BJT - Bipolar Junction Transistor (BJT) PDF

© Chor EF BJT-1 Bipolar Junction Transistor (BJT) Bipolar Junction Transistor (BJT) 1. Introduction 2. Modes of Operation: Forward Active, Saturation, Cut- Off and Reverse Active 3. Fo...

© Chor EF BJT-1 Bipolar Junction Transistor (BJT) Bipolar Junction Transistor (BJT) 1. Introduction 2. Modes of Operation: Forward Active, Saturation, Cut- Off and Reverse Active 3. Forward Active Current-Voltage Characteristic 4. Large-Signal Model and dc Analysis 5. Amplifier Circuit 6. Small-Signal Model (Simplified Hybrid-π) and ac Analysis 7. Other Hybrid-π Small-Signal Models Reference  Sedra and Smith, Microelectronic Circuits, Fifth Edition, Oxford (2004), pp. 159 – 168, 173 - 183, 203 - 241. © Chor EF BJT-2 BJT – Introduction (Structure) B E C Active region of npn BJT: Emitter (n++) ++ Emitter Base Collector Emitterp-type (n ) E C (n++) (p+) (n) Base (p+) Silicon B Collector (n) C Circuit symbol of npn BJT: B Active region E  Bipolar junction transistor (BJT) is a 3-terminal device made using a single crystal semiconductor (typically silicon), just like the pn-junction diode.  BJT is made with 3 doped semiconductor regions, namely emitter, base and collector, corresponding to the 3 terminals.  Left figure above shows a schematic cross-sectional view of an npn BJT, where the emitter is n-type, base is p-type and collector is n-type.  The active region of the BJT is the region under (and including) the emitter. This is the part of the device that provides, for example, the amplifying function of the BJT. The rest of the structure is to facilitate the movement of the currents into or out of the transistor. We will focus on the active region of BJT. © Chor EF BJT-3 BJT – Introduction (Structure) B E C Active region of npn BJT: Emitter (n++) ++ Emitter Base Collector Emitterp-type (n ) E C (n++) (p+) (n) Base (p+) Silicon B Collector (n) C Circuit symbol of npn BJT: B Active region E  BJT is not a symmetrical device, in particular, impurities added to the emitter is at a much higher concentration than that added to collector. Hence, emitter is indicated as n++-type (very heavily doped and has many more electrons) and collector as n-type.  Concentration of impurities (of a different type from that for emitter/collector) added to base is in between those of emitter and collector. Base is thus indicated as p+-type.  BJT comprises two back-to-back pn-junctions*: emitter-base junction and base- collector junction. The two pn-junctions must be close enough to each other to interact, and this requires the base to be thin. * BJT cannot be made by connecting two pn-junctions back-to-back. © Chor EF BJT-4 BJT – Introduction (npn vs pnp BJT) Active region of npn BJT: Active region of pnp BJT: Emitter Base Collector Emitter Base Collector E (n++) (p+) (n) C E C (p++) (n+) (p) B VBC - C B VCB + C + Circuit symbol of npn BJT: B Circuit symbol of pnp BJT: B - + - VBE - E VEB + E  The emitter-base junction and base-collector junction can be either forward biased or reverse biased.  There are two types of BJT: npn and pnp (as shown above). Note the difference between the 2 structures and circuit symbols.  For pnp BJT, the emitter is p++-type, base is n+-type and collector is p-type.  Both npn and pnp BJTs perform similar functions: amplification and switching. Their difference lies in the biasing polarity of the two junctions when operating. For a given mode of operation, the current directions and voltage polarities are exactly opposite between npn and pnp BJTs.  Note that BJT is a nonlinear device, just like the pn-junction. © Chor EF BJT-5 Bipolar Junction Transistor (BJT) Bipolar Junction Transistor (BJT) 1. Introduction 2. Modes of Operation: Forward Active, Saturation, Cut- Off and Reverse Active 3. Forward Active Current-Voltage Characteristic 4. Large-Signal Model and dc Analysis 5. Amplifier Circuit 6. Small-Signal Model (Simplified Hybrid-π) and ac Analysis 7. Other Hybrid-π Small-Signal Models Reference  Sedra and Smith, Microelectronic Circuits, Fifth Edition, Oxford (2004), pp. 159 – 168, 173 - 183, 203 - 241. © Chor EF BJT-6 BJT – Modes of Operation  BJT has 2 pn-junctions (emitter-base junction and collector- iC base junction), and each pn-junction can be either forward VBC - C biased or reverse biased, hence there are 4 possible iB B + + permutations of the biasing arrangement for a BJT. These VCE + correspond to different modes of operation of the BJT, as VBE - - shown in the table below for an npn BJT*: E iE Modes of operation of the npn bipolar junction transistor Mode of Emitter-Base Collector-Base Application Operation Junction Junction Cut-off Reverse biased Reverse biased Logic - (VBE < 0 for npn) (VBC < 0 for npn) OFF State Forward Active Forward biased Reverse biased Amplifier (VBE > 0 for