Resistors dissipate electric energy but do not store it. The two basic kinds of circuit elements (idealized components) that store energy are capacitors and inductors and are called reactances. They are duals of each other because the behavior of one is the same as the other with voltage and current interchanged. Reactances are inherently dynamic and their behavior depends on the rate of change of their current and voltage waveforms.
CapacitorsCapacitors are components that store electric charge, q. The charge is stored in electrically insulating (or dielectric) material that is between two conducting plates or sheets. A capacitor is analogous to a charge storage tank and its value is defined as
where C is capacitance, with units of farads (F), v is voltage, and d is "an infinitesimal change in." In other words, capacitance is the change in charge over the change in voltage. (Using notation from calculus, C is the derivative of q with respect to v; that is, it is the instantaneous slope of q plotted against v. For a constant capacitance, the slope of the q-v line, Dq/Dv, is C. dq/dv is the slope of a line tangent at a given point on any continuous curve.) For constant C, C = q/v.
A farad is a coulomb/volt. But a coulomb is an ampere-second (A×s) so a farad is an A×s/V. By Ohm's Law, a farad is the same as F º s/W, or a second per ohm.
Capacitors can be made by wrapping a plastic film (such as polycarbonate, polyester or polypropylene) between two sheets of aluminum foil. The sheets are offset on the film so that each extends beyond the film at opposite ends. The ends are crushed together and attached to metal leads. A capacitor can be constructed out of the plastic and aluminum films found in kitchens. Capacitance in terms of geometry is:
where e is the permittivity (or dielectric constant), A is the area of the dielectric and l is its thickness. The permittivity has units of F/m and is a measure of the ability of the insulator to store charge. The permittivity of vacuum is e0 = 1/36× p nF/m or about 8.85 pF/m. Relative permittivity, er, is a factor multiplied to e0 to produce total permittivity:
e = er×e0
For example, kitchen plastic wrap (polyvinylidene chloride) has er » 3.5 (e » 31 pF/m) and a typical thickness of 15 mm (0.6 mil). Then a 1 inch (25.4 mm) wide by 1 foot (304.8 mm) long piece will make a capacitor of 16 nF.
Just as Ohm's Law gives the v-i relationship for resistors, the v-i relationship for capacitors can be found from its definition and the definition of current, i = dq/dt:
Solving for dv/dt, the rate of change of voltage across the capacitor is i/C. For a constant current flowing into C, the voltage increases linearly. The voltage ramps up for a step in current applied to it.
Although the current instantaneously steps to its value, I, the voltage "lags" in its response. For a resistor, v would have the same waveform as I, a voltage step of R× I. This time dependence of capacitance is reflected in its unit (F = s/W), which includes time as a basic quantity.
As voltage increases, the electric field across the dielectric also increases until its breakdown voltage is reached and the dielectric material fails structurally. Capacitors are specified for a maximum allowable voltage.
The highest charge density is achieved in electrolytic capacitors. They are also polarized and marked for proper voltage polarity. Reversing the voltage on an electrolytic capacitor can destroy it explosively. Two of them in series with opposing polarities can be used for bipolar (± ) applied voltages.
Capacitors in parallel add and in series, the total capacitance is like parallel resistors. These formulas can be derived from the above equations.
Cparallel = C1 + C2
Cseries = C1× C2/(C1 + C2)
InductorsInductors store magnetic flux, f, in a magnetic field that is created by closed loops of current flowing in conductors (usually wires or circuit-board traces). Each loop produces a given flux and is related to the magnetic flux linkage, l, by the number of turns:
l = N× f
Flux linkage is the dual of current and is defined as
Magnetic flux and flux linkage have units of V× s.
The definition of inductance is:
For constant L, L = l/i. Inductance has units of henries, H, which is the same as a volt-second/ampere. Because V/A is W, H º W× s. Inductance also has a time-dependent unit and, like capacitance, will have a time-related circuit response.
