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Single-stage BJT dynamics Dennis Feucht - February 11, 2013 This article goes through the circuit analysis of a generalized single-transistor amplifier for the purpose of finding the open-circuit time constants (OCTCs) of the two BJT capacitances. The derivations of the open-circuit resistances take some work but only have to be done once. The result can then be applied as a template to any single-stage transistor amplifier. Along the way, some further clarification is given of behavior in the high-frequency region of BJT operation, above the fβ of the BJT. Textbook CE Stage Transfer Function A classic derivation from basic circuit equations of the “textbook” CE stage using the hybrid-pi BJT incremental model without external emitter resistance, RE, is outlined as follows. The circuit is shown below. When vbe of KCL at the collector is substituted into the base-node equation and simplified, Noting that rπ||rm = re, then after much algebra, A different derivation thevenizes the Rb resistors with Cπ to form the following equivalent circuit. Thevenize the collector side of the upper circuit and reduce to the lower circuit equivalent. After much more algebra, this reduces to the same result as in the first derivation. Although it is tedious to work these equations through from KCL and ΩL, once completed the result is useful in confirming the correctness of results obtained by easier means using the higher-level circuit methods. The general BJT stage open-circuit resistances readily reduce for RE = 0 Ω to what is in the linear term of D(s) above. Once accomplished, the circuit solutions of general, often-used circuits such as this one provide a template with memorable formulas for use in future analyses. General Single-Stage BJT Amplifier The BJT single-stage amplifier in a generalized form can be used for any of the three BJT configurations. It is shown below in incremental model form. The goal is to find the OCTCs of its capacitances. The incremental emitter resistance of the BJT T model is found directly from the static emitter current as re = VT/|IE| ≈ 26 mV/|IE|, and β0 (the low-frequency or quasistatic β) is found readily in BJT specifications. The hybrid-π model is equivalent to the T model and results from referring the shunt re, Ce elements to the base using the β transform. Otherwise, it is the same BJT model for which The shunt rπ, Cπ base impedance of the hybrid-π BJT model is For the T model, β = β(s); Frequency-dependent β has a breakpoint at 1/τβ or fβ. Then Zπ is referred to the emitter by the β transform as Then the time constant, The correspondence of the dynamic T model with the hybrid-π parameters is thus Cc = Cμ ; Ce = Cπ ; re = rπ/(β0 + 1) A pole is associated with each of the capacitors. Base-Emitter Open-Circuit Resistance The open-circuit resistances across Ce and Cc of the single-stage BJT are needed to find the OCTCs and their corresponding poles. From the following circuit with Ce and Cc removed, we can derive the open-circuit Rbe. The circuit is redrawn to the right to make the b-e port more obvious. The port resistance has re shunted by a circuit that is reduced by thevenizing the current source and RB, as shown below. Then re is in parallel with a branch having a dependent source; The ground is included only for circuit reference and does not affect the b-e port resistance. Then solving, When the parallel formula is invoked, This can be rewritten in a slightly different form that has a circuit interpretation: The first term in the braces has a fraction that is a voltage divider. The resistances of the divider are all referred to the emitter circuit. It is the fraction of the emitter loop voltage that is developed across the base resistance of the circuit, referred to the emitter. The re outside the brackets is multiplied by β0 + 1 to refer it to the base as rπ by this fraction. The second term is a voltage divider that is the fraction of emitter-loop voltage across RE. Rbe is then the fraction of emitter resistance referred to the base (by β0 + 1) added to the fraction of re due to the emitter branch. In other words, Rbe is a combination of base-referred and emitter-referred re, and the voltage dividers give the fractions of each. Another expression for Rbe can be derived more directly from the hybrid-π model and it is derived from the base-referred circuit, with rπ across the b-e terminals instead of re; The equivalent circuit is the same as that above with re changed to rπ and with the current source driving the emitter instead of the base terminal. Then a thevenized voltage source in series with RE has its positive terminal connected to the emitter terminal. The general formula for τbe = Rbe·Ce is Now that we have a general formula for τbe, we can see what effect RB and RE have on the time constant. As RE is made large, Then the b-e time constant, τbe → α0·τT. We can use the fact that whenever RE >> re, we can disregard the effect of τbe on frequency response because it is near τT, the high end of the useable frequency range of the BJT. With RE = 0 Ω, Rbe = Rb = rπ||RB. For RB >> re or RB → ∞, Rbe → rπ and the slowest time constant results: τbe = τβ. The value of a large RE or small RB is apparent; either reduces τbe and results in a faster amplifier. Base-Collector Open-Circuit Resistance The b-c capacitance, Cc, forms a time constant with the open-circuit resistance of the b-c port. What is different about this resistance is that whereas Rbe involves only the base and emitter - the input circuit loop - Rbc involves both input and output loops. The general one-stage circuit for Rbc analysis is shown below. The current source between collector and base is applied to the b-c port to find the resistance. KCL (as the node-voltage method) is applied at the base node; The emitter voltage divider gives us vbe; Noting that re = α0·rm, the current source value is Substituting into the KCL equation and solving for i, having applied Solving for vb, vb = Rb·i. At the collector node, Finally, Rbc includes elements in both the input loop (Rb, rE) and output loop (RL) of the amplifier stage. Rbc is the resistance of a single pole across the b-c nodes. Rbc is composed of two time-constant terms, one associated with the collector node, and the other with the base node, where the Miller effect from the base-to-collector voltage gain effectively increases Cc at the base node by (1 + Kv) as given in Rbc. These are not, however, two separable time constants but are components of the one time constant, When RB = 0 Ω, τc = τcc and Rbc = RL , RB = 0 Ω As RB increases to infinity, in the limit For the case of RE = 0 Ω, Kv has high gain without RE and the Miller effect significantly increases open-circuit resistance. As RE → ∞ instead, in the limit, Rbc = RL + RB , RE → ∞ Kv decreases with increasing RE and the loading of the emitter branch referred to the base becomes an open circuit, leaving base node resistance Rb = RB. Rbc has two maximum values, one under the same conditions that Rbe is a maximum: RB → ∞, RE = 0 Ω: The other is when RB → ∞ and RE → ∞. Then Rbc → ∞, a circuit condition that has no useful static solution and is not of practical importance. A minimum occurs at In the case of a grounded base, Rbc is isolated to the output circuit and the time constant of Cc is entirely at the collector node. Then τcb = 0 and τc = τcc. For a finite RB, the minimum Rbc(RE) asymptotically approaches a minimum of RL + RB as RE → ∞. Closure The determination of the general BJT stage OCTCs can be rather involved but it presents a clear picture of the effects of both BJT capacitances and associated circuit resistances on dynamic response. We now have the BJT time constants that are needed to approximate bandwidth and risetime for a single amplifier stage, though stage interaction has yet to be taken into account, a topic for a subsequent article.