npn) (VBC < 0 for npn) Saturation Forward biased Forward biased Logic – (VBE > 0 for npn) (VBC >0 for npn) ON State Reverse Active Reverse Biased Forward Biased Not used (VBE < 0 for npn) (VBC > 0 for npn) * A similar table for pnp bipolar junction transistor is shown in the Appendix A. © Chor EF BJT-7 BJT – Modes of Operation (for reference) Some definitions: iC Voltage at the base with-respect-to the emitter (i.e., VBC - C the base-emitter junction voltage) - VBE iB B + + + VCE Voltage at the base with-respect-to the collector VBE - - (i.e., the base-collector junction voltage) - VBC E Voltage at the collector with-respect-to the emitter - iE VCE (= VBE - VBC)  Cut-off operation (used as the OFF-state in logic circuit): Both emitter-base junction and base-collector junction are reverse biased. VBE < 0 and VBC < 0 for npn BJT. Practically no current flows in the BJT. Between collector and emitter is practically an open circuit.  Saturation operation (Used as the ON-state in logic circuit): Both emitter-base junction and base-collector junction are forward biased. VBE > 0 (~ 0.7 V) and VBC > 0 (~0.7 V) for npn BJT. Large current flows in the BJT. Between collector and emitter is practically a short circuit, and VCE is a small voltage. © Chor EF BJT-8 BJT – Modes of Operation (for reference)  Forward active operation (used in amplifier circuit): iC Emitter-base (n++p+) junction is forward biased and base- VBC - C collector (p+n) junction is reverse biased. iB + + B + VCE VBE > 0 and VBC < 0 for npn BJT. VBE - - BJT works as a controlled current regulator. The main E current through the BJT flows between the emitter and iE collector, iC , and it is controlled by the base-emitter junction voltage, vBE (or equivalently, the base current, iB). This is transistor action and is made possible by an n++-type emitter and p+-type base, and a thin base. See slides BJT-11 and -12.  Reverse active operation (not used): Emitter-base (n++p+) junction is reverse biased and base-collector (p+n) junction is forward biased. VBE < 0 and VBC > 0 for npn BJT. This mode of operation is not useful, as it does not provide amplification, unlike the forward active operation. This is owing to the non-symmetrical nature of BJT, specifically, collector being n-type while emitter is n++-type. © Chor EF BJT-9 BJT – Modes of Operation  The circuit below shows an npn BJT in common-emitter configuration. The common-emitter IV characteristics are also plotted, which shows the relationships between the collector current, iC, and the collector-emitter voltage, vCE, for different base currents, iB, (or equivalently different vBE).  The regions corresponding to the forward active, saturation and cut-off modes of operation are as indicated in the plot below.  We shall be focusing on BJT in forward active operation. Saturation Forward active iC RC VBC - C iB6 RB iB B + + iB5 + VCE iB4 VCC VBE - - iB3 VBB E iB2 (or vBE) iE iB1 Cut-off, iC ~ 0 © Chor EF BJT-10 Bipolar Junction Transistor (BJT) Bipolar Junction Transistor (BJT) 1. Introduction 2. Modes of Operation: Forward Active, Saturation, Cut- Off and Reverse Active 3. Forward Active Current-Voltage Characteristic 4. Large-Signal Model and dc Analysis 5. Amplifier Circuit 6. Small-Signal Model (Simplified Hybrid-π) and ac Analysis 7. Other Hybrid-π Small-Signal Models Reference  Sedra and Smith, Microelectronic Circuits, Fifth Edition, Oxford (2004), pp. 159 – 168, 173 - 183, 203 - 241. © Chor EF BJT-11 BJT – Forward Active IV Characteristic  Discussion will be based on an npn BJT, which applies to a pnp BJT as well, except for the bias voltage polarities and current directions (see Appendix B).  In forward active operation, emitter-base junction is forward biased and base- collector junction is reverse biased: vBE > 0 and vBC < 0 for an npn BJT. The collector current, iC, which flows into the collector and through the reverse- biased base-collector junction is controlled by vBE - iC 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑣𝑣𝐵𝐵𝐵𝐵/𝑉𝑉𝑇𝑇. (3.1) VBC - C  The base current, iB, which flows into the base and iB B + + through the forward-biased base-emitter junction is + VCE VBE - 𝐼𝐼𝑆𝑆 𝑣𝑣 /𝑉𝑉 - E 𝑖𝑖𝐵𝐵 = 𝑒𝑒 𝐵𝐵𝐵𝐵 𝑇𝑇. (3.2) 𝛽𝛽 iE  In the above, IS is the saturation current; VT is the thermal voltage (𝑉𝑉𝑇𝑇 = 𝑘𝑘𝑘𝑘/𝑞𝑞 ≈ 0.025 V at T = 300 K); and β is the common-emitter current gain (𝛽𝛽 = 𝑖𝑖𝐶𝐶 /𝑖𝑖𝐵𝐵 ).  From equations (3.1) and (3.2) - 𝑖𝑖𝐶𝐶 = 𝛽𝛽𝑖𝑖𝐵𝐵 (3.3) © Chor EF BJT-12 BJT – Forward Active IV Characteristic  From equation (3.1): BJT works as a controlled current regulator, where the main current, iC, that flows through the reverse-biased base-collector junction does not depend on vBC, but is controlled by the voltage across a close by forward-biased base-emitter junction, vBE. This is transistor action.  