The magnetic field of an inductor is concentrated within the current loop(s) and the amount of flux stored depends on the permeability of the material containing the field. The permeability of vacuum or air is m0 = 400× p nH/m » 1.26 mH/m. The relative permeability, mr, of ferromagnetic materials, such as iron or ferrites, is very high - typically several thousand. Total permeability is:
m = mr× m0
A magnetic field will concentrate in (be contained by) high-permeability materials while being weak in air. High-m materials consequently can be used to shield circuits from stray magnetic fields.
The flux produced by a given current, i, is proportional to the per-turn-square inductance, or permeance, L:
f = L×(N× i)
The terminal current is multiplied by the number of current-loops, or turns. A winding with N turns is equivalent to N individual loops with current i. The magnetic circuit "sees" a current of (N× i).
Permeance is also related to the geometry of the inductor by a formula like that for capacitance:
where A is the current-loop area containing the flux, and l is the closed-path length of the flux.
These equations can be combined to find L by substituting for f in the defining equation:
Inductance varies by the square of the number of turns. Substituting for the permeance,
A constant voltage applied across the terminals of an inductor will cause the current to increase linearly or ramp up. This will continue until the flux exceeds the ability of the magnetic (core) material to sustain it, and the inductor saturates, with a sharp decrease in inductance.
The v-i relation for inductance is found by substituting for dl in its definition:
Then solving for v,
Inductors in series add and in parallel are like resistors, assuming they share no flux in common.
Lparallel = L1× L2/( L1 + L2)
Lseries = L1 = L2
ImpedanceThe concept of resistance can be generalized to include reactances. This more general "resistance" is called impedance,
where R is resistance, j is 
and X is reactance. (We use j instead of i to avoid confusion with the symbol for current.) Impedance is a complex number, with real and imaginary values. Resistance is the real part of impedance; reactance is the imaginary part. Complex numbers can be represented as two-dimensional vectors. When plotted, the horizontal axis is the real axis and the vertical is the imaginary.
Shown above, Z1 = 2.2 kW + j1.5 kW . When added to Z2, the sum is:
To add complex numbers in rectangular form, add the real and imaginary components individually.
Complex numbers can also be represented in polar coordinates, with a magnitude (length of vector) and phase angle. To convert impedance from the rectangular form to polar form,
The above impedance is converted to polar form as
The impedance vectors shown in the above diagram have R and jX values as rectangular-form components. The polar-form magnitude is geometrically the length of the Z vectors and their angle from the R-axis is the phase angle.
To multiply complex numbers in polar form, multiply the magnitudes and add the angles:
To add complex numbers in polar form, they must be converted to rectangular form first:
This rectangular form for Z can also be written as a single mathematical expression by use of Euler's formula, which relates trigonometry to complex numbers:
Applying Euler's formula to Z above, the polar form for Z results in a single expression:
Impedance allows us to extend our existing circuit analysis techniques to circuits with reactances. We now use impedance as we have resistance to solve circuits once we have the reactances of capacitors and inductors. The unit of reactance is the same as resistance (W), or v/i. To solve the v-i relations for C and L, we first need the concept of complex frequency.
Complex frequency, s, is:
s = s + jw
Complex frequency is related to the derivative of a quantity with respect to time:
s× x Û dx/dt
where x is voltage or current. That is, whenever s is multiplied by a variable, x, it can be transformed to the time domain as the derivative of x with respect to time; s× x thus represents the rate of x. Using this approach, apply it to the v-i relations above for C and L to get:
From Ohm's Law, we know that v/i is a resistance. Impedance is expressed in the complex-frequency domain (s-domain) by solving the above equations for v/i.
These impedances can be used in circuit analysis in the same way R is used for resistors. Because s represents a frequency, reactances are frequency-dependent. From them, at a frequency of zero (dc), capacitors are open circuits and inductors are short circuits. At infinite frequency, capacitors are shorts and inductors open.
