Electronics Guide

Frequency Response of Linear Time-Invariant Systems

Drive a stable linear time-invariant system with A cos ωt, wait for the transient to die away, and the output is

y(t) = A|G(jω)| cos(ωt + φ)

where G(jω) is the transfer function G(s) evaluated at s = jω and φ is the angle of that complex number. At each frequency the system does exactly two things to a sinusoid: it scales the amplitude by |G(jω)| and shifts the phase by φ. Taken over all frequencies, the pair is the frequency response. The steady state exists only if the system is stable, so an unstable plant cannot be measured open loop, although its G(jω) remains well defined for calculation.

In factored form, G(s) = K(s − z₁)(s − z₂) ... / [(s − p₁)(s − p₂) ...], the magnitude at s = jω is the product of the factors' magnitudes, and the phase is the sum of the zeros' angles minus the sum of the poles' angles. In decibels the product also becomes a sum, so a complicated response is a sum of simple curves, one per pole or zero.

A feedback loop is analyzed through its loop gain L(s), the product of every transfer function around the loop: the controller, the plant, the sensor, and the feedback network. For an amplifier with forward gain A(s) and feedback factor β, L(s) = A(s)β. With unity feedback the closed-loop response is

T(s) = L(s) / [1 + L(s)]

and the closed-loop poles are the roots of 1 + L(s) = 0. Here L(s) excludes the sign inversion at the summing junction, so the loop is on the edge of oscillation where L(jω) = −1: a magnitude of 1 at a phase of −180°. The loop gain is the quantity to plot, because it is a product of known blocks, it can be measured with the loop closed, and stability depends on how near it comes to −1.

Decibels and Logarithmic Frequency Axes

The magnitude of a voltage ratio, a current ratio, or a dimensionless loop gain is expressed in decibels as

|G| in dB = 20 log₁₀|G|

Power ratios use 10 log₁₀. Gains multiply along a signal path, so their decibel values add.

The frequency axis is logarithmic. A decade is a factor of 10 in frequency and an octave a factor of 2. On these axes a response proportional to ωⁿ is a straight line with a slope of 20n dB per decade, so a single pole's roll-off of −20 dB per decade is also −6.02 dB per octave. Corner frequencies may be given in radians per second or in hertz, with ω = 2πf; mixing the units misplaces a corner by a factor of 6.28.

The phase axis is linear in degrees. Instruments and many programs wrap phase into a range of 360°, such as −180° to +180°, so a curve that appears to jump by 360° has only wrapped. Unwrap it, following the curve continuously from low frequency, before reading any crossover.

Building a Bode Plot from Asymptotes

Construction starts by writing the transfer function in standard form, with every factor normalized to 1 at low frequency:

G(s) = K sⁿ × [(1 + s/ω_z1) ...] / [(1 + s/ω_p1) ...] × 1/(1 + 2ζs/ω_n + s²/ω_n²) × e^−sT

Here n is positive for zeros at the origin and negative for poles there, which are integrators; ω_z and ω_p are real corner frequencies; the quadratic stands for any complex pair; and T is a time delay.

Asymptotic Versus Exact Response

The exact curve bends away from the straight lines near every corner, and the size of each error shows when a sketch is good enough.

Minimum-Phase and Non-Minimum-Phase Systems

Reading Gain and Phase Crossover

Two frequencies organize a loop-gain Bode plot. The gain crossover frequency ω_c is where |L(jω)| falls through 1, or 0 dB. The phase crossover frequency ω₁₈₀ is where the phase of L(jω) reaches −180°. Stability Analysis and Compensation defines the phase margin as 180° plus the phase at ω_c and the gain margin as 1/|L(jω₁₈₀)|, and it discusses target values.

The familiar Bode test is a special case of the Nyquist criterion. If L(s) has no poles in the right half-plane and its magnitude falls through 0 dB exactly once, the closed loop is stable exactly when the phase margin is positive, with the phase followed continuously up from low frequency. Loops with right-half-plane poles, or with several 0 dB crossings near a resonance, need a Nyquist plot. When the phase crosses −180° more than once, read the magnitude at every crossing; a crossing below ω_c sets how much gain reduction the loop tolerates.

Consider the position loop L(s) = 50/[s(1 + s/50)(1 + s/500)], the servo of Time-Domain Response of Control Systems with an added lag at 500 rad/s from, for example, a sensor filter.