Alternatively, iC is seen to be controlled by iB in equation (3.3). Typically, iC is much greater than iB, e.g., 100 times. iC  By Kirchhoff’s Current Law (KCL), the emitter current is VBC - C iB B + + 𝑖𝑖𝐸𝐸 = 𝑖𝑖𝐶𝐶 + 𝑖𝑖𝐵𝐵 (3.4) VCE + VBE -  𝑖𝑖𝐸𝐸 flows out of emitter for npn BJT. - E iE  The arrow head at the emitter in the BJT circuit symbol indicates the direction of current flow in forward active operation. © Chor EF BJT-13 BJT – Forward Active IV Characteristic (Ideal) Ideal common-emitter IV characteristics of an npn BJT - Saturation Forward active iC RC iB6 VBC - C iB5 RB iB B + + iB4 + VCE VCC VBE - - iB3 VBB E iB2 (or vBE) iE iB1 Cut-off, iC ~ 0 In forward active operation, emitter-base junction is forward biased, hence vBE ≈ 0.7 V; while base-collector junction is reverse biased, vBC < 0 for npn BJT.  From eqns. (3.1) and (3.3), iC, is a strong function of vBE (or iB).  𝑣𝑣𝐶𝐶𝐶𝐶 = 𝑣𝑣𝐶𝐶𝐵𝐵 + 𝑣𝑣𝐵𝐵𝐸𝐸 ≈ 𝑣𝑣𝐶𝐶𝐶𝐶 = −𝑣𝑣𝐵𝐵𝐶𝐶 for 𝑣𝑣𝐶𝐶𝐶𝐶 ≫ 𝑣𝑣𝐵𝐵𝐵𝐵 ≈ 0.7 V.  From eqn. (3.1), iC, is not dependent on vBC, hence not dependent on vCE in forward active operation, where 𝑣𝑣𝐶𝐶𝐶𝐶 ≫ 𝑣𝑣𝐵𝐵𝐵𝐵 ≈ 0.7 V. © Chor EF BJT-14 BJT – Forward Active IV Characteristic (Non-Ideal)  In a real (non-ideal) BJT, iC is not independent on vBC (hence vCE) in forward active operation. This is known as the Early Effect*.  The iC -vCE characteristics would then have a slight upward slope, as shown in the figure below.  When all iC -vCE characteristics are projected to the negative vCE-axis, they meet approximately at the same voltage, -VA.  VA is known as the Early voltage, and is typically large, e.g., ~100 V. 𝑣𝑣  Expression for iC -vCE is then modified to: 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑣𝑣𝐵𝐵𝐵𝐵 /𝑉𝑉𝑇𝑇 1 + 𝐶𝐶𝐶𝐶 (3.5) 𝑉𝑉𝐴𝐴 𝑣𝑣𝐶𝐶𝐶𝐶  The additional factor, , accounts for the 𝑉𝑉𝐴𝐴 dependence of iC on vCE, or Early effect. *Early Effect is also known as the base-width modulation. © Chor EF BJT-15 Bipolar Junction Transistor (BJT) Bipolar Junction Transistor (BJT) 1. Introduction 2. Modes of Operation: Forward Active, Saturation, Cut- Off and Reverse Active 3. Forward Active Current-Voltage Characteristic 4. Large-Signal Model and dc Analysis 5. Amplifier Circuit 6. Small-Signal Model (Simplified Hybrid-π) and ac Analysis 7. Other Hybrid-π Small-Signal Models Reference  Sedra and Smith, Microelectronic Circuits, Fifth Edition, Oxford (2004), pp. 159 – 168, 173 - 183, 203 - 241. © Chor EF BJT-16 BJT – Large-Signal Model (npn BJT)  For an npn BJT in forward active operation: vBE > 0 and vBC < 0. iC 𝑣𝑣𝐵𝐵𝐵𝐵 /𝑉𝑉𝑇𝑇 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 (3.1) VBC - C 𝑖𝑖𝐵𝐵 = 𝐼𝐼𝑆𝑆 𝑣𝑣 /𝑉𝑉 𝑒𝑒 𝐵𝐵𝐵𝐵 𝑇𝑇 (3.2) iB B + + 𝛽𝛽 VCE + 𝑖𝑖𝐶𝐶 = 𝛽𝛽𝑖𝑖𝐵𝐵 (3.3) VBE - - E  We can model an npn BJT using iE 𝐼𝐼𝑆𝑆 𝑣𝑣 /𝑉𝑉 𝑖𝑖𝐵𝐵 = 𝑒𝑒 𝐵𝐵𝐵𝐵 𝑇𝑇 a diode (between the emitter-base junction) 𝛽𝛽 𝑖𝑖𝐶𝐶 = 𝛽𝛽𝑖𝑖𝐵𝐵 a dependent current source C iC iB iC B C iB + B + vBE βiB VBE iE is equivalent to - - E iE E © Chor EF BJT-17 BJT – Large-Signal Model (npn BJT)  As the emitter-base junction is forward biased, vBE ≈ 0.7 V, it can be modeled by a constant voltage source (using the constant-voltage-drop model for a forward biased pn-junction). iB iC B C npn BJT + C vBE ≈ 0.7 V βiB iC - iB iE B E + VBE iE is equivalent to iB iC - B C E + vBE ≈ 0.7 V βiB - The forward biased base- emitter junction is modeled by a iE constant-voltage-drop of ~0.7 V E © Chor EF BJT-18 BJT – Large-Signal Model (pnp BJT)  Large-signal model of a pnp BJT: polarities of voltages and directions of currents are opposite to those of an npn BJT. iB iC B C pnp BJT - vEB ≈ 0.7 V βiB C + iC iE iB E B - iB iC is equivalent to VEB iE + B C E - vEB ≈ 0.7 V βiB + The forward biased emitter - iE base junction is modeled by a E constant-voltage-drop of ~0.7 V © Chor EF BJT-19 BJT – Operating Point (Bias Point): DC Analysis β= 100 IC RC = 220 Ω RB = VBC - C 10 kΩ + B+ + VCE + IB + VCC = 10 V VBB = VBE - - E - 3.7 V - IE output input circuit circuit (Assume VBE ≈ 0.7 V, as emitter- base junction is forward biased)  We want to determine the dc operating point (also known as bias point) voltages and currents of the common-emitter amplifier circuit shown above.  Note: all currents and voltages in above circuit are dc values and are denoted by capital letter symbol and capital subscript: for examples, IB for base current and VBE for base-emitter junction voltage. 𝑉𝑉𝐵𝐵𝐵𝐵 −𝑉𝑉𝐵𝐵𝐵𝐵 3.7 −0.7  Input circuit, using KVL: 𝐼𝐼𝐵𝐵 = ≈ = 0.3 mA = 300 μA 𝑅𝑅𝐵𝐵 10k  Output circuit, using KVL: 𝑉𝑉𝐶𝐶𝐶𝐶 = 𝐼𝐼𝐶𝐶 𝑅𝑅𝐶𝐶 + 𝑉𝑉𝐶𝐶𝐶𝐶 (known as the load line) © Chor EF BJT-20 BJT – Operating Point (Bias Point): DC Analysis β= 100 IB= 300 µA IC load line: 𝑉𝑉𝐶𝐶𝐶𝐶 = 𝐼𝐼𝐶𝐶 𝑅𝑅𝐶𝐶 + 𝑉𝑉𝐶𝐶𝐶𝐶 RC = RB = C 220 Ω 10 kΩ B + + VCE Q + VCC = + VBE - VBB = - E 10 V 3.7 V - - IE output input circuit circuit  IC and VCE are also governed by the IC -VCE characteristics of BJT shown above for different IB).  