Time and Frequency ResponsesThe quickness of response of amplifiers and other circuits can be characterized by their output when a step waveform is applied to the input. This is sometimes called the step response and is a function of time (in the time domain). The step response of a capacitor to a current step input is a voltage ramp output.
For a step input, a circuit that is underdamped will output a step waveform that overshoots the final step level and oscillates or "rings" about it for a while. An overdamped response does not overshoot but takes excessive time coming up to the final value of the step. A response that rises in minimal time without overshooting is called critically damped. The time that a step waveform takes to go from 10 % to 90 % of its final value is called the risetime.
Steps are usually generated repetitively as square-waves. Each repetition of the square-wave lasts long enough for circuit behavior to reach constant values, as though the square-wave lasted forever, like a step function. Square-waves are obtained from electronic instruments that are waveform sources, such as function generators or pulse generators. The square-wave period is set to be long enough (low enough square-wave frequency) to give the step response adequate time to decay away so that the full response can be viewed on an oscilloscope.
Step response is one way of observing the effect of circuit self-behavior, which is called the transient response or natural response. When a reactive component in a circuit contains energy, the response of the circuit to that energy is the transient response. Unless the circuit is purely reactive (no resistance), then this energy will eventually be dissipated, and the transient response will decay away in time. The transient response is the behavior of the circuit with no continual input applied.
Another approach to analysis of circuit dynamics is in the frequency domain. Sinusoids (sine-waves) of constant amplitude are applied to the input of a system and their frequency is varied. As frequency increases, the limited quickness of circuit response will cause the amplitude to decrease, or "roll off" with increasing frequency. The frequency at which roll-off becomes significant is called the bandwidth. In addition, the output sinusoids lag behind the input by some number of degrees of a cycle - by some amount of phase (angle).
Both amplitude (magnitude) and phase are affected by frequency and characterize the sinusoidal response. The magnitude of (vo/vi) is the gain of the circuit as a function of frequency. Its phase versus frequency is also significant. When plotted, they are called Bode or frequency response plots. Because the frequency response depends on frequency, not time, it is the steady-state response.
The total dynamic response of a circuit is the sum of the two responses:
Total response = transient + steady-state
After enough time, the transient response decays away and the steady-state response alone is left. When the input waveform is a sinusoid, the steady-state response is the frequency response. In practice, frequency response can be measured by network analyzers (expensive instruments), spectrum analyzers (now under $1000, but good for frequencies above 100 kHz to 1 GHz), audio sine-wave generators and the sine function of function generators. Sweeping function generators vary (or "sweep") their frequency linearly or exponentially (to give a log plot), and the output amplitude of the swept sine-waves traces the magnitude of the frequency response on an oscilloscope. A slower way of obtaining frequency response is to use a fixed-frequency (non-sweeping) sine-wave generator and measure output amplitudes at several frequencies with constant input amplitude. Then plot the points.
The transient response is caused by initial energy in reactive components: voltages across capacitors or currents through inductors. The circuit responds to this energy and its behavior is the transient response. The remaining steady-state response is caused by a periodic input waveform, and is also called the forced response. In summary, the two responses are:
 | transient: time-domain response |
| steady-state: frequency-domain response |
An elegant aspect of the use of complex frequency is that the transient response results from the real component and the frequency response from the imaginary component. By substituting s = jw into the impedance formulas for C and L above, the reactance values of each are obtained:
where w = 2× p × f and f is frequency, in Hertz (Hz); w is the "radian frequency."
Furthermore, because s is in the denominator of ZC, its imaginary component contains 1/j = - j. An imaginary number is plotted on the vertical (imaginary) axis of a complex-number plot and consequently has a phase angle of 90 deg. And a negative imaginary number (such as - j) has an angle of - 90 deg. In polar form, j = 1, Ð 90 deg and - j = 1, Ð - 90 deg.
