Section 5.6 Small Signal Model & Analysis

Quiz No 1 DE 27 (CE) 06-03-07 State the purpose and four steps (each) that are to be taken for carrying out DC Analysis (b) Small signal Analysis

The operation of the transistor as an amplifier.

Conceptual circuit with the signal source eliminated . (vbe =0) sedr42021_0548a.jpg

DC Analysis Signal source eliminated Active Mode Verification VC>VB-0.4 V

The collector Current & Trans-conductance

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

Linear operation of the transistor under the small-signal condition: sedr42021_0549.jpg

Base Current & Input Resistance at the Base

Emitter Current & Input Resistance @ Emitter

Voltage Gain

DC-AC Models

Large Signal Model Small Signal Model

The amplifier circuit sedr42021_0550.jpg

Figure 5.51 Two slightly different versions of the simplified hybrid-p model for the small-signal operation of the BJT. sedr42021_0551a.jpg

Figure 5.52 Two slightly different versions of what is known as the T model of the BJT. sedr42021_0552a.jpg

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.

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

Figure 5.53 Example 5.14: (a) circuit; (b) dc analysis; sedr42021_0553a.jpg Β = 100 Find Voltage Gain

Figure 5.53 Example 5.14: (a) circuit; (c) small-signal model.

Figure 5.54 Signal waveforms in the circuit of Fig. 5.53. sedr42021_0554a.jpg

Figure 5.55 Example 5.16: (a) Common Base circuit; (b) dc analysis; sedr42021_0555a.jpg

Figure 5.55 Example 5.16: (a) circuit; (c) small-signal model; sedr42021_0555a.jpg

Figure 5.55 Example 5.16: (a) circuit; (d) small-signal analysis performed directly on the circuit. sedr42021_0555a.jpg

Figure 5. 56 Distortion in output signal due to transistor cutoff Figure 5.56 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

Figure 5. 57 Input and output waveforms for the circuit of Fig. 5. 55 Figure 5.57 Input and output waveforms for the circuit of Fig. 5.55. Observe that this amplifier is noninverting, a property of the common-base configuration. sedr42021_0557.jpg

The Early Effect In real world (a) Collector current does show some dependence on collector voltage (b) Characteristics are not perfectly horizontal line

Figure 5.19 (a) Conceptual circuit for measuring the iC –vCE characteristics of the BJT. (b) The iC –vCE characteristics of a practical BJT. sedr42021_0519.jpg

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

The Early Effect vCE < - 0.4 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 axis @ vCE = -VA Typical value of VA ranges 50-100v & called early voltage, after the name of english scientist JM Early

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

The Early Effect

Figure 5.20 Large-signal equivalent-circuit models of an npn BJT operating in the active mode in the common-emitter configuration. sedr42021_0520a.jpg

Figure 5.58 The hybrid-pi small-signal model, in its two versions, with the resistance ro included. sedr42021_0558a.jpg

The hybrid- small-signal model, with the resistance ro included.

Figure E5.40 sedr42021_e0540.jpg

Table 5.4 sedr42021_tb0504a.jpg

Figure 5.59 Basic structure of the circuit used to realize single-stage, discrete-circuit BJT amplifier configurations. sedr42021_0559.jpg

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.

Figure E5.41 sedr42021_e0541a.jpg

Solution Signal swing: for β = 100, +8V, -3.4V; for β= 50, +8V, -4.4V;

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

Figure P5.112 sedr42021_p05112.jpg

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

Figure P5.115 sedr42021_p05115.jpg

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?

Figure P5.116 sedr42021_p05116.jpg

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?

Figure P5.124 sedr42021_p05124.jpg

sedr42021_p05126.jpg Figure P5.126

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-Π model, and find the value of Rm, the voltage gain and the current gain

Figure P5.130 sedr42021_p05130.jpg

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

Figure P5.130 DC Analysis β = 100 , α = 0.99 VA = 100V IE = ?, Rin = ?, overall gain vo/vsig, io/i1 sedr42021_p05130.jpg

Solution P5.130 DC Values 1.92 mA 9.64 KΩ 3.21 V 1.94 mA

Solution P5.130 Check for Mode 3.21 V ACTIVE MODE VCB > - 0.4 V 1.94 mA 1.92 mA ACTIVE MODE VCB > - 0.4 V