BJT is operating at IB = 300 µA, as determined earlier.  Operating point of BJT (point Q) is the intersection between its IC -VCE characteristic for IB = 300 µA and the output circuit load line: 𝑉𝑉𝐶𝐶𝐶𝐶 = 𝐼𝐼𝐶𝐶 𝑅𝑅𝐶𝐶 + 𝑉𝑉𝐶𝐶𝐶𝐶 or 𝐼𝐼𝐶𝐶 = 𝑉𝑉𝐶𝐶𝐶𝐶 − 𝑉𝑉𝐶𝐶𝐶𝐶 /𝑅𝑅𝐶𝐶  At the operating point, Q: IC = 30 mA and VCE = 3.4 V (determined graphically). © Chor EF BJT-21 BJT – Calculation of Operating Point Example 1 (Thevenin equivalent method) Find the currents IC, IB, and IE , and the voltage VC using the large-signal model for the BJT, assuming that it is operating in the forward active region. The value of β for the BJT is 100. 10 V 10 V RC R1 3 kΩ 600 kΩ IC C IR1 IB + B VC R2 E 300 kΩ IE - © Chor EF BJT-22 BJT – Calculation of Operating Point Example 1 (Thevenin equivalent method)  First, we obtain the Thevenin Equivalent Circuit of the base-biasing circuit, shown in the dashed grey box: 10 V 10 V RC R1 3 kΩ 600 kΩ IC C + RTHV = IR1 IB VTHV = - 200 kΩ + is equivalent B 3.33 V to VC E R2 300 kΩ IE - R2 VTHV = × 10 = 3.33 V R1 + R2 RTHV = R1 // R2 = 200 kΩ. Base biasing circuit Note: With the base biasing circuit replaced by its Thevenin’s equivalent circuit, the above circuit has the same topography as the circuit of slide BJT-20. © Chor EF BJT-23 BJT – Calculation of Operating Point Example 1 (Thevenin equivalent method)  Next, we replace the npn BJT by its large-signal model (Forward active). 10 V 10 V Thevenin equivalent of the 10 V RC base biasing circuit R1 3 kΩ RC 600 kΩ IC 3 kΩ C IR1 IB IC + C B + B 0.7 V VC RTHV = IB R2 E VTHV = + 200 kΩ 300 kΩ IE 3.33 V - - β IB IE E - Large-Signal Model for npn BJT © Chor EF BJT-24 BJT – Calculation of Operating Point Example 1 (Thevenin equivalent method) 10 V RC 3 kΩ 3.33 V – 0.7 V IB = = 0.0132 mA. IC 200 kΩ IB C + B IC = β IB = 100×0.0132 mA RTHV = 0.7 V + = 1.32 mA. VTHV = + + 3.33 V - 200 kΩ - β IB V C IE = (β +1)IB = 101×0.0132 mA VB IE = 1.33 mA. E - - VC = 10 V – 3 k × (1.32 mA) = 6.04 V. Check: Base voltage, VB = VBE = 0.7 V. Base-collector voltage VBC = VB - VC = 0.7 V - 6.04 V = - 5.34 V, Since the base-emitter pn junction is forward biased, and the base- collector pn junction is reverse biased, the npn BJT is in operating in the forward active region. © Chor EF BJT-25 BJT – Calculation of Operating Point Example 2 (Voltage divider method) The circuit below has VCC = 20 V, RC = 5 kΩ, RE = 1 kΩ, R1 = 20 kΩ, and R2 = 3 kΩ. The value of β for the BJT is 100. Assume BJT is operating in the forward active region, determine the values of IC and IB using the large-signal model for the BJT V CC RC R1 IC C IR1 IB B IR2 E R2 IE RE © Chor EF BJT-26 BJT – Calculation of Operating Point Example 2 (Voltage divider method) VCC Given : VCC = 20 V, RC = 5 kΩ, RE = 1 kΩ, R1 = 20 kΩ, and R2 = 3 kΩ. RC R1 Assuming that IB is small compared to IR1 and IR2, IC we apply the voltage division rule to obtain C IR1 IB B VCC 20 V IR2 + I R1 ≈ I R 2 ≈ = = 0.87 mA E R1 + R2 20 kΩ + 3 kΩ R2 IE VB RE R2 3 kΩ VB = VCC = 20 V = 2.61 V R1 + R2 (20 + 3) kΩ - Base biasing circuit © Chor EF BJT-27 BJT – Calculation of Operating Point Example 2 (Voltage divider method) VCC  Replace the npn BJT by its large-signal model. VE = VB – VBE = 2.61 – 0.7 = 1.91 V. RC R1 Thus, IE = VE /RE = VE /1 kΩ = 1.91 mA. IR1 IC IB IC = [β / (β + 1)]IE = [100/101] ×1.91 mA C B + = 1.89 mA. IR2 0.7 V IB = IC / β = 0.0189 mA = 18.9 µA R2 - βIB Check: IB (0.0189 mA), is indeed much smaller IE E than IR1 and IR2 (≈ 0.87 mA). Voltage divider assumption is valid. RE Note: If IB is not small compared to the Large-signal currents flowing through R1 and R2, the Model for npn BJT voltage divider rule cannot be applied. We would then need to use the Thevenin equivalent method. © Chor EF BJT-28 BJT – Calculation of Operating Point Example 2 (Voltage divider method) VCC Check that the BJT is indeed in the forward- RC active region of operation: R1 IR1 VC = VCC – IC RC = 20 – 1.89 mA × 5 kΩ IB IC = 10.55 V. C B + IR2 0.7 V VBC = VB – VC = 2.61 – 10.55 = - 7.94 V. R2 - βIB Since VBC < 0 and VBE > 0, the npn BJT is in the IE E forward active region of operation. RE Large-Signal Model for npn BJT © Chor EF BJT-29 Bipolar Junction Transistor (BJT) Bipolar Junction Transistor (BJT) 1. Introduction 2. Modes of Operation: Forward Active, Saturation, Cut- Off and Reverse Active 3. Forward Active Current-Voltage Characteristic 4. Large-Signal Model and dc Analysis 5. Amplifier Circuit 6. Small-Signal Model (Simplified Hybrid-π) and ac Analysis 7. Other Hybrid-π Small-Signal Models Reference  Sedra and Smith, Microelectronic Circuits, Fifth Edition, Oxford (2004), pp. 159 – 168, 173 - 183, 203 - 241. © Chor EF BJT-30 BJT – Amplifier Circuit  In an amplifier circuit, BJT is biased in the forward active region. In addition to dc sources, VBB and VCC, an ac (small-signal) source, vs, is applied to the input circuit, as shown in the common-emitter amplifier circuit below. vs can be considered as a small add-on to VBB.  