B39SE Semiconductor Electronics AC analysis- BJTs Week 9 Outline BJT transistor modelling The transistor model CE fixed bias configuration CE emitter bias configuration Frequency response BJT transistor modelling AC analysis To perform the AC analysis of a circuit, small signal analysis is used where an equivalent circuits is employed. Because we are interested only in the ac response of the circuit, all dc supplies can be replaced by a zero-potential equivalent (short circuit). The coupling capacitors are usually chosen to be very small at the frequency of application and can be replaced by a short circuit. Summary for obtaining ac equivalent network Setting all dc sources to zero and replacing them by a short-circuit equivalent Replacing all capacitors by a short-circuit equivalent Removing all elements bypassed by the short circuit equivalent introduced by steps 1 and 2 Redrawing the network in a more convenient and logical form Example Fig. 3.1 Transistor circuit under study Fig. 3.2 New network following removal of the dc supply and insertion of the short circuit equivalent capacitors Redrawn for small signal ac analysis The transistor model The transistor model A model is a combination of circuit elements, properly chosen, that best approximates the actual behavior of semiconductor device under specific operating conditions. CE configuration Fig. 3.3 Finding the input equivalent circuit for a BJT transistor The equivalent circuit for CE configuration will be constructed using the device characteristics and a number of approx. Starting with the input side, we find the applied voltage Vi is equal to the voltage Vbe with the input current being the base current IB. CE configuration The current through the forward biased junction of the transistor is IE, the characteristics for the input side appear in Fig. 3.4. for various levels of VCB. Taking the average value for the curves will result in a single curve 3.4b, which is simply that of a forward biased diode. Fig. 3.4 Defining the average curve for the characteristics of (a) CE configuration For the equivalent circuit, the input side is simply a single diode with a current Ie, as shown in Fig. 3.5 using the output characteristics. Fig. 3.5 Equivalent circuit for the input side of a BJT transistor CE configuration If we redraw the collector characteristics to have a constant as shown in Fig. 2.6 (another pproximation), the entire characteristics at the output section can be replaced by a controlled source whose magnitude is beta times the base current. Fig. 3.6 Constant characteristics CE configuration Because all the input and output parameters of the original configuration are now present, the equivalent network for the CE configuration is shown in Fig. 3.7. Fig. 3.7 BJT equivalent circuit CE configuration The equivalent model of Fig. 3.7 can be awkward to work with due to the direct connection between the input and output networks. It can be improved by first replacing the diode by its equivalent resistance as determined by the level IE, as shown in Fig. 3.8. Diode resistance is determined by rD = 26mV / ID. * Using the subscript e because the determining current is the emitter current will result in re = 26 mV / IE *Vt=kT/q at Room Temp Vt=26mV CE