9.1.2 Sub-Harmonic Oscillation

Peak current mode regulator can exhibit unstable behavior when operating above 50% duty cycle. This behavior is known as sub-harmonic oscillation and is characterized by alternating wide and narrow pulses at the SW pin. Sub-harmonic oscillation can be prevented by adding an additional slope voltage ramp (slope compensation) on top of the sensed inductor current. By choosing K ≥ 0.82~1, the sub-harmonic oscillation is eliminated even with wide varying input voltage.

In time-domain analysis, the steady-state inductor current starting from an initial point returns to the same point. When the amplitude of an end cycle current error (dI₁) caused by an initial perturbation (dI₀) is less than the amplitude of dI₀ or dI₁/dI₀ > –1, the perturbation naturally disappears after a few cycles. When dl₁/dl₀ < –1, the initial perturbation no longer disappear, it results in sub-harmonic oscillation in steady-state.

Figure 27. Effect of Initial Perturbation when dl₁/dl₀ < –1

dI₁/dI₀ can be calculated as:

Equation 19.

The relationship between dI₁/dI₀ and K factor is illustrated graphically in Figure 28.

Figure 28. dl₁/dl₀ vs K Factor

The absolute minimum value of K is 0.5. When K < 0.5, the amplitude of dl₁ is greater than the amplitude of dl₀ and any initial perturbation results in sub-harmonic oscillation. If K = 1, any initial perturbation is removed in one switching cycle. This is known as one-cycle damping. When –1 < dl₁/dl₀ < 0, any initial perturbationis under-damped. Any perturbation is over-damped when 0 < dl₁/dl₀ < 1.

In the frequency-domain, Q, the quality factor of sampling gain term in modulator transfer function, is used to predict the tendency for sub-harmonic oscillation, which is defined as:

Equation 20.

The relationship between Q and K factor is shown in Figure 29.

Figure 29. Sampling Gain Q vs K Factor

The recommended absolute minimum value of K is 0.5. High gain peaking when K is less than 0.5 results sub-harmonic oscillation at f_SW/2. A higher value of K factor may introduce additional phase shift near the crossover frequency, but has the benefit of reducing noise susceptibility in current loop. The maximum allowable value of K factor can be calculated by the maximum crossover frequency equation in frequency analysis formulas in Table 2.

Table 2. Boost Regulator Frequency Analysis

	SIMPLIFIED FORMULA	COMPREHENSIVE FORMULA(1)
MODULATOR TRANSER FUNCTION
Modulator DC gain (2)
RHP zero (2)
ESR zero
ESR pole	Not considered
Dominant load pole
Sampled gain inductor pole	Not considered	or
Quality factor	Not considered
Sub-harmonic double pole	Not considered	or
K factor	K = 1
FEEDBACK TRANSFER FUNCTION
Feedback DC gain
Mid-band Gain
Low frequency zero
High frequency pole
OPEN LOOP RESPONSE
Crossover frequency (3) (Open loop band width)		Use graphic tool
Maximum cross over frequency(4)		or , whichever is smaller

(1) Comprehensive equation includes an inductor pole and a gain peaking at f_SW / 2, which is caused by sampling effect of the current mode control. Also, it assumes that a ceramic capacitor C_OUT2 (No ESR) is connected in parallel with C_OUT1. R_ESR1 represents ESR of C_OUT1.

(2) With multiphase configuration, , , , and C_OUT = C_OUT of each phase x n, where n = number of phases. As is the current sense amplifier gain.

(3) Assuming , , , , and .

(4) The frequency at which 45° phase shift occurs in modulator phase characteristics.