Two types of operation: dc and ac (small-signal) operations.  The dc sources, VBB and VCC bias the BJT to operate in the forward active region and set the bias point (point Q) around which the small ac signal operates. VBB vs (t) iC Output RC t vBC - C Circuit RB B+ + iB + v - CE VCC VBB vBE - E Input Circuit iE vs © Chor EF BJT-31 BJT – Amplifier Circuit  The dc bias point (Q) currents and voltages (𝐼𝐼𝐵𝐵 , 𝐼𝐼𝐶𝐶 , 𝑉𝑉𝐶𝐶𝐶𝐶 ) can be determined using the BJT large-signal model, in the absence of the ac (small-signal) source, vs.  In the presence of ac (small-signal) source, vs, the total base current, 𝑖𝑖𝐵𝐵 , in the input circuit has an ac (small-signal) value, 𝑖𝑖𝑏𝑏 , in addition to the dc value, 𝐼𝐼𝐵𝐵 :  𝑖𝑖𝐵𝐵 = 𝐼𝐼𝐵𝐵 + 𝑖𝑖𝑏𝑏  As a result, total collector current, 𝑖𝑖𝐶𝐶 , and total collector-emitter voltage, 𝑣𝑣𝐶𝐶𝐶𝐶 , of the output circuit will be similarly changed, with the addition of a small-signal 𝑖𝑖𝑐𝑐 and 𝑣𝑣𝑐𝑐𝑒𝑒 , respectively:  𝑖𝑖𝐶𝐶 = 𝐼𝐼𝐶𝐶 + 𝑖𝑖𝑐𝑐 and 𝑣𝑣𝐶𝐶𝐸𝐸 = 𝑉𝑉𝐶𝐶𝐸𝐸 + 𝑣𝑣𝑐𝑐𝑒𝑒 VBB vs (t) iC Output RC t vBC - C Circuit RB B+ + iB + v - CE VCC VBB vBE - E Input Circuit iE vs © Chor EF BJT-32 BJT – Amplifier Circuit Notations for dc and ac values (examples) dc current: IC Capital symbol Capital subscript ac current: ic Lower case symbol Lower case subscript Total current: 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝐶𝐶 + 𝑖𝑖 c Lower case symbol Capital subscript VBB vs (t) iC Output RC t vBC - C Circuit RB B+ + iB + v - CE VCC VBB vBE - E Input Circuit iE vs © Chor EF BJT-33 BJT – Amplifier Circuit The common-emitter amplifier circuit below will demonstrate signal amplification, graphically with the help of the IV-characteristics.  Without the ac (small-signal) source, i.e., vs = 0 -  No ac (small-signal) values: ib = 0, ic = 0, and vce = 0.  iB = IB = 300 μA, iC = IC = 30 mA, and vCE = VCE = 3.4 V (see slides BJT 19 & BJT-20 for dc operation analysis). These are dc biasing currents and voltages.  Point Q, the dc bias point. β = 100 iC RC = 220 Ω RB = vBC - C 10 kΩ + B+ vCE iB + - vBE - E VCC = VBB = 10 V 3.7 V iE vs (Assume VBE = 0.7 V) © Chor EF BJT-34 BJT – Amplifier Circuit With ac source vs -  When ib = 100 μA, iB = IB + ib= 300 μA +100 μA = 400 μA.  Point A: iC = IC + ic = 40 mA, and vCE = VCE + vce = 1.2 V.  When ib = -100 μA, iB = IB + ib= 300 μA -100 μA = 200 μA.  Point B: iC = IC + ic = 20 mA, and vCE = VCE + vce = 5.6 V.  Small-signal ib = 100 μA leads to small-signal ic = 10 mA, which is 100 (β) times bigger. This is signal amplification. β = 100 iC RC = 220 Ω RB = vBC - C 10 kΩ + B+ vCE iB + - vBE - E VCC = VBB = 10 V 3.7 V iE vs (Assume VBE = 0.7 V) © Chor EF BJT-35 BJT – Amplifier Circuit (The Principles)  Amplifier circuit is a 2-port dc voltage source network with the dc voltage source that supplies power to output the entire circuit, while the ac (amplified) signal is applied to the input ac signal port. Amplifier circuit  The transistor in the amplifier circuit acts as a control device, input ac signal which controls the flow of power from the dc source, to produce an enhanced (amplified) ac signal at the output port.  The output ac signal is of the same waveform as the input ac input port output port signal, but is of larger amplitude (in current, voltage, or power). © Chor EF BJT-36 BJT – Amplifier Circuit (Inappropriate Bias Point) With RB reduced to 7.5 kΩ -  dc base current, IB = (VBB - VBE ) / RB = (3.7 – 0.7)/7.5k = 400 μA. Bias point Q.  With ac source vs - when ib = 100 μA, iB = IB + ib= 400 μA +100 μA = 500 μA.  Intersection between load line and BJT characteristics moves to point A, where the BJT no longer operates in the forward active region.  The ac signal waveforms of ic and vce are distorted. β = 100 iC RC = 220 Ω RB = vBC - C 7.5 kΩ + B+ vCE iB + - vBE - E VCC = VBB = 10 V 3.7 V iE vs (Assume VBE = 0.7 V) © Chor EF BJT-37 BJT – Amplifier Circuit (Inappropriate Bias Point) With RB increased to 30 kΩ -  dc base current, IB = (VBB - VBE ) / RB = (3.7 – 0.7)/30k = 100 μA. Bias point Q.  With ac source vs, assuming ib = 200 μA. At some point during the negative half- cycle of ib the intersection between load line and BJT characteristics is at point B.  ac signal ic is clipped at 0 A, as total iC cannot be negative.  ac signal vce is clipped at 10 V, as total vCE cannot exceed the power supply voltage. β = 100 iC RC = 220 Ω RB = vBC - C 30 kΩ + B+ vCE iB + - vBE - E VCC = VBB = 10 V 3.7 V iE vs (Assume VBE = 0.7 V) © Chor EF BJT-38 Bipolar Junction Transistor (BJT) Bipolar Junction Transistor (BJT) 1. Introduction 2. Modes of Operation: Forward Active, Saturation, Cut- Off and Reverse Active 3. Forward Active Current-Voltage Characteristic 4. Large-Signal Model and dc Analysis 5. Amplifier Circuit 6. Small-Signal Model (Simplified Hybrid-π) and ac Analysis 7. Other Hybrid-π Small-Signal Models Reference  Sedra and Smith, Microelectronic Circuits, Fifth Edition, Oxford (2004), pp. 159 – 168, 173 - 183, 203 - 241. © Chor EF BJT-39 BJT – Small-Signal Model To develop a small-signal model of BJT operating in the forward active region.  