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Exponentials simplify linear circuit analysis (part one)
( 01 Sep 2008 )
by Joshua Israelsohn, Contributing Editor
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Despite (or, perhaps, because) engineering as a family of disciplines greatly depends on rigorous mathematical modeling and analysis, its practitioners have become remarkably adept at making the math go away. Over the centuries engineers, mathematicians, and physicists have contrived all manner of technique to reduce intractably complex expressions into simple solvable ones while maintaining proper accuracy. We use symmetry, superposition, and segmentation to cancel or disassemble unwieldily complex expressions into piecewise solvable bits and wefve been doing so since long before the existence of the first computational circuit simulator.
Similarly, good design-engineering practice also teaches us to take advantage of circuit concepts that reduce analytic complexity. Half-circuit analysis in differential signal chains is a good example of this sort of complexity reduction.A danger posed by our casual use of these powerful shortcuts is that we can lose track of their origins and, in so doing, lose track of their limitations and of the assumptions on which they depend. This installment of Engineering on Purpose examines the origins of a family of shortcuts that allow us to reduce complex linear circuit analysis to reasonably simple algebraic expressions.
ASSUME A PERIODIC SIGNAL
The defining characteristic of an LTIC (linear time-invariant circuit) is the effect it has on a periodic signal. A linear circuitfs time-domain response takes the form
where f is the circuitfs time-dependent gain function, Vout(t) is the circuitfs time-domain output, and Vin(t) is the periodic input. In the frequency domain, the circuitfs transfer function takes a form that parallels that of Equation 1:
where H(s) is the frequency-domain transfer function and s represents the signalfs complex frequency + j.
Periodic signals are sums of harmonically related sine waves:
where o is the amplitude of the waveformfs DC offset, n and n are coefficients to the in-phase and quadrature components of each harmonic, and o is the frequency of the fundamental in radians per second. Because complex periodic waveforms comprise sums of harmonically related sine waves, the spectral properties of LTICs reveal themselves as the superposition of the responses to individual equal-amplitude pure sine waves, the sum of which expresses the spectrum of interest.
There are several ways to describe individual sine waves with respect to a temporal standard, each requiring two data. One provides in-phase and quadrature amplitude pairs similar to Equation 3. Another expresses the information as a scalar amplitude and a phase angle. The two relate to each other through Eulerfs theorem:
where both and t are real numbers and j is the complex operator equal to the square root of -1.
Separate from descriptors of signalsf spectral content or instantaneous amplitudes, signals can also exhibit exponential envelopes that take the form et where is a real number. Negative values of indicate that the signal envelope is decaying as is the case, for example, with a damped oscillation. Positive values of indicate that the signal envelope is growing exponentially. The = 0 case equates to the multiplicative identity, 1, so the envelope factor has no effect on the signal.
Combining these methods of describing both a periodic signal and its amplitude envelope results in a description of a complex frequency, est, as
Generalizing the equation allows for arbitrary amplitude and phase:
where A is a complex number that embodies both an amplitude scalar and a phase rotation, |A| is the amplitude component of A, and is the phase-rotation component of A. One more step isolates the real and imaginary parts of Equation 4:
START WITH THE BEGINNING OF TIME
As a practical matter, analyses of signal sources and of LITCs that process them cannot start with the beginning of time or even with the instants of devicesf manufacture. Analysis methods require a switch with which to establish the beginning of a time interval of interest.
This switch is better known as the Heaviside unit-step function, U-1(t), the derivation of which begins with the Dirac-delta or unit-impulse function, (t):
where x is a positive real number.
The Dirac-delta function generates a Gaussian of unit area (Figure 1). For decreasing values of x, the Dirac delta functionfs width shrinks and its amplitude grows to maintain constant area. As x approaches zero, the amplitude of (t) approaches . Integrating (t) yields the Heaviside step function (Figure 2):
Note that in discrete-interval calculations, such as you might use in a simulator, deriving the Heaviside function from its historic origins is not necessary and, indeed, can be problematic because the value of x is limited by the size of your time-step. In such cases, too small a value of x can create an error in the Heaviside function, which can reflect as a gain error in higher-level functions such as complex signal sources. To avoid this, a discrete-time version of the Heaviside unit step function simply sets its result to 0 for t < t0 and to 1 otherwise. An advantage of a computationally derived Heaviside unit step in some applications is that its artifacts occur within a narrower bandwidth than the instantaneously switching discrete-time version.
EXPONENTIALS SIMPLIFY THE ANALYSIS
Beyond describing signals, these basic equations also serve as a foundation to LTIC analysis. A simple RC low-pass filter section (Figure 3) demonstrates many of the key concepts.
The I/V relationship for a capacitor is
This expression states that the voltage on a capacitor at time T is equal to its initial voltage at t = 0 plus the charge added between t = 0 and t = T scaled by the reciprocal of the capacitance. At any given instant, this expression is equivalent to V = Q/C.
Taking the derivative of both sides of Equation 5 gives the capacitor current in terms of the time rate of change of the capacitor voltage:
In this simple circuit, the capacitor current and the resistor current are identical. Applying KVL (Kirchhofffs Voltage Law) to Figure 3 yields:
In this form, the capacitor voltage, Vc, is the difference between the source and the voltage across the resistor. Correspondingly, the dynamic response of the circuit is the sum of the forced response and the natural response.
The forced response is that of the source, which, in this case, is a Heaviside unit step scaled by a gain factor, k. Expressing the source in generic terms,
but because the input in this case is just the scaled step, Sf must equal zero.
The natural responses of LTICs are exponentials in the form
Postulating Vc = kest, the time derivative of the capacitor voltage is ksest or simply sVc. Substituting this relation into Equation 6 reveals that
The advantage to expressing our signals and circuit behaviors in terms of exponentials is that taking their derivatives reduces to the essentially trivial act of multiplying by s. Note, however, that if we had chosen some other way of describing these entities, the full complexity of calculating the derivative would have remained. As an exercise to appreciate better this simplification, substitute Equation 5 into Equation 6 and compare the result with Equation 9.
The calculus having conveniently collapsed, substituting Equation 9 into Equation 2 yields the transfer function of the simple low-pass RC section:
The capacitorfs initial charge sets the boundary condition that the natural response must satisfy. That is, at t = 0+, Vc = 0. At the same moment, Vin = k. The scaling factor to the natural response that the circuit requires to satisfying the boundary condition is
The natural response represents the circuitfs behavior when the source is set to zero but charge may exist on the capacitor. Setting Equation 9 to reflect these conditions,
Satisfying this equation requires that s = -1/RC and substituting that value into Equation 8 identifies = RC.
As was stated earlier, the total response equals the sum of the forced and natural responses. Summing Equations 7 and 10
The forced, natural, and total responses appear in Figure 4.
The next installment of Engineering on Purpose will pick up from here, building on the simplification that exponential expressions bring to analyzing LTICs by reducing the differential equations in t to simple multiplications in s.
About the author
Joshua Israelsohn is a co-founder of JAS Technical Media where he manages the companyfs technical-communication consultancy practice. You may find his contact information at www.jas-technicalmedia.com.
Click here for the illustrations:
Figure 1, Figure 2, Figure 3, Figure 4 |
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