Solution P5.130 Small Signal Model IC = 1.92 mA VT = 25 mV β = 100 , α = 0.99 VA = 100 V

Solution P5.130

CE with pi Model sedr42021_p05130.jpg

CE with ‘T’ Model sedr42021_p05130.jpg

Comparison ‘pi’ Vs ‘T’ Model sedr42021_p05130.jpg

sedr42021_p05134.jpg Figure P5.134

Single Stage BJT Amplifier Three Configurations Common Emitter (CE) Common Emitter (CE) with Emitter Resistance Common Base (CB) Common Collector (CC)

Figure 5. 60 (a) A common-emitter amplifier using the structure of Fig sedr42021_0560a.jpg

Amplifiers Configurations Common Emitter sedr42021_tb0506a.jpg

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

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

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

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

Figure 5.61 (a) A common-emitter amplifier with an emitter resistance Re. sedr42021_0561a.jpg

Quiz No 2 (DE 27 CE) 13-03-2007 Redraw the circuit for Small Signal pi model analysis Redraw the circuit for DC analysis

Figure 5.61 (a) A common-emitter amplifier with an emitter resistance Re. sedr42021_0561a.jpg

Figure 5.61 (a) A common-emitter amplifier with an emitter resistance Re. sedr42021_0561a.jpg

Small Signal Analysis : CE with Emitter Resistance Input Resistance Multiplication by a factor (1+β) is known as the Resistance Reflection Rule. Analysis

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

Small Signal Analysis : CE with Emitter Resistance Current Gain

Small Signal Analysis : CE with Emitter Resistance Summary Re introduces negative feedback gives it the name emitter degenerative resistance

Comparison ‘T’ Vs ‘pi’ Model

A common-base amplifier sedr42021_0562a.jpg

A common-base amplifier with its T model. sedr42021_0562a.jpg

Small Signal Analysis : CB CB has low input resistance CB is non-inverting amplifier sedr42021_0562a.jpg

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.

An emitter-follower circuit : Common Collector sedr42021_0563a.jpg Non-unilateral Amplifier Input Resistance depends upon RL Output Resistance depends upon Rsig

Common Collector An emitter-follower circuit : T model sedr42021_0563a.jpg

An equivalent circuit of the Emitter Follower - CC sedr42021_0564.jpg

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

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

Common Collector : Output Resistance Output Resistance is low

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

An equivalent circuit of the emitter follower sedr42021_0566.jpg

BJT Configurations sedr42021_tb0506a.jpg

Common Emitter sedr42021_tb0506a.jpg

Common Emitter with Emitter Resistance sedr42021_tb0506a.jpg

Common Base : Current Buffer sedr42021_tb0506a.jpg

Common Collector : Voltage Follower sedr42021_tb0506a.jpg

Summary & Comparison

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

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

Figure P5.135 sedr42021_p05135.jpg

Solution P5-135 DC Analysis Suppress the AC (independent Sources) Short Circuit Voltage Sources Open Circuit the Capacitors Calculate DC Node Voltages & Loop Currents

Solution P5-135 DC Analysis β=100, α=0.99

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

Small Signal Model

Small Signal Model

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

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

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

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

Small Signal Analysis C vo ic=αie B ie ve E Rin

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

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

Figure P5.143 Common Collector ( Emitter follower) Calculate DC Node Voltages & Loop Currents β=40 sedr42021_p05143.jpg

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

Small Signal Model P5-143 sedr42021_p05143.jpg

Small Signal Model P5-143 vb vo (β+1)ib Ri

Figure P5.144 sedr42021_p05144.jpg

Problem

Problem

Problem 5.147 For the circuit in Fig P5.147, called a boot-strapped follower: Find the dc emitter current and gm, re, and rΠ Use 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.

Boot-Strapped Follower sedr42021_p05147.jpg

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

Figure P5.147 DC Analysis Calculate DC Node Voltages & Loop Currents sedr42021_p05147.jpg

Solution 5-147 DC Analysis

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

Figure P5.147 Small Signal Model B E C αie re sedr42021_p05147.jpg

Solution 5-147 B E C αie B E C αie ie i Rin

B E C αie ie i Rin Solution 5-147 vo

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.

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