We seek linear relationships* among the small-signal ac components of the base current, collector current, base-emitter voltage and collector-emitter voltage - ib, ic, vbe, and vce (at a bias point). npn BJT C is equivalent to B E ib ic B + ??? + C pnp BJT vce vbe C - - E is equivalent to B E * In the development of the small-signal model of a pn-junction diode, we obtain an approximate linear relationship between its small-signal current, id, and its small-signal voltage, vd, at a bias point (see slides pn-30 to pn-35, in particular, equations (2.12) and (2.14)). © Chor EF BJT-40 BJT – Small-Signal Model Parameter - gm  From equation (3.1), collector current - 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑣𝑣𝐵𝐵𝐵𝐵/𝑉𝑉𝑇𝑇  At the dc bias point (point Q), dc collector current - 𝐼𝐼𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑉𝑉𝐵𝐵𝐵𝐵 /𝑉𝑉𝑇𝑇  A linear relationship between a small iC Gradient at Q = 𝜕𝜕𝑖𝑖𝐶𝐶 change in the collector current, ∆iC, | = 𝜕𝜕𝑣𝑣𝐵𝐵𝐵𝐵 𝑉𝑉𝐵𝐵𝐵𝐵 𝑔𝑔𝑚𝑚 and a small change in the base-emitter Bias point Q(IC,VBE) voltage, ∆vBE, can be found by IC ∆iC = ic linearizing the iC-vBE characteristic.  A small change in iC w.r.t a small VBE change in vBE, can be approximated 0 vBE by the derivative of iC w.r.t vBE – ∆vBE = vbe ∆𝑖𝑖𝐶𝐶 𝑖𝑖𝑐𝑐 𝜕𝜕𝑖𝑖𝐶𝐶 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑉𝑉𝐵𝐵𝐵𝐵 /𝑉𝑉𝑇𝑇 𝐼𝐼𝐶𝐶 | = | ≈ | = = = 𝑔𝑔𝑚𝑚 (3.6) ∆𝑣𝑣𝐵𝐵𝐵𝐵 𝑉𝑉𝐵𝐵𝐵𝐵 𝑣𝑣𝑏𝑏𝑏𝑏 𝑉𝑉𝐵𝐵𝐵𝐵 𝜕𝜕𝑣𝑣𝐵𝐵𝐵𝐵 𝑉𝑉𝐵𝐵𝐵𝐵 𝑉𝑉𝑇𝑇 𝑉𝑉𝑇𝑇 𝑖𝑖𝑐𝑐 𝐼𝐼𝐶𝐶  Transconductance : 𝑔𝑔𝑚𝑚 = = (3.7) 𝑣𝑣𝑏𝑏𝑏𝑏 𝑉𝑉𝑇𝑇  Transconductance, gm, models a small change in the collector current, ic, caused by a small change in base-emitter voltage, vbe. © Chor EF BJT-41 BJT – Small-Signal Model Parameter - rπ 𝐼𝐼𝑆𝑆 𝑣𝑣 /𝑉𝑉  From equation (3.2), base current - 𝑖𝑖𝐵𝐵 = 𝑒𝑒 𝐵𝐵𝐵𝐵 𝑇𝑇 𝛽𝛽 𝐼𝐼𝑆𝑆 𝑉𝑉 /𝑉𝑉  At the dc bias point (point Q), dc base current - 𝐼𝐼𝐵𝐵 = 𝑒𝑒 𝐵𝐵𝐵𝐵 𝑇𝑇 𝛽𝛽  A small change in iB w.r.t a small change in vBE, can be approximated by the derivative of iB w.r.t vBE – ∆𝑖𝑖𝐵𝐵 𝑖𝑖𝑏𝑏 𝜕𝜕𝑖𝑖𝐵𝐵 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑉𝑉𝐵𝐵𝐵𝐵 /𝑉𝑉𝑇𝑇 𝐼𝐼𝐶𝐶 | = | ≈ | = = = 𝑔𝑔𝜋𝜋 (3.8) ∆𝑣𝑣𝐵𝐵𝐵𝐵 𝑉𝑉𝐵𝐵𝐵𝐵 𝑣𝑣𝑏𝑏𝑏𝑏 𝑉𝑉𝐵𝐵𝐵𝐵 𝜕𝜕𝑣𝑣𝐵𝐵𝐵𝐵 𝑉𝑉𝐵𝐵𝐵𝐵 𝛽𝛽𝑉𝑉𝑇𝑇 𝛽𝛽𝑉𝑉𝑇𝑇  Taking the reciprocal of 𝑔𝑔𝜋𝜋 , input resistance: Gradient at Q iB 𝜕𝜕𝑖𝑖𝐵𝐵 1 𝑣𝑣𝑏𝑏𝑏𝑏 𝛽𝛽𝑉𝑉𝑇𝑇 𝛽𝛽 𝑣𝑣𝑏𝑏𝑏𝑏 = | 𝑟𝑟𝜋𝜋 = = = = = (3.9) 𝜕𝜕𝑣𝑣𝐵𝐵𝐵𝐵 𝑉𝑉𝐵𝐵𝐵𝐵 𝑔𝑔𝜋𝜋 𝑖𝑖𝑏𝑏 𝐼𝐼𝐶𝐶 𝑔𝑔𝑚𝑚 𝑖𝑖𝑏𝑏 1 Bias point Q(IB,VBE) = 𝑔𝑔𝜋𝜋 = 𝑟𝑟𝜋𝜋  Hence, 𝛽𝛽 = 𝑔𝑔𝑚𝑚 𝑟𝑟𝜋𝜋 (3.10) IB ∆iB = ib  Input resistance, rπ, models the relationship between a small change in the base VBE current, ib, and a small change in base- 0 vBE ∆vBE = vbe emitter voltage, vbe. © Chor EF BJT-42 BJT – Small-Signal Model Parameters (Summary)  Equation (3.7) - Transconductance, gm, models a small change in the collector current, ic, caused by a small change in base-emitter voltage, vbe – 𝐼𝐼𝐶𝐶 𝑖𝑖𝑐𝑐 = 𝑔𝑔𝑚𝑚 𝑣𝑣𝑏𝑏𝑏𝑏 [ 𝑔𝑔𝑚𝑚 = ] (3.7) 𝑉𝑉𝑇𝑇 Small change in Small change in base- collector current emitter voltage  Equation (3.9) - Input resistance, rπ, models the relationship between a small change in the base current, ib, and a small change in base-emitter voltage, vbe – 𝛽𝛽𝑉𝑉𝑇𝑇 𝑣𝑣𝑏𝑏𝑏𝑏 = 𝑟𝑟𝜋𝜋 𝑖𝑖𝑏𝑏 [𝑟𝑟𝜋𝜋 = 𝐼𝐼𝐶𝐶 ] (3.9) Small change in Small change in base-emitter voltage base current  Equation (3.10) - 𝛽𝛽 = 𝑔𝑔𝑚𝑚 𝑟𝑟𝜋𝜋 (3.10)  Combine equations (3.7), (3.9) and (3.10) - 𝑖𝑖𝑐𝑐 = 𝑔𝑔𝑚𝑚 𝑣𝑣𝑏𝑏𝑏𝑏 = 𝑔𝑔𝑚𝑚 𝑟𝑟𝜋𝜋 𝑖𝑖𝑏𝑏 = 𝛽𝛽𝑖𝑖𝑏𝑏 (3.11) © Chor EF BJT-43 BJT – Small-Signal Model (Simplified Hybrid-π)  Small-signal model of BJT is developed using equations (3.7), (3.9), (3.11) - Eqn. (3.9): 𝑣𝑣𝑏𝑏𝑏𝑏 = 𝑟𝑟𝜋𝜋 𝑖𝑖𝑏𝑏 - 𝑣𝑣𝑏𝑏𝑏𝑏 is linearly related to 𝑖𝑖𝑏𝑏 - base-emitter junction is modeled by a resistor, 𝑟𝑟𝜋𝜋 BJT Voltage-Controlled Version Current-Controlled Version iC C B ib ic C B ib ic C iB + + rπ vbe rπ vbe B - gmvbe - β ib E iE ie ie E E Eqn. (3.7): 𝑖𝑖𝑐𝑐 = 𝑔𝑔𝑚𝑚 𝑣𝑣𝑏𝑏𝑏𝑏 - Eqn. (3.11): 𝑖𝑖𝑐𝑐 = βib – ic is dependent on vbe – ic is dependent on ib - modeled by a voltage- modeled by a current- dependent current source. dependent current source. © Chor EF BJT-44 BJT – Forward Active Mode Models (Summary) Npn BJT iC C Voltage-Controlled Version iB Replace B C ib + ic B BJT by E hybrid-π vbe iE Model for rπ - gmvbe Replace BJT by large -signal small-signal ie model for dc analysis E analysis Current-Controlled Version B C IB + IC or B C VBE ib + ic - β IB vbe rπ - β ib IE E ie E Large-Signal Model of the npn BJT Small-Signal model (Simplified hybrid-π) of the npn BJT © Chor EF BJT-45 BJT – Amplifier Circuit Operation & Analysis  The operation of an amplifier circuit is shown in the 𝑖𝑖𝐶𝐶 -𝑣𝑣𝐶𝐶𝐶𝐶 curve below, where the dc bias point (Q) and the small signals 𝑖𝑖𝑏𝑏 , 𝑖𝑖𝑐𝑐 and 𝑣𝑣𝑐𝑐𝑐𝑐 are indicated.  