Plots of impedance magnitude versus frequency are called reactance plots and are conveniently overlaid on reactance charts which have values of R, L and C already drawn. (See reactance chart.) Values of R are horizontal lines. Values of XL increase with frequency and are parallel lines with a slope of 1 (+45 deg) on a log-log scale; values of XC have a slope of –1 (- 45 deg) and decrease with frequency.
Circuit Analysis in the s-DomainAn example of circuit analysis in the s-domain is the voltage divider consisting of a resistor and capacitor, as shown below.
The divider formula is applied using the above impedance for C:
What is the transmittance? It is a complex number that varies with frequency, s. Complex gain Av(s) can be represented as having a magnitude, Av, and an angle, Ð q. Frequency response is found by substituting s = jw and reducing to polar form.
Note that angles follow the rules of exponents. An angle in the denominator is negated when moved to the numerator. When the magnitude and phase given here are plotted versus w (or f), the following frequency-response plots result.
The frequency at which the amplitude decreases by 
is called the bandwidth, fbw, and is a measure of the speed of a two-port network.
The frequency response plots for the RC integrator can be closely approximated with line segments on a log-log chart of magnitude versus frequency, w , and a semi-log chart of phase versus w . These "asymptotic approximations" are shown below for the RC integrator.
The straight-line approximation for magnitude is horizontal (or "flat") out to the break frequency, fb, and then decreases (or "rolls off") at a slope of –1. The phase is flat to a decade before fb, then decreases at –45 ° /decade, crossing –45 ° at fb, and becoming flat at –90 ° at 10× fb.
Having solved for the transmittance (out/in) of a circuit, the general form is a rational algebraic expression in s of the form:
For the RC integrator, K = 1, p1 = –1/RC, and there are no z1, z2, ¼ The rational expression is what results algebraically when the polynomials of both numerator and denominator are factored. The zi are called zeros and are the roots of the numerator polynomial; the pi are called poles and are roots of the denominator polynomial. The poles characterize the transient response of the circuit and the zeros characterize the steady-state response. K is the dc response, the frequency-independent amplification or attenuation of the circuit.
Poles cause the frequency response to decrease with frequency, or "roll off." Zeros cause the opposite effect. This can be demonstrated by the following circuit.
The transmittance of this circuit can be determined by applying basic circuits laws, using 1/sC for the impedance of the capacitor. After some algebraic manipulation, the result is
where R1|| R2 = Rp is the value of R1 and R2 in parallel. The circuit has a dc gain equal to the voltage divider formula, for without C it is a resistive divider.
This "lead-lag" circuit has a single pole at the complex frequency, p = –1/Rp× C, and one zero at frequency z = –R1× C. Because resistors have positive resistances, Rp < R1, and the pole will always be at a higher frequency than the zero. The asymptotic approximation of its frequency response is shown below. The zero has the opposite effect of a pole; it causes the magnitude plot to increase at a +1 slope at the zero frequency, fz = 1/2× p × R1× C. The phase increases at +45 ° /decade until it reaches the influence of the pole phase, at fp. The slopes of zero and pole cancel and the phase is flat until 10× fz, where the zero loses its influence. The pole then rolls the phase off until a decade past the pole frequency, at 10× fp. The pole also cancels the +1 magnitude slope of the zero, resulting in a flat response from fp to higher frequencies.
By using the pole and zero approximation rules, magnitude and phase plots can be constructed for any combination of poles and zeros. But there is one complication. From algebra, a polynomial can be factored into products that can contain not only real roots but also complex ones which appear in pairs, symmetric about the real axis, in the form:
s = –a ± jw
where –a is the real component and jw is the imaginary component. Complex poles and/or zeros can appear in the transmittances of circuits with two or more reactances. The number of poles will equal the number of reactive components and also be the degree of the denominator polynomial. For arbitrarily complicated circuits, both numerator and denominator of the transmittance will consist of products of first and second-degree factors. The first-degree factors will be real and the (reduced) second-degree factors will be complex or imaginary.