1. Sketch. The asymptote falls at 20 dB per decade to 0 dB at the 50 rad/s corner, suggesting a crossover there. The straight-line phase at 50 rad/s is −90° − 45° − 0° = −135°, suggesting a phase margin of 45°.
2. Correct. The crossover sits on a corner, where the exact curve lies 3.05 dB below the asymptote, so the exact crossover falls to 39.2 rad/s and the phase margin is 47.4°.
3. Find the phase crossover. The phase reaches −180° where arctan(ω/50) + arctan(ω/500) = 90°, which requires ω² = 50 × 500, so ω₁₈₀ = 158.1 rad/s, where the gain margin is 20.8 dB against the asymptote's 20.0 dB.
4. Attribute the lag. Without the 500 rad/s pole the phase margin would be 51.8°, the value for the second-order prototype with ζ = 0.5. The extra pole, more than a decade above crossover, still costs 4.4°.

Constructing a Nyquist Plot

A Nyquist plot takes the numbers of a Bode plot and places each as a point in the complex plane, at distance |L(jω)| from the origin and at an angle equal to its phase. As ω rises the points trace a curve. For a transfer function with real coefficients, L(−jω) is the complex conjugate of L(jω), so the curve for negative frequencies is the mirror image of the positive-frequency curve in the real axis.

The complete plot is the image of the Nyquist contour, a closed path in the s-plane that encloses the whole right half-plane. The contour runs up the imaginary axis from −j∞ to +j∞ and returns clockwise along a semicircle of infinite radius. For a proper L(s) the large semicircle maps to a single point, the origin when L(s) has more poles than zeros. A pole on the imaginary axis, such as an integrator, would sit on the contour, so the contour detours to its right on a small semicircle, leaving the pole outside; m poles at the origin map that detour into an arc of infinite radius sweeping m × 180° clockwise.

1. Tabulate magnitude and phase across the frequency range, and convert each pair to a real part |L| cos φ and an imaginary part |L| sin φ.
2. Find the low-frequency end. A loop without integrators starts on the real axis at L(0); a loop with m integrators starts infinitely far out, at an angle of −90° times m.
3. Find the high-frequency end. A loop with more poles than zeros ends at the origin, and a minimum-phase loop arrives at an angle of −90° times the excess of poles over zeros.
4. Mark the crossings of the negative real axis and of the unit circle, the points that set the margins.
5. Add the mirror image for negative frequencies, close the curve with any infinite arcs, and mark the direction of increasing ω.

The curve starts at +4, swings clockwise through the fourth and third quadrants, and crosses the negative real axis where 3 arctan ω = 180°, at ω = √3 rad/s, with |L| = 4/8. It then curls through the second quadrant into the origin. The crossing at −0.5 gives a gain margin of 2, or 6.0 dB, and the unit-circle crossing at 1.233 rad/s gives a phase margin of 27.1°.

The Nyquist Stability Criterion

Conditionally Stable Systems

A loop is conditionally stable when it is stable over a band of loop gain but becomes unstable if the gain rises above the band or falls below it. Its phase dips below −180° at frequencies where the magnitude still exceeds 0 dB, so its Nyquist plot crosses the negative real axis on both sides of −1, and at the nominal gain the resulting encirclements cancel.

Consider L(s) = K(1 + s)²/[s³(1 + s/30)²]. Three integrators give −270° at low frequency, the double zero at 1 rad/s adds up to 180° of lead, and the double pole at 30 rad/s takes it back. The phase rises to a maximum of −131.4° at ω = √30 = 5.48 rad/s, and it lies above −180° only between the roots of ω² − 29ω + 30 = 0, which are 1.074 and 27.93 rad/s. With K = √30, the crossover falls at the phase peak, the phase margin is 48.6°, and the plot crosses the negative real axis at −9.50 and at −0.105.

The nominal gain has 19.6 dB of margin in each direction: raising K to 52.1 moves the crossing at 27.93 rad/s out to −1, and lowering K to 0.576 moves the crossing at 1.074 rad/s in to −1. A Bode plot read only at crossover shows a healthy 48.6° and hides the lower limit, so read the magnitude wherever the phase passes −180°.

The lower limit matters because a saturating stage lowers the effective gain: in describing-function terms, a clipped sinusoid carries less fundamental than the unclipped drive. During start-up or after a large disturbance, a conditionally stable loop can be pushed into its unstable range, where saturation can sustain a large oscillation. High-order delta-sigma modulators, servo loops with several integrators, and voltage-mode converters whose filter resonance drags the phase below −180° can all be conditionally stable. Remedies include clamping or resetting integrators, limiting how fast a reference may change, and compensation that keeps the phase above −180° wherever the gain exceeds 0 dB.