In the presence of an ac (small-signal) source, vs, -  𝑖𝑖𝐵𝐵 = 𝐼𝐼𝐵𝐵 + 𝑖𝑖𝑏𝑏 , 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝐶𝐶 + 𝑖𝑖𝑐𝑐 , and 𝑣𝑣𝐶𝐶𝐶𝐶 = 𝑉𝑉𝐶𝐶𝐶𝐶 + 𝑣𝑣𝑐𝑐𝑐𝑐 𝑖𝑖𝐵𝐵 , 𝑖𝑖𝐶𝐶 and 𝑣𝑣𝐶𝐶𝐶𝐶 all have a small-signal component and they vary with time. A common-emitter VCC amplifier RC IC + ic C + RB B VCE + vce VBB IB + ib - E IE + ie vs © Chor EF BJT-46 BJT – Amplifier Circuit Operation (dc Analysis)  The dc bias point of the amplifier can be determined using the large-signal equivalent circuit, in the absence of the ac (small signal) source, vs. A common-emitter VCC Large-Signal Equivalent amplifier Circuit of Amplifier RC VCC IC + ic C RC + RB B IC VCE + vce RB IB B C VBB IB + ib + - VBB VBE βIB E IE + ie - vs IE E S.C Replace BJT by large-signal model for dc analysis © Chor EF BJT-47 BJT – Amplifier Circuit Operation (ac Analysis)  The small-signal ac analysis of the amplifier is carried out using the small- signal equivalent circuit, and dc voltage sources need to be ac short. Small-Signal Equivalent Circuit of Amplifier A common-emitter VCC amplifier dc source is replaced RC by short-circuit (ac- short) IC + ic RC C + RB B RB ib B ic VCE + vce C IB + ib + VBB - vbe β ib or E - gmvbe IE + ie vs vs ie E dc source is replaced by Replace BJT by small-signal Need to ac short dc short-circuit (ac-short) hybrid-π model for ac analysis voltage source. © Chor EF BJT-48 BJT – Small-Signal Model (pnp) npn BJT pnp BJT The change in collector current IC+ic C for the npn BJT is positive when there is an increase in the IC+ic collector current flowing into the C IB+ib collector IB+ib ic = gmvbe= βib The change in collector B B current for the pnp BJT is E E positive when there is an increase in the collector Replace the BJT by current flowing out of the IE+ie the simplified hybrid-π IE+ie collector model for small-signal analysis ic = gmveb= βib Simplified Hybrid-π Model for npn BJT or pnp BJT B ib ic C + vbe rπ gmvbe This Hybrid-π Model is in its - or βib simplest form here. We will use this simplest form by default. ie E © Chor EF BJT-49 BJT – Small-Signal Model (pnp vs npn) pnp BJT IC+ic C B ib ic C IB+ib - veb rπ gmveb B E + or βib IE+ie ie E B ib ic C B ib ic C + + vbe rπ gmvbe vbe rπ -gmvbe - or βib - or -βib There is no difference between ie the small-signal models of npn ie E E BJT and pnp BJT. © Chor EF BJT-50 Bipolar Junction Transistor (BJT) Bipolar Junction Transistor (BJT) 1. Introduction 2. Modes of Operation: Forward Active, Saturation, Cut- Off and Reverse Active 3. Forward Active Current-Voltage Characteristic 4. Large-Signal Model and dc Analysis 5. Amplifier Circuit 6. Small-Signal Model (Simplified Hybrid-π) and ac Analysis 7. Other Hybrid-π Small-Signal Models Reference  Sedra and Smith, Microelectronic Circuits, Fifth Edition, Oxford (2004), pp. 159 – 168, 173 - 183, 203 - 241. © Chor EF BJT-51 BJT – Small-Signal Model (Simplified Hybrid-π)  In simplified hybrid-π model, Early effect was ignored and collector current, 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑣𝑣𝐵𝐵𝐵𝐵/𝑉𝑉𝑇𝑇 , (3.1) is a function of only the base-emitter junction voltage, 𝑣𝑣𝐵𝐵𝐸𝐸 : 𝑖𝑖𝐶𝐶 = 𝑓𝑓 𝑣𝑣𝐵𝐵𝐵𝐵.  Small-signal collector current, 𝑖𝑖𝑐𝑐 , is directly due to small-signal base-emitter junction voltage, 𝑣𝑣𝑏𝑏𝑏𝑏 : 𝜕𝜕𝑖𝑖𝐶𝐶 𝑖𝑖𝑐𝑐 = | × 𝑣𝑣𝑏𝑏𝑏𝑏 = 𝑔𝑔𝑚𝑚 × 𝑣𝑣𝑏𝑏𝑏𝑏 (see slide BJT-40, equation (3.6)) 𝜕𝜕𝑣𝑣𝐵𝐵𝐵𝐵 𝑉𝑉𝐵𝐵𝐵𝐵 Saturation Forward active Simplified Hybrid-π model B C ib + ic vbe rπ - gmvbe ie or βib (or vBE) E © Chor EF BJT-52 BJT – Small-Signal Model (Hybrid-π)  With Early effect (see slide BJT-14), collector current becomes 𝑣𝑣𝐶𝐶𝐶𝐶 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑣𝑣𝐵𝐵𝐵𝐵/𝑉𝑉𝑇𝑇 1 + , (3.5) 𝑉𝑉𝐴𝐴 is also a function of collector-emitter voltage, 𝑣𝑣𝐶𝐶𝐶𝐶 , in addition to 𝑣𝑣𝐵𝐵𝐵𝐵 : 𝑖𝑖𝐶𝐶 = 𝑓𝑓 𝑣𝑣𝐵𝐵𝐵𝐵 , 𝑣𝑣𝐶𝐶𝐶𝐶.  Small-signal collector current, 𝑖𝑖𝑐𝑐 , will then be contributed by the small-signal collector-emitter voltage, 𝑣𝑣𝑐𝑐𝑐𝑐 , as well, in addition to that contributed by 𝑣𝑣𝑏𝑏𝑒𝑒 - 𝜕𝜕𝑖𝑖𝐶𝐶 𝜕𝜕𝑖𝑖𝐶𝐶 ∆𝑖𝑖𝑐𝑐 = 𝑖𝑖𝑐𝑐 ≈ | × 𝑣𝑣𝑏𝑏𝑏𝑏 + | × 𝑣𝑣𝑐𝑐𝑐𝑐 (3.12)* 𝜕𝜕𝑣𝑣𝐵𝐵𝐵𝐵 𝑉𝑉𝐵𝐵𝐵𝐵 𝜕𝜕𝑣𝑣𝐶𝐶𝐶𝐶 𝑉𝑉𝐶𝐶𝐶𝐶 Contribution of 𝑣𝑣𝑏𝑏𝑒𝑒 Contribution of 𝑣𝑣𝑐𝑐𝑒𝑒 * Mathematical proof of equation IC Q (3.12) is given in Appendix C VCE © Chor EF BJT-53 BJT – Small-Signal Model (Hybrid-π) 𝑣𝑣𝐶𝐶𝐶𝐶  Equation (3.5) - 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑣𝑣𝐵𝐵𝐵𝐵/𝑉𝑉𝑇𝑇 1 + 𝑉𝑉𝐴𝐴  For given VBE and VCE (point Q), when vCE changes, iC will change. Change in iC owing to a small change in vCE is related to the slope (or derivative) of the corresponding iC-vCE curve as follows – ∆𝑖𝑖𝐶𝐶 𝑖𝑖𝑐𝑐 𝜕𝜕𝑖𝑖𝐶𝐶 𝐼𝐼𝑆𝑆 𝑉𝑉 /𝑉𝑉 𝐼𝐼𝐶𝐶 1 = = | = 𝑒𝑒 𝐵𝐵𝐵𝐵 𝑇𝑇 ≈ = (3.13) ∆𝑣𝑣𝐶𝐶𝐶𝐶 𝑣𝑣𝑐𝑐𝑐𝑐 𝜕𝜕𝑣𝑣𝐶𝐶𝐶𝐶 𝑉𝑉𝐶𝐶𝐶𝐶 , 𝑉𝑉𝐵𝐵𝐸𝐸 𝑉𝑉𝐴𝐴 𝑉𝑉𝐴𝐴 𝑟𝑟𝑜𝑜 𝑣𝑣𝑐𝑐𝑐𝑐 𝑉𝑉𝐴𝐴  Output resistance: 𝑟𝑟𝑜𝑜 = = (3.14) 𝑖𝑖𝑐𝑐 𝐼𝐼𝐶𝐶  Output resistance, 𝑟𝑟𝑜𝑜 , models a small change in the collector current, 𝑖𝑖𝑐𝑐 , caused by a small change in collector- Q IC emitter voltage, 𝑣𝑣𝑐𝑐𝑐𝑐. VCE © Chor EF BJT-54 BJT – Small-Signal Model (Hybrid-π)  From equation (3.12) – 𝜕𝜕𝑖𝑖𝐶𝐶 𝜕𝜕𝑖𝑖𝐶𝐶 𝑖𝑖𝑐𝑐 = | × 𝑣𝑣𝑏𝑏𝑏𝑏 + | × 𝑣𝑣𝑐𝑐𝑐𝑐 𝜕𝜕𝑣𝑣𝐵𝐵𝐵𝐵 𝑉𝑉𝐵𝐵𝐵𝐵 𝜕𝜕𝑣𝑣𝐶𝐶𝐶𝐶 𝑉𝑉𝐶𝐶𝐶𝐶 or, 𝑖𝑖𝑐𝑐 = 𝑔𝑔𝑚𝑚 𝑣𝑣𝑏𝑏𝑏𝑏 + 𝑣𝑣𝑐𝑐𝑐𝑐 /𝑟𝑟𝑜𝑜 = 𝛽𝛽𝑖𝑖𝑏𝑏 + 𝑣𝑣𝑐𝑐𝑐𝑐 /𝑟𝑟𝑜𝑜 (3.15)  The term 𝑣𝑣𝑐𝑐𝑐𝑐 /𝑟𝑟𝑜𝑜 accounts for the dependence of iC on vCE (or Early effect).  