuit current gain is unity Current Buffer, it accept an input signal current at a low input resistance and delivers equal current at a very high output resistance.
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An emitter-follower circuit : Common Collector
sedr42021_0563a.jpg Non-unilateral Amplifier Input Resistance depends upon RL Output Resistance depends upon Rsig
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Common Collector An emitter-follower circuit : T model
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An equivalent circuit of the Emitter Follower - CC
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Overall Voltage Gain is less than unity:
An equivalent circuit of the Emitter Follower - CC sedr42021_0564.jpg Overall Voltage Gain is less than unity: RB>>Rsig, (β+1)(re+(ro||RL))>>(Rsig||RL) The voltage at the emitter (vo) follows very closely the voltage at the input thus give the circuit the name Emitter Follower
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The emitter follower : Reflecting resistance into emitter
sedr42021_0565.jpg For RB>> Rsig & ro >> RL Gain approaches Unity when Rsig/(1+β)<<RL89 Short Circuit Current Gain = 1+β
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Common Collector : Output Resistance
Output Resistance is low
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Summary : Common Collector
Non-unilateral Amplifier Input Resistance depends upon RL Output Resistance depends upon Rsig High Input Resistance Low out Resistance Voltage Gain ≈ unity Relatively Large Current = 1+β sedr42021_0563a.jpg
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An equivalent circuit of the emitter follower
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BJT Configurations sedr42021_tb0506a.jpg
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Common Emitter sedr42021_tb0506a.jpg
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Common Emitter with Emitter Resistance
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Common Base : Current Buffer
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Common Collector : Voltage Follower
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Summary & Comparison
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Comparison of Transistor Configurationsж
 Quantity Common Emitter (CE) Common Collector (CC) Common Base (CB) AI Current Gain High (-50) High (50) Low (0.98) AV Voltage Gain High (-136) Low (0.99) High (1.4) Ri Input Resistance Medium (1 kΩ) High (154 kΩ) Low (21 Ω) Ro Output Resistance High (∞) Low (80 Ω) ж re = 1.1 kΩ, β = 50, RL= Rs = 3kΩ
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Problem 5.135 The amplifier of Fig. P5.135 consists of two identical common-emitter amplifier connected in cascade. Observe that the input resistance of the second stage, Rin2, constitutes the load resistance of the first stage. For Vcc = 15V, R1 = 100kΩ , R2 = 47kΩ , RE = 3.9kΩ , Rc = 6.8kΩ , and β = 1000, determine the dc collector current and dc collector voltage of each transistor. Draw the small-signal equivalent circuit of the entire amplifier and give the values of all its components. Neglect ro1 and ro2 Find Rin1 and vb1/vsig for Rsig = 5 kΩ Find Rin2 and vb2/vb1. For RL = 2kΩ , find vo /vb2 Find the overall voltage gain vo /vsig
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Figure P5.135 sedr42021_p05135.jpg
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Solution P5-135 DC Analysis
Suppress the AC (independent Sources) Short Circuit Voltage Sources Open Circuit the Capacitors Calculate DC Node Voltages & Loop Currents
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Solution P5-135 DC Analysis
β=100, α=0.99
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Small Signal Model Suppress the DC (independent Sources) Short Circuit Voltage Sources Open Circuit Current Sources Short Circuit the Capacitors Draw the Small Signal Model
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Small Signal Model
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Small Signal Model
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Figure P5.141 Common Base For the circuit shown, Assume β=100
Find the input resistance Rin Find the voltage gain vo/vsig sedr42021_p05141.jpg
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Figure P5.141 (Common Base) DC Analysis
Suppress the AC (independent Sources) Short Circuit Voltage Sources Open Circuit the Capacitors Calculate DC Node Voltages & Loop Currents sedr42021_p05141.jpg
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Figure P5.141 (Common Base) DC Analysis
Calculate DC Node Voltages & Loop Currents β =100 I = IB +IC=IE=0.33 mA IB IC sedr42021_p05141.jpg IE
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Figure P5.141 (Common Base) Small Signal Analysis
Suppress the DC (independent Sources) Short Circuit Voltage Sources Open Circuit Current Sources Short Circuit the Capacitors Draw the Small Signal Model sedr42021_p05141.jpg
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Small Signal Analysis C vo ic=αie B ie ve E Rin
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Figure P5.143 Common Collector ( Emitter follower)
sedr42021_p05143.jpg For the circuit shown, Assume β=40 Find IE,VE,& VB Find the input resistance Rin Find the voltage gain vo/vsig
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Figure P5.143 Common Collector ( Emitter follower)
Suppress the AC (independent Sources) Short Circuit Voltage Sources Open Circuit the Capacitors Calculate DC Node Voltages & Loop Currents sedr42021_p05143.jpg
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Figure P5.143 Common Collector ( Emitter follower)
Calculate DC Node Voltages & Loop Currents β=40 sedr42021_p05143.jpg
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Figure P5.143 Common Collector ( Emitter follower)
Small Signal Analysis Suppress the DC (independent Sources) Short Circuit Voltage Sources Open Circuit Current Sources sedr42021_p05143.jpg Short Circuit the Capacitors Draw the Small Signal Model
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Small Signal Model P5-143 sedr42021_p05143.jpg
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Small Signal Model P5-143 vb vo (β+1)ib Ri
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Figure P5.144 sedr42021_p05144.jpg
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Problem
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Problem
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Problem 5.147 For the circuit in Fig P5.147, called a boot-strapped follower: Find the dc emitter current and gm, re, and rΠUse β = 100. Replace the BJT with its T model (neglecting ro), and analyze the circuit to determine the input resistance Rin and the voltage gain vo/vsig. Repeat (b) for the case when capacitor CB is open –circuited. Compare the results with those obtained in (b) to find the advantages of bootstrapping.
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Boot-Strapped Follower
sedr42021_p05147.jpg
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Figure P5.147 DC Analysis Suppress the AC (independent Sources)
Short Circuit Voltage Sources Open Circuit the Capacitors sedr42021_p05147.jpg Calculate DC Node Voltages & Loop Currents
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Figure P5.147 DC Analysis Calculate DC Node Voltages & Loop Currents
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Solution 5-147 DC Analysis
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Figure P5.147 Small Signal Model Suppress the DC (independent Sources)
Short Circuit Voltage Sources Open Circuit Current Sources sedr42021_p05147.jpg Short Circuit the Capacitors Draw the Small Signal Model
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Figure P5.147 Small Signal Model B E C αie re sedr42021_p05147.jpg
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Solution 5-147 B E C αie B E C αie ie i Rin
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B E C αie ie i Rin Solution 5-147 vo
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Without Boot-Strap Capacitor
Solution 5-147 Without Boot-Strap Capacitor Rin Rib The value of overall voltage gain and Rin obtained by using Bootstrap capacitor is higher than cct ,without Bootstrapping Bootstrapping is used to avoid loading of the input cct and to have higher gain.
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Comparison of Transistor Configurations
 Quantity Common Emitter (CE) Common Collector (CC) Common Base (CB) AI Current Gain High Low AV Voltage Gain Ri Input Resistance Medium Ro Out Resistance
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