configuration Now, for the input side: 𝑉𝑖 𝑉𝑏𝑒 𝑍𝑖 = = 𝐼𝑏 𝐼𝑏 Solving for Vbe 𝑉𝑏𝑒 = 𝐼𝑒 𝑟𝑒 = 𝐼𝑐 + 𝐼𝑏 𝑟𝑒 = 𝛽𝐼𝑏 + 𝐼𝑏 𝑟𝑒 = (𝛽 + 1)𝐼𝑏 𝑟𝑒 𝑉𝑏𝑒 (𝛽 + 1)𝐼𝑏 𝑟𝑒 𝑍𝑖 = = 𝐼𝑏 𝐼𝑏 𝑍𝑖 = (𝛽 + 1)𝑟𝑒 ≅ 𝛽𝑟𝑒 Eq. 3.1 CE configuration The result is that the impedance seen “looking into” the base of the network is a resistor equal to beta times the value of re as shown in Fig. 3.8. The collector output current is still linked to the input current by beta. Fig. 3.8 Improved BJT equivalent circuit Early voltage We have now a good representation for the input circuit, but aside from the collector output current being defined by the level of beta and IB, we do not have a good representation for the output impedance of the device. In reality the characteristics do not have the ideal appearance of Fig. 3.6. Early voltage Rather, they have a slope as shown in Fig. 3.9 That defines the output impedance of the device. If the slope of the curves is extended until they reach the horizontal axis, they will intersect at a voltage called the Early voltage. Fig. 3.9 Defining the Early voltage and the output impedance of a transistor Early voltage For a particular collector and base current, the output impedance can be found using the following equation: 𝑟𝑜 = ∆𝑉 ∆𝐼 = 𝑉𝐴 +𝑉𝐶𝐸𝑄 𝐼𝐶𝑄 Eq. 3.2 Typically the Early voltage is sufficiently large compared with the applied collector to emitter voltage to permit the following approx. 𝑟𝑜 ≅ 𝑉𝐴 𝐼𝐶𝑄 Eq. 3.3 Early voltage Clearly since VA is a fixed voltage, the larger the collector current, the less the output impedance. For situations where the Early voltage is not available the output impedance can be found from the characteristics at any base or collector using the following equation: ∆𝑦 ∆𝐼𝐶 1 𝑆𝑙𝑜𝑝𝑒 = = = ∆𝑥 ∆𝑉𝐶𝐸 𝑟𝑜 𝑟𝑜 = ∆𝑉𝐶𝐸 ∆𝐼𝐶 Eq. 3.4 The transistor model The output impedance can now be defined that will appear as a resistor in parallel with the output as shown in the equivalent circuit Fig. 3.10. Fig. 3.10 re model for the CE transistor configuration including effects of r o. CE fixed bias configuration CE fixed bias configuration Note: 𝐼𝑖 ≠ 𝐼𝐵 𝑎𝑛𝑑 𝐼𝑜 = 𝐼𝐶 Fig. 3.11 CE fixed bias configuration. Fig. 3.12 Network after the removal of the effects of VCC, C1 and C2 CE fixed bias configuration Fig. 3.13 Substituting the re model into the network Zi 𝑍𝑖 = 𝑅𝐵 ∕∕ 𝛽𝑟𝑒 ohms Eq. 3.5 For the majority of situations RB is greater than re by more than a factor of 10, permitting the following approx. 𝑍𝑖 ≅ 𝛽𝑟𝑒 ohms RB 10re Eq. 3.6 Zo Recall that the output impedance of any system is defined as the impedance Zo determined when Vi =0. For Fig. 3.13, when Vi = 0, Ii = Ib = 0, resulting in an open circuit equivalence for the current source. This result is the configuration of Fig. 3.14. Fig. 3.14 Determining Zo for the network of Fig. 3.13 Zo We have 𝑍𝑜 = 𝑅𝑐 //𝑟𝑜 ohms Eq. 3.7 If ro 10RC, the approximation RC // ro ≅ RC is frequently applied and 𝑍𝑜 ≅ 𝑅𝐶 ro 10RC Eq. 3.8 Av The resistor ro and RC are in parallel 𝑉𝑜 = −𝛽𝐼𝑏 (𝑅𝑐 ∥ 𝑟𝑜 ) But 𝑉𝑖 𝐼𝑏 = 𝛽𝑟𝑒 So 𝑉𝑜 = −𝛽 𝑉𝑖 𝛽𝑟𝑒 𝑅𝐶 ∥ 𝑟𝑜 And 𝐴𝑣 = 𝑉𝑜 𝑉𝑖 = (𝑅𝑐 ∥𝑟𝑜 ) − 𝑟𝑒 Eq. 3.9 Av If ro 10RC , so that the effect of ro can be ignored 𝐴𝑣 = 𝑅𝐶 − 𝑟𝑒 ro 10RC Eq. 3.10 Phase relationship The negative sign in the resulting equation for Av reveals that a 180 phase shift occurs between the input and output