# E a 1 then output at he l o oin i is equ to a op was

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Unformatted text preview: Agothe feedback so op and taking the output at the point n l = A(s)B ( l ) where the lo op was opened. For example, if the lo op is opened at the summing junction, then the lo op gain is Vf /Vi as shown in Figure 5.2. 139 140 CHAPTER 5. OSCILLATORS B Vf Vi Σ A Vo Figure 5.2: Lo op gain at summing junction Loop gain is calculated by opening the feedback loop and taking the uppose that ttheopp gatnwherealthe1lot some freopency ω i.e. A (ω ) = 1. Then output at he l o oin i is equ to a op was qu ened, , i.e., S o lo o the voltage transfer function in Figure 5.1 is singular (inﬁnite), which can be interpreted Vf s as ﬁnite output for zero input. InAthe=word. , the circuit is a potential source of radio or lo frequency energy at the frequency where theVo op gain is 1, even in the absence of any input li excitation Vi . RoThe condition forationy-state oscillation (Alo (ωo ) = 1) is intuitively satisfying - when ots of the equ stead the lo op gain is 1, a sinusoidal excitation presented to the input of the circuit traverses the feedback lo op and appears back 1 t the lo (s)t = ith the same amplitude and phase that it a − A inpu w 0 started with. This re-circulation of the disturbance pro ceeds indeﬁnitely, with the circuit “oscillating” in a steady state. In practice the smal l-signal lo op gain is set to a value are the pole locations. somewhat larger than 1. This means that the disturbance is ampliﬁed after each pass through the lo op, and the output grows as the disturbance passes repeatedly through the lo op. In most radio frequency oscillators the lo op gain is equal to 1 (or some real number When poles of the transfer function are on the j ω axis the system supports steady-state oscillation (neither decaying or growing). A pole is on the j ω axis if the following equation is satisﬁed for some real value of frequency, denoted by ωo : Alo (j ωo ) = 1. If Alo (j ωo ) = 1 can be satisﬁed for some real ωo , then the system supports steady-state oscillation at frequency ωo . Alo is a complex quantity, so two conditions must be satisﬁed for steady-state oscillation to occur. They are called the Barkhausen Criteria for oscillation: arg[Alo (ωo ] = 2nπ , for n an integer |Alo (ωo )| = 1 Practical oscillator circuits are designed so that poles are actually in the RHP. The circuit is then unstable, and any small perturbation will result in an oscillation that grows exponentially, i.e. like eαt , α > 0. Thermal noise, or the transient caused by turning on the supply voltage provides the initial excitation that excites the growing oscillation. When the oscillation amplitude is suﬃciently large, nonlinear saturation of the ampliﬁer (often a single transistor) eﬀectively reduces the loop gain, moving the pole onto the j ω axis, and oscillation is limited at a ﬁnite value. Initial oscillation will start and grow if arg[Alo (ωo )] = 2nπ , for n an integer and |Alo (ωo )| > 1 Summary of procedure for loop gain design of oscillator circuits. 1. Identify feedback loop. Break the loop and terminate with the impedance that the feedback output normally sees when the loop is closed. Solve for loop gain function, Alo (s). 2. Solve arg[Alo (ω )] = 2nπ to determine the potential frequency (or frequencies) of oscillation, ωo . This will yield an expression for ωo in terms of circuit parameters. Choose the parameters so that ωo is the desired value. 3. Set |Alo (ωo )| = 1. This will yield an expression for the minimum gain required for the ampliﬁer. For example, when the ampliﬁer is a single transistor, this will yield an expression for gm,ss , which is the transconductance required for steady-state oscillation. Bias the transistor so that the actual transconductance is larger than gm,ss to ensure |Alo (ωo )| > 1. |Alo (ω )|ω=ωo ≥ 1 (5.3) In practice we are usually able to apply condition (5.2) to solve for the potential frequency of oscillation, ωo . Then applying condition (5.3) will determine how much gain is necessary in order to make the lo op gain larger than 1 at ωo . Circuits with the topology shown in Figure 5.3a are commonly employed as oscillators. The active device could be a BJT or an FET. This circuit can be analyzed as a feedback Z3 Z3 Z2 Z1 Z2 Z1 (a) (b) Figure 5.3: (a) Topology of one class of oscillator circuits. (b) Same as (a), redrawn to show the feedback path from output to input through Z3 . lo op. The circuit is redrawn in Figure 5.3b to explicitly show that the feedback from output to input is through the element Z3 . The lo op gain is easily computed for circuits of the type where IC Q is the quiescent collector current. VT β = A useful linear mo del for the behavior of small high frequency signals superimposed on IC Q the DC bias point is the small-signal hybrid-pi mo del shown in Figure A.2. Typically ro ∼ .025β rπ IC Q rµ 6 and the small-signal transconductance B rx APPENDIX A. CµIRCUIT MODELS FOR BJT AND FET C C + e A.2) V - E rπ Cπ rπ gm ∂ IC Vb =V ∂ Vbe Ce beq...
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