Feedback Loops

Introduction

Some of the most important design criteria in dc/dc power conversion design are the calculations and methodologies involved in the feedback loop compensation. If the feedback loop parameters are not properly calculated, the converter can exhibit instability or regulation failure.

The function of a feedback loop in a DC/DC converter is to maintain the output at a fixed value which is dependent on a reference value only – i.e. it is independent of load, input voltage or environmental variations. This sounds simple and is relatively simple for static or slowly changing conditions, but to handle dynamic or step changes the feedback loop design becomes very complex. One of the most important compromises that has to be made is the balance between a smooth output during static operating conditions ( low jitter, small deadband and good accuracy) and the response to dynamic operating conditions ( fast reaction time, quick settling time and low overshoot). In addition, the control loop must be stable under all operating conditions, including low load or even no load conditions. The feedback loop design is therefore one of the main factors defining the overall performance of the converter.

Open Loop Design

Not all DC/DC converters use feedback. The basic Royer relaxation oscillator used in the example shown in Figure 1.30 has no feedback network. The self-oscillating circuit runs at a frequency which is determined by the physical characteristics of the transformer and the input voltage only, according to the following relationship:



Where NP is the number of primary turns, B is the saturation flux and AE is the cross sectional area of the transformer. The formula can be rewritten to give the free-running frequency, ƒ:



The factor “4” differs from the standard