Complex poles and zeros cause resonances in circuits. The two kinds of resonances are series and parallel resonance. The linear approximations for resonances can be very inaccurate around the resonant frequency when the resonance is highly underdamped. The approximate frequency response plots for resonances are shown below.
The phase slope depends on how underdamped the resonance is, as does the magnitude peak at resonance. A parallel resonance changes from a +1 to a –1 magnitude slope through the resonant frequency, fr, and a series resonance changes from –1 to +1 slope.
A series resonance can occur when an inductor and capacitor are in series, and a parallel resonance when in parallel. The resonant impedance in either case is
A series resonance is critically damped when a series resistance, Rs = 2× Zr; a parallel resonance is damped when a parallel resistance, Rp = Zr/2. In both cases, the resistance must equal the value of the combined reactance of the L and C.
A critically damped circuit has poles that are real but border on being complex. The step response for critical damping rises as quickly as possible without overshoot (no ringing).
The resonant frequency of an LC circuit (both series and parallel) is:
For circuits with more than two reactances, the algebraic method of finding the transmittance consists of solving n-degree polynomials. This can be difficult even for a third-degree polynomial, and are often solved numerically for higher-degree polynomials. But before resorting to a computer, there is a graphical method that often produces adequate results using the reactance chart. Series and parallel combinations of RC and RL circuits are shown, with reactance charts.
The method is demonstrated using the following circuit.
The upper and lower impedances of the divider are plotted on the reactance chart. The upper branch is R1 and is a flat line on the chart. The parallel combination of R2 and C is plotted by plotting each separately.
Where R2 crosses C, C dominates (or "swamps out") R2 so that its impedance is approximately the combined impedance above the frequency where the impedances are equal (which is 1/2× p × R2× C). The combined impedance is then the composite plot labeled R2 || C.
The divider transmittance is found by the voltage-divider formula, Z2/(Z1 + Z2) = Z2/Zin. Zin is plotted, beginning at R1 + R2 and extending to break frequency 1/R2× C. It then decreases to R1. Where it meets R1, a dotted line is extended down and intersects R2 || C at the parallel equivalent resistance of the two resistors.
Because the plot is log-log, division is accomplished by subtraction. Z2 and Zin maintain equal vertical separation until Zin flattens out at R1 while Z2 continues to roll off. This results in a frequency response magnitude plot that is flat out to 1/(R1 || R2)× C, then rolls off with a –1slope.
The reactance chart impedances and transmittance for the lead-lag circuit are shown below.
The difference between R2 and Zin decreases at frequency 1/R1× C; the denominator of the impedance divider, Zin, is decreasing, causing vo/vi to increase, until Zin flattens out. At that frequency, 1/(R1 || R2)× C, the capacitor is essentially a short and passes vi to vo. The transmittance is then one.
Time-Domain ResponseThe time-domain response can also be approximated from the s-domain circuit transmittance. The real component of a pole causes an exponential response of the form e–t/t , where t is the time constant,
t = 1/a
and a is the negative real component of the pole. For a real pole of an RC circuit, t = R× C, and for an RL circuit, t = L/R.
The step response of a real (negative) pole is an exponential rise with time constant t . After one time constant, the step has risen to e–1 @ 63 % of the final value. After 5× t , the exponential is within 1 % of the target value.
A measure of the speed of response to a step is the risetime, the time the response takes to go from 10 % to 90 % of the target value. For single-pole circuits with time constant, t , the risetime is
tr @ 2.2× t
A 4 V step applied to an RC integrator results in the oscilloscope waveform shown below, where R = 1.00 kW and C = 10 nF. The time constant is
t = R× C = 10–5 s = 10 m s
The waveform rises to about 63 % of 4 V, or 2.5 V in 10 m s, and after 5× t , or 50 m s, it has reached the target value of 4 V.