signals, as shown in Fig. 3.15. This is a result of the fact that Ib establishes a current through RC that will result in a voltage across RC, the opposite of that defined by Vo. Fig. 3.15 Demonstrating the 180 phase shift between input and output waveforms CE emitter bias configuration CE emitter bias configuration The networks examined in this section include an emitter resistor that may or may not be bypassed in the ac domain. We first consider the unbypassed situation and then modify the resulting equations for the bypassed configuration. Unbypassed Fig. 3.18 CE emitter bias configuration Fig. 3.19 Substituting the re equivalent circuit into the ac equivalent network of Fig. 3.18 Unbypassed Note the absence of ro. The effect of ro is to make the analysis a great deal more complicated and considering the fact that in most situations its effect can be ignored, it will not be included in this analysis. Unbypassed Applying Kirchhoff’s voltage law to the input side of Fig. 3.19 results in 𝑉𝑖 = 𝐼𝑏 𝛽𝑟𝑒 + 𝐼𝑒 𝑅𝐸 𝑉𝑖 = 𝐼𝑏 𝛽𝑟𝑒 + (𝛽 + 1)𝐼𝑏 𝑅𝐸 And the input impedance looking into the network to the right of RB is 𝑉𝑖 𝑍𝑏 = = 𝛽𝑟𝑒 + (𝛽 + 1)𝑅𝐸 𝐼𝑏 Unbypassed The result as displayed in Fig. 3.19 reveals that the input impedance of a transistor with an unbypassed resistor RE is determined by 𝑍𝑏 = 𝛽𝑟𝑒 + (𝛽 + 1)𝑅𝐸 Eq. 3.17 Because is normally much greater than 1, the approximate equation is 𝑍𝑏 ≅ 𝛽𝑟𝑒 + 𝛽𝑅𝐸 𝑍𝑏 ≅ 𝛽(𝑟𝑒 + 𝑅𝐸 ) Eq. 3.18 Unbypassed Because RE is usually greater than re, Eq. 3.18 can be further reduced 𝑍𝑏 ≅ 𝛽𝑅𝐸 Eq. 3.19 Zi Returning to Figure 3.19 we have 𝑍𝑖 = 𝑅𝐵 ∥ 𝑍𝑏 Eq. 3.20 Zo With Vi set to zero, Ib = 0 and Ib can be replaced by an open circuit equivalent. The result is: 𝑍𝑜 = 𝑅𝐶 Eq. 3.21 Av 𝑉𝑖 𝐼𝑏 = 𝑍𝑏 𝑉𝑖 𝑉𝑜 = −𝐼𝑜 𝑅𝐶 = −𝛽𝐼𝑏 𝑅𝐶 = −𝛽( )𝑅𝐶 𝑍𝑏 With 𝐴𝑣 = 𝑉𝑜 𝑉𝑖 = 𝛽𝑅𝐶 − 𝑍𝑏 Eq. 3.22 Av Substituting 𝑍𝑏 ≅ 𝛽(𝑟𝑒 + 𝑅𝐸 ) gives 𝐴𝑣 = 𝑉𝑜 𝑉𝑖 ≅ 𝑅𝐶 − 𝑟𝑒 +𝑅𝐸 Eq. 3.23 And for the approximation 𝑍𝑏 ≅ 𝛽𝑅𝐸 𝐴𝑣 = 𝑉𝑜 𝑉𝑖 ≅ 𝑅𝐶 − 𝑅𝐸 Eq. 3.24 Phase relationship The negative sign in Eq. 3.22 again reveals a 180 phase shift between Vo and Vi. Bypassed If RE of Fig. 3.18 is bypassed by an emitter capacitor CE, the complete re equivalent model can be substituted, resulting in the same equivalent network as Fig. 3.13. Eq. 3.5 to 3.10 are therefore applicable. Summary Table Frequency response - BJTs Low frequency analysis In the low-frequency region of the single BJT amplifier, it is the RC combinations formed by the network capacitors CC, CE and CS and the network resistive parameters that determine the cutoff frequencies. Miller effect capacitance In the high-frequency region, the capacitive elements of importance are the interelectrode (betweenterminals) capacitances internal to the active device and the wiring capacitance between leads of the network. The large capacitors of the network that controlled the low-frequency response are all replaced by their shortcircuit equivalent due to their very low reactance levels. Miller effect capacitance For inverting amplifiers (phase shift of 180 between input and output, resulting in a negative value for Av), the input and output capacitance is increased by a capacitance level sensitive to the interelectrode capacitance between the input and output terminals of the device and the gain of the amplifier.
