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Circuit Analysis Demysti ed
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EXAMPLE 14-6 Consider a series LC circuit and describe when the circuit is BIBO stable Assume the circuit is excited with a voltage source v(t) = A cos t SOLUTION The differential equation describing this circuit is LC d 2vc + v c = A cos t dt 2
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Taking the Laplace transform of both sides we obtain (LCs 2 + 1)Vc (s) = As + 2
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Now recalling that the natural frequency is de ned via
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We can write the solution as Vc (s) = The transfer function is H (s) = with poles at s = j 0 If the frequencies don t match, then the system is stable For a trivial example suppose Vc (s) = 4 + 16 s +8 0 2 s 2 + 0 0 2 + 0 As + 2
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Circuit Stability
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Fig 14-9 A plot of v c (t) = 1 (cos 2 2t cos 4t) 2
Then the voltage across the capacitor is 1 v c (t) = (cos 2 2t cos 4t) 2 This is a stable voltage A plot is shown in Fig 14-9 On the other hand, suppose the frequencies match (there is a resonance) Continuing the example suppose instead that Vc (s) = then 1 v c (t) = t sin 4t 2 This function oscillates but grows linearly it grows without bound Hence, at resonance the circuit is unstable This is shown in Fig 14-10 where you can see the voltage across the capacitor start to grow s2 4 + 16 s2 s + 16
Summary
Often we need to consider the stability of a circuit under different conditions For example, do the voltages and currents remain nite as time progresses We can characterize stability behavior by looking at the impulse response or
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Circuit Analysis Demysti ed
Fig 14-10 The unstable case of resonance
transfer function A circuit is impulse response stable if the transfer function remains nite, that is, if lim |h(t)| < We often look at zero-input response
stability (that is, no sources) This is done by looking at the poles of the transfer function H (s) When the poles of H (s) are real and negative, the circuit is stable; this indicates that the currents and voltages in the circuit will decay to zero as t The nal type of stability we considered was bounded input-bounded output or BIBO stability A system is BIBO stable if a bounded input results in a bounded output
Quiz
1 Is the function h(t) = t sin 2t stable In the following questions, determine the stability of the following transfer functions and nd their time domain representation s 2 H (s) = s 2 +16 3 H (s) = 4 H (s) = 5 H (s) =
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6 Consider an LC circuit with the capacitor and inductor in series If C = 1/12, L = 4, and the initial voltage across the capacitor is 1 V and all other initial voltages and currents are zero, determine whether the circuit is zero input stable
Bode Plots and Butterworth Filters
The frequency response of a system can be plotted using a logarithmic scale in the following manner Given the frequency response H ( ), we calculate |H ( )|dB = 20 log10 |H ( )| (151)
We call this quantity the magnitude of the frequency response in decibels (dB), or sometimes we denote (151) by ( j ) and call it the gain function A decibel is a dimensionless unit based on the ratio of two quantities The reader is probably familiar with the use of decibels in the study of sound In that case, we can characterize how loud a sound is by comparing the intensity I of a given sound wave to the threshold for human hearing, which we denote as Io We then compute the intensity of the sound in decibels as IdB = 10 log10 I Io (152)