The RC differentiator, under similar conditions, behaves similarly, as shown below. In this case, the waveform steps to 4 V, then decays to its 0 V target.
For complex poles at –a + jw , the real (–a ) component (a > 0) causes a decaying exponential response as before, but the imaginary component causes a sinusoidal response. Mathematically, this follows from Euler's equation:
More generally, the response of a pole at s = –a +jw is:
where f is the phase angle of the sinusoid. This time response is a decaying sinewave. The following RLC circuit was built and the step response acquired.
The response to a 4 V step at vo is shown below, and is underdamped. The waveform overshoots the 4 V target and oscillates (or "rings") around it. The ringing is a decaying (or "damped") sinewave. The pulse generator that supplied the step had a 100 W output resistance, and this generator resistance was in series with the LC circuit.
The ringing is no longer discernible after about two cycles. The complex pole can be expressed in polar form, as a magnitude and angle, f , relative to the (negative) real axis. For negative real poles, f = 0, and the response is a decaying exponential. For f = 90 degrees, the pole is imaginary and the response is a non-decaying sinusoid. On the jw axis, a = 0, and the time constant of a pole on the jw axis is 1/0 or infinite. In other words, the decay time is forever. For poles with angles in between 0 and 90 degrees, the closer the angle is to 90 degrees, the more underdamped is the response. By counting the cycles on an oscilloscope, Ns, the pole angle can be estimated.
| Ns , cycles | f , degrees |
| 0 | 0 |
| 0.3 | 30 |
| 0.6 | 45 |
| 1 | 60 |
| 2 | 75 |
| 5 | 84 |
The pole angle of a series resonant circuit is
and for a parallel resonant circuit, it is
The pole angle for this series resonant circuit is calculated by first finding the resonant impedance,
Then f is cos–1{0.275} @ 74 ° .
The resonant frequency is
Because period is the reciprocal of frequency,
Tr = 1.14 m s
When a sinuosid is damped, its frequency decreases to
In this case, sinf @ 0.99 and Td @ 1.15 m s. By noting that the time scale of the oscillograph is 500 ns per division (spacing between dotted vertical lines), or 500 ns/div, the period of the oscillation is about 1 m s.
The waveform peaks at about 5.5 V, or has an overshoot of 5.5 V – 4 V = 1.5 V. The ratio of overshoot to target voltage is the fractional overshoot, Mp. The pole angle can also be calculated from it as
In this case, Mp @ 1.5 V/4.0 V = 0.375, and f @ 73 ° , in good agreement with the other calculation.
Finally, because time and frequency responses are related, risetime and bandwidth are related for circuits near critical damping by the approximation,
tr @ 0.35/fbw
Closure
The concept of resistance was extended to include reactive circuit elements, capacitance and inductance, as impedance, a complex number involving complex frequency, s. With reactances expressed in s, we could apply circuit laws (Ohm's Law and Kirchhoff's laws) and develop transmittances for two-port circuits.
Because the algebraic expressions for transmittances involve solving nth-degree polynomials for their roots (poles and zeros), a graphical method was introduced based on the reactance plots of circuit impedances.
For single-pole circuits, various circuit properties were developed, such as the time constant and bandwidth. For a single complex-pole pair, the concept of resonance, with its resonant impedance, Zr, and resonant frequency, fr, were expressed in circuit component values L and C.
Time-response characteristics such as risetime can be related to frequency-response characteristics, such as bandwidth.
The principles illustrated here all involved passive circuits (no transistors or op-amps, etc.), but can also be applied directly to active circuits.
This article has merely introduced the analysis of circuit dynamic response theory, but these basic concepts and analysis methods should be adequate for the simpler (and most common) circuits encountered in electronics applications, and for understanding the terminology applied to them in the literature.
Publicado por: Geraldine F. Linares M. /CRF
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