Lead-Lag Compensator Calculator

Compensator Transfer Function (pole-zero and time-constant forms)

Gc(s) = Kc · (s + 1/T) / (s + 1/(α·T)) | equivalently | Gc(s) = Kc·α · (T·s + 1) / (α·T·s + 1) | lead: α < 1 | lag: α > 1 | zero at s = −1/T | pole at s = −1/(α·T)

Variables: Gc = compensator transfer function (dimensionless gain ratio), s = Laplace variable (rad/s), Kc = compensator gain, the high-frequency gain Gc(∞) (dimensionless), T = compensator time constant (s), α = the lead/lag ratio, alpha (dimensionless) — α < 1 for a phase-LEAD network, α > 1 for a phase-LAG network, α·T = time constant of the second corner (s), 1/T = zero corner frequency (rad/s), 1/(α·T) = pole corner frequency (rad/s). The two forms are algebraically identical: Kc·α·(Ts+1)/(αTs+1) = Kc·(s+1/T)/(s+1/(αT)). In the pole-zero form the DC gain is Gc(0) = Kc·α and the high-frequency gain is Gc(∞) = Kc, so a lead network written this way ATTENUATES at DC unless Kc·α is set to the loop gain you want to keep. For lead the zero is nearer the origin than the pole (1/T < 1/(αT)); for lag the order reverses. Source: Ogata, Modern Control Engineering, 5th ed., Ch. 6 (Control Systems Design by the Root-Locus Method) and Ch. 7 (Design by the Frequency-Response Method) — lead, lag, and lag-lead compensation techniques

Maximum Phase Shift and Alpha

sin(φₘ) = (1 − α) / (1 + α) | → | α = (1 − sin φₘ) / (1 + sin φₘ) = tan²(45° − φₘ/2) | 0° < |φₘ| < 90°

Variables: φₘ = maximum (peak) phase shift contributed by the compensator (degrees) — POSITIVE for lead, NEGATIVE for lag; α = lead/lag ratio (dimensionless); sin/tan take their arguments in degrees here (convert with rad = deg·π/180 before calling a radian-based library). The tan²(45° − φₘ/2) form is the exact algebraic inverse and gives the standard hand-checkable values: φₘ = 30° → α = 1/3; φₘ = 45° → α = 0.171573; φₘ = 60° → α = 0.071797; α = 0.1 → φₘ = 54.90°. The relation is singular at |φₘ| = 90° (α → 0 or ∞), so ONE stage cannot deliver ±90°; about 60–65° is the practical single-stage ceiling and α below roughly 0.05 is not physically realizable with a passive network. In frequency-response design, add 5–12° to the phase you actually need, because inserting the lead network raises the gain-crossover frequency where the plant contributes more phase lag. Source: Ogata, Modern Control Engineering, 5th ed., Ch. 7 (Design by the Frequency-Response Method), lead compensation design procedure; Nise, Control Systems Engineering, 7th ed., Ch. 11 (Design via Frequency Response)

Frequency of Maximum Phase and the Corner Frequencies

ωₘ = 1 / (T·√α) | → | T = 1 / (ωₘ·√α) | zero: 1/T = ωₘ·√α | pole: 1/(α·T) = ωₘ/√α | ωₘ = √( (1/T) · (1/(α·T)) ) | log ωₘ = ½·[ log(1/T) + log(1/(α·T)) ]

Variables: ωₘ = frequency at which the phase shift peaks at φₘ (rad/s) — the design input, i.e. the frequency where the extra phase is needed (normally the intended gain-crossover frequency of the compensated loop); T = compensator time constant (s); α = lead/lag ratio (dimensionless); 1/T = zero corner frequency (rad/s); 1/(α·T) = pole corner frequency (rad/s). ωₘ is the GEOMETRIC mean of the two corners, i.e. the arithmetic midpoint on a logarithmic (Bode) frequency axis — this is why the phase bump is symmetric on a Bode plot. Worked check: φₘ = 45° at ωₘ = 10 rad/s gives √α = 0.414214, T = 0.241421 s, α·T = 0.041421 s, zero at s = −4.142136 rad/s, pole at s = −24.142136 rad/s, and (4.142136)(24.142136) = 100 = ωₘ². Convert to Hz with f = ω/(2π). Source: Ogata, Modern Control Engineering, 5th ed., Ch. 7 (Design by the Frequency-Response Method) — the maximum phase-lead angle occurs at the geometric mean of the two corner frequencies

Magnitude at ωₘ (dB) and the Frequency Response

|Gc(jω)| = Kc · √(ω² + (1/T)²) / √(ω² + (1/(α·T))²) | |Gc(jω)|_dB = 20 · log₁₀ |Gc(jω)| | at ω = ωₘ: | |Gc(jωₘ)| / Gc(0) = 1/√α | boost_dB = 20 · log₁₀(1/√α) = −10 · log₁₀(α) | |Gc(jωₘ)| = Kc·√α | ∠Gc(jω) = [ arctan(ω·T) − arctan(ω·α·T) ] · 180/π

Variables: |Gc(jω)| = compensator magnitude, an amplitude (not power) ratio, so decibels are 20·log₁₀, NEVER 10·log₁₀ (dimensionless before conversion, dB after); ω = angular frequency (rad/s); Kc = compensator gain (dimensionless); T, α·T = the two time constants (s); Gc(0) = Kc·α = DC gain (dimensionless); 1/√α = the gain at ωₘ RELATIVE to the DC gain (dimensionless) — this is the quantity textbooks quote, e.g. φₘ = 45° → α = 0.171573 → 1/√α = 2.414214 = 7.6555 dB, φₘ = 30° → √3 = 4.7712 dB, φₘ = 60° → 2+√3 = 11.4390 dB, α = 0.1 → √10 = exactly 10.00 dB; boost_dB = that same figure in decibels, positive for lead and negative (attenuation) for lag; ∠Gc = compensator phase (degrees) — a difference of two principal-branch arctangents with strictly positive arguments, so it is inherently continuous and confined to (−90°, +90°): there is no atan2 branch cut and nothing to unwrap, and the peak is exactly φₘ at ω = ωₘ. In frequency-response lead design the 1/√α boost is precisely what shifts the loop's 0 dB crossing to ωₘ, which is why the required φₘ must include the 5–12° allowance noted above. Source: Ogata, Modern Control Engineering, 5th ed., Ch. 7 (Design by the Frequency-Response Method) — magnitude of the lead network at ωₘ is 1/√α; Nise, Control Systems Engineering, 7th ed., Ch. 11 (Design via Frequency Response)