The Nichols Chart

The Nichols chart plots loop-gain magnitude in decibels against phase in degrees as a single curve, with frequency marked along it. Nathaniel B. Nichols developed it for servomechanism design, and it was published in 1947 in Theory of Servomechanisms, volume 25 of the MIT Radiation Laboratory Series. The chart has three useful properties:

• A change in loop gain slides the curve vertically without changing its shape, and a compensator adds its decibels and degrees point by point.
• The critical point −1 becomes the point at 0 dB and −180°. The phase margin is the horizontal distance from the curve's 0 dB crossing to −180°, and the gain margin is the vertical distance below 0 dB where the curve crosses −180°.
• With unity feedback the closed-loop response L/(1 + L) depends on L alone, so the chart carries contours of constant closed-loop magnitude and phase; the highest magnitude contour the curve touches gives the closed-loop peak. With a constant feedback factor β, the contours give the response relative to its ideal value 1/β.

The 0 dB closed-loop contour, where |L| = |1 + L|, is the vertical line through −1/2 on a Nyquist plot, and on the Nichols chart it passes through −6.02 dB at −180°. One exact relation ties margin to peaking. At crossover |L| = 1 and the phase is −180° plus the phase margin PM, so |1 + L| = 2 sin(PM/2), and the closed-loop magnitude at that frequency is

|T(jω_c)| = 1 / [2 sin(PM/2)]

The closed-loop peak can be no lower than this value, so a loop with a 45° margin peaks by at least 2.3 dB. For the position loop above, with its 47.4° margin, the formula gives 1.90 dB, and the computed closed-loop peak is also 1.90 dB, at 38.4 rad/s. The chart does not show encirclements, so a loop with right-half-plane poles still needs the Nyquist count.

Measuring a Loop Response

No model contains every parasitic, so a loop that matters should be measured. Opening a high-gain loop would upset its operating point and saturate it, so instead a small sinusoid is injected at one point with the loop closed, and the loop gain is computed from the voltages on either side of that point.

Worked Example: An Op-Amp Loop

A model op-amp has the open-loop gain

A(s) = 100,000 / [(1 + s/ω_p1)(1 + s/ω_p2)]

with poles at 10 Hz and 2 MHz, so its gain-bandwidth product is 1 MHz, and an open-loop output resistance R_o of 100 Ω. It drives a load capacitance C_L, and a feedback network with an impedance much larger than R_o returns a fraction β of the output. The load adds a pole inside the loop:

L(s) = A(s)β / (1 + sR_oC_L)

Real op-amps have more complicated output impedances, so the numbers illustrate the method rather than any part.

1. Draw the open-loop gain. The asymptote is flat at 100 dB to 10 Hz, falls at 20 dB per decade through 0 dB at 1 MHz, and steepens to 40 dB per decade above 2 MHz.
2. Subtract the noise gain. In decibels the loop gain is the open-loop gain minus the noise gain 1/β, so the crossover lies where the open-loop curve meets the horizontal 1/β line: 0 dB for a follower and 20 dB for a noninverting gain of 10. Without a load, the follower crosses over at 910 kHz with 65.5° of phase margin, and the gain-of-10 amplifier at 99.9 kHz with 87.1°.
3. Add the load. A 10 nF load puts a pole at 1/(2πR_oC_L) = 159 kHz. Above it the follower's asymptote falls at 40 dB per decade, as (1 MHz/f)(159 kHz/f), and crosses 0 dB at √(1 MHz × 159 kHz) = 399 kHz, where the phase leaves a margin of about 10°.
4. Read the exact curve. The crossover is at 380 kHz and the phase margin 12.0°. Because the 10 Hz pole already contributes almost −90°, the phase reaches −180° where arctan(f/2 MHz) + arctan(f/159 kHz) = 90°, at f = √(2 MHz × 159 kHz) = 564 kHz; the magnitude there gives a gain margin of 6.7 dB.

Raising the noise gain moves the crossover below the load pole, which is why an amplifier that rings as a follower can be well behaved at a gain of 10. An isolation resistor R_iso between the output and the capacitor, with feedback taken from the op-amp side, makes the load factor (1 + sR_isoC_L)/[1 + s(R_iso + R_o)C_L]: with 50 Ω the pole drops to 106 kHz, but a zero at 318 kHz restores the phase near crossover. The price is a capacitor voltage that lags outside the loop and, with any DC load current, a drop across R_iso.

The loop also shows an injection point that fails. Between the op-amp output and the load, Z₁ is the 100 Ω output resistance and Z₂ the capacitor, 159 Ω at 100 kHz and 42 Ω at crossover. By Middlebrook's relation, the measured phase at 100 kHz would read −93.0° instead of −125.0°, and the magnitude at 380 kHz would read −5.9 dB instead of 0 dB. At the inverting input the output node drives the op-amp's input capacitance instead; taking that as 5 pF, |Z₁/Z₂| at 380 kHz is below 0.001.

Worked Example: A Buck Converter Voltage Loop

A voltage-mode buck converter makes 3.3 V from 12 V, switching at 500 kHz. Its output filter has L = 4.7 µH and C = 47 µF, the effective value of ceramic capacitors under DC bias, with 3 mΩ of ESR and 20 mΩ of series resistance r in the inductor and switches. The load R is 1.1 Ω (3 A) or 33 Ω (0.1 A), and the PWM ramp V_M is 1 V peak to peak. Well below the switching frequency, the averaged control-to-output transfer function is

G_vc(s) ≈ G₀(1 + s/ω_esr) / [1 + s/(Qω₀) + s²/ω₀²]

with G₀ = V_inR/[V_M(R + r)] = 11.8 (21.4 dB) at full load and Q ≈ 1/[ω₀(L/R + (r + R_ESR)C)], which is 2.8 at full load and 12 at light load. The resonance f₀ is 10.8 kHz at full load, close to 1/(2π√(LC)) = 10.7 kHz. The ESR zero, at 1.13 MHz, lies beyond the range of interest.

A type III compensator, including the feedback divider, has

G_c(s) = (ω_I/s)(1 + s/ω_z1)(1 + s/ω_z2) / (1 + s/ω_p)²

with zeros at 5.4 kHz and 10.8 kHz, a double pole at 250 kHz, half the switching frequency, and ω_I = 12,900 s⁻¹, which sets the crossover at 50 kHz, a tenth of the switching frequency. The error amplifier supplies the loop's inversion. Switching Converter Control covers how such placements are chosen; the task here is to draw and read the loop gain T(s) = G_c(s)G_vc(s).

1. Low frequencies. The integrator and G₀ give |T| ≈ ω_IG₀/ω = 152,000/ω, which is 47.7 dB at 100 Hz and falls at 20 dB per decade.
2. First zero. At 5.4 kHz the asymptote levels off at 152,000/(2π × 5,400) = 4.48, or 13.0 dB.
3. Second zero and resonance. At 10.8 kHz the zero adds +20 dB per decade and the LC pair −40, so the asymptote falls at 20 dB per decade and reaches 0 dB at 10.8 kHz × 4.48 = 48.4 kHz.
4. Double pole. Above 250 kHz the slope steepens to −60 dB per decade.
5. Correct at the resonance. At 10.8 kHz, add 20 log₁₀2.8 = 8.9 dB for the pair, 3.0 dB for the zero at its corner, and 1.0 dB for the zero an octave below, 12.9 dB in all.
6. Sum the phase. The integrator contributes −90°, the zeros add up to 180° of lead, the LC pair removes 180° around 10.8 kHz, and the double pole takes phase away from about 25 kHz upward.

At full load the loop crosses over at 50.0 kHz with a phase margin of 56.2°, and at light load at 50.2 kHz with 52.6°. Above the resonance the power stage's gain approaches G₀ω₀²/ω², independent of load, so the crossover barely moves; the load changes the resonance instead. At light load the phase falls to −146° at 13.5 kHz, where the loop gain is still 22 dB. The phase reaches −180° only near 315 kHz, above half the switching frequency, where the averaged model fails, so a gain margin must come from measurement. At a 6 V input G₀ halves, and the crossover falls to about 29.5 kHz, with margins of 56.1° at full load and 49.6° at light load.

Moving both zeros up an octave, to 10.8 kHz and 21.6 kHz, and retuning ω_I for the same crossover shows how a resonance creates conditional stability. The phase margin drops to 39.0° at full load and 35.5° at light load. At light load the phase also crosses −180° at 12.3 kHz and 15.1 kHz, where the loop gain is 31.9 dB and 22.0 dB. The loop is stable, because both crossings of the negative real axis lie beyond −1 and cancel, but losing between 22 and 32 dB of loop gain would make it unstable.

On the bench, inject through a 10 Ω resistor between the output node and the top of the divider, where the output capacitor's 68 mΩ at 50 kHz faces kilohms, and sweep from about 100 Hz to just below 250 kHz. An analyzer that displays v_y/v_x should read a phase of about 56° at the 50 kHz crossover at full load.

Summary

A Bode plot shows loop gain as magnitude and phase against logarithmic frequency. Its asymptotes add factor by factor and err by 3 dB at a real corner and by 20 log₁₀Q at a complex pair, while a time delay adds phase lag without changing the magnitude. For a minimum-phase loop the magnitude slope implies the phase, which is why a crossover at −20 dB per decade is well damped; right-half-plane poles and zeros and delays break that link.

A Nyquist plot shows the same data as a curve around −1, and Z = N + P turns its encirclements into a count of unstable closed-loop poles. It handles what simple Bode readings miss: open-loop unstable plants, multiple crossovers, and conditionally stable loops. The Nichols chart reads closed-loop peaking directly. On the bench, voltage injection measures the loop while it stays closed, provided the injection point has a low source impedance and a high load impedance and the injected level keeps the loop linear.

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