To model the increase in iC owing to an increase in vCE, a resistance ro is included between nodes C and E in the hybrid-π model. iC C B ib ic’ = gmvbe (or βib ) ic C iB + ro + rπ vbe vce B - βib or E Replace BJT by this gmvbe - ic” = vce/ro Hybrid-π Model for Small-Signal Analysis iE ie E Note: In EE2027, you can assume that ro is infinite by default. However, if the Early voltage VA is given, then ro should be calculated and included in the small-signal analysis. © Chor EF BJT-55 BJT – Full Hybrid-π Model at High Frequency  BJTs have capacitances associated with base-emitter and base-collector pn- junctions, which have charge storage.  At high frequencies, two parasitic capacitors have to be included: Cπ and Cµ, respectively across the base-emitter and base-collector junctions. Cµ ic B ib C + ro + Cπ rπ vbe vce - β ib or gmvbe - ie E Note: We introduce the existence of these capacitances Cµ and Cπ here, and their inclusion in the Hybrid-π Model is only necessary when high frequency response is discussed. We will ignore these capacitances (in the pF range) for low frequency signals. © Chor EF BJT-56 BJT - Topics Discussed  Structure: 3 regions/terminals - emitter, base and collector  Operation regions: Forward active, saturation, cut-off  𝒊𝒊𝑪𝑪 -𝒗𝒗𝑪𝑪𝑪𝑪 curves  Forward active IV characteristics (npn BJT): 𝐼𝐼𝑆𝑆 𝑣𝑣 /𝑉𝑉 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑣𝑣𝐵𝐵𝐵𝐵/𝑉𝑉𝑇𝑇 , 𝑖𝑖𝐵𝐵 = 𝑒𝑒 𝐵𝐵𝐵𝐵 𝑇𝑇 , 𝑖𝑖𝐶𝐶 = 𝛽𝛽𝑖𝑖𝐵𝐵 𝛽𝛽 𝑣𝑣𝐶𝐶𝐶𝐶  Forward active iC with Early Effect (npn BJT): 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑣𝑣𝐵𝐵𝐵𝐵/𝑉𝑉𝑇𝑇 1 + 𝑉𝑉𝐴𝐴  npn vs pnp BJT: same functions, opposite bias voltage polarities and current directions  Forward active large-signal model & dc analysis Thevenin equivalent vs voltage divider method for base biasing circuit 𝐼𝐼𝐶𝐶 𝛽𝛽𝑉𝑉𝑇𝑇 𝑉𝑉𝐴𝐴  Forward active small-signal model: 𝑔𝑔𝑚𝑚 = 𝑉𝑉𝑇𝑇 , 𝑟𝑟𝜋𝜋 = 𝐼𝐼𝐶𝐶 , 𝑟𝑟𝑜𝑜 = 𝐼𝐼𝐶𝐶 © Chor EF BJT-57 APPENDICES © Chor EF BJT-58 APPENDIX A – Operation of pnp BJT iC VCB + C iB B - - - VEC VEB + + E Modes of operation of the pnp bipolar junction transistor iE Mode of Emitter-Base Collector-Base Applications Operation Junction Junction Cut-off Reverse biased Reverse biased Logic- (VEB < 0 for pnp) (VCB < 0 for pnp) OFF State Forward Active Forward biased Reverse biased Amplifier (VEB > 0 for pnp) (VCB < 0 for pnp) Saturation Forward biased Forward biased Logic – (VEB > 0 for pnp) (VCB >0 for pnp) ON State Reverse Active Reverse Biased Forward Biased Not used (VEB < 0 for pnp) (VCB > 0 for pnp) * For the same mode of operation, the current directions and voltage polarities of pnp BJT are exactly opposite from those of npn BJT. © Chor EF BJT-59 APPENDIX B – IV Characteristic of pnp BJT  For pnp BJT in forward active operation – iC  Emitter-base junction is forward biased: vEB > 0. VCB + C iB B - -  Collector-base junction is reverse biased: vCB < 0. - VEC VEB +  Collector current, iC, is controlled by vEB – + E iE 𝑣𝑣𝐸𝐸𝐵𝐵 /𝑉𝑉𝑇𝑇 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 (flows out of the collector), (B.1)  Base current, iB, is given by 𝐼𝐼𝑆𝑆 𝑣𝑣 /𝑉𝑉 𝑖𝑖𝐵𝐵 = 𝑒𝑒 𝐸𝐸𝐵𝐵 𝑇𝑇 (flows out of the base), (B.2) 𝛽𝛽 where IS is the saturation current, VT is the thermal voltage and 𝑉𝑉𝑇𝑇 = 𝑘𝑘𝑘𝑘/𝑞𝑞 ≈ 0.025 at T = 300 K, and β is the common-emitter current and 𝛽𝛽 = 𝑖𝑖𝐶𝐶 /𝑖𝑖𝐵𝐵.  From equations (B.1) and (B.2): 𝑖𝑖𝐶𝐶 = 𝛽𝛽𝑖𝑖𝐵𝐵 (B.3)  Emitter current: 𝑖𝑖𝐸𝐸 = 𝑖𝑖𝐶𝐶 + 𝑖𝑖𝐵𝐵 (flows into the emitter). (B.4) © Chor EF BJT-60 APPENDIX C – Non-Ideal 𝒊𝒊𝑪𝑪 -Expression  With Early effect (see slide BJT-14), collector current in forward active region is given by 𝑣𝑣𝐶𝐶𝐶𝐶 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝑆𝑆 𝑒𝑒 𝑣𝑣𝐵𝐵𝐵𝐵/𝑉𝑉𝑇𝑇 1 + = 𝑓𝑓 𝑣𝑣𝐵𝐵𝐵𝐵 , 𝑣𝑣𝐶𝐶𝐸𝐸 (C.1) 𝑉𝑉𝐴𝐴  We can express the total (instantaneous) currents and voltages in terms of the dc (operating point) and ac (signal) currents and voltages, that is  𝑖𝑖𝐶𝐶 = 𝐼𝐼𝐶𝐶 + 𝑖𝑖𝑐𝑐  𝑣𝑣𝐵𝐵𝐸𝐸 = 𝑉𝑉𝐵𝐵𝐸𝐸 + 𝑣𝑣𝑏𝑏𝑒𝑒  𝑣𝑣𝐶𝐶𝐶𝐶 = 𝑉𝑉𝐶𝐶𝐶𝐶 + 𝑣𝑣𝑐𝑐𝑐𝑐  Hence, 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝐶𝐶 + 𝑖𝑖𝑐𝑐 = 𝑓𝑓 𝑣𝑣𝐵𝐵𝐵𝐵 , 𝑣𝑣𝐶𝐶𝐶𝐶 = 𝑓𝑓 𝑉𝑉𝐵𝐵𝐵𝐵 + 𝑣𝑣𝑏𝑏𝑏𝑏 , 𝑉𝑉𝐶𝐶𝐶𝐶 + 𝑣𝑣𝑐𝑐𝑐𝑐 (C.2)  Expand equation (C.2) as a Taylor series and neglecting higher order terms for small 𝑣𝑣𝑏𝑏𝑏𝑏 and 𝑣𝑣𝑐𝑐𝑐𝑐 - 𝑖𝑖𝐶𝐶 = 𝐼𝐼𝐶𝐶 + 𝑖𝑖𝑐𝑐 = 𝑓𝑓 𝑉𝑉𝐵𝐵𝐵𝐵 + 𝑣𝑣𝑏𝑏𝑏𝑏 , 𝑉𝑉𝐶𝐶𝐶𝐶 + 𝑣𝑣𝑐𝑐𝑐𝑐 𝜕𝜕𝑖𝑖𝐶𝐶 𝜕𝜕𝑖𝑖𝐶𝐶 ≈ 𝑓𝑓 𝑉𝑉𝐵𝐵𝐵𝐵 , 𝑉𝑉𝐶𝐶𝐶𝐶 + | × 𝑣𝑣𝑏𝑏𝑏𝑏 + | × 𝑣𝑣𝑐𝑐𝑐𝑐 𝜕𝜕𝑣𝑣𝐵𝐵𝐵𝐵 𝑉𝑉𝐵𝐵𝐵𝐵 𝜕𝜕𝑣𝑣𝐶𝐶𝐶𝐶 𝑉𝑉𝐶𝐶𝐶𝐶  As 𝐼𝐼𝐶𝐶 = 𝑓𝑓 𝑉𝑉𝐵𝐵𝐵𝐵 , 𝑉𝑉𝐶𝐶𝐶𝐶 , small signal collector current 𝜕𝜕𝑖𝑖𝐶𝐶 𝜕𝜕𝑖𝑖𝐶𝐶 𝑖𝑖𝑐𝑐 ≈ | × 𝑣𝑣𝑏𝑏𝑏𝑏 + | × 𝑣𝑣𝑐𝑐𝑐𝑐. (C.3) 𝜕𝜕𝑣𝑣𝐵𝐵𝐵𝐵 𝑉𝑉𝐵𝐵𝐵𝐵 𝜕𝜕𝑣𝑣𝐶𝐶𝐶𝐶 𝑉𝑉𝐶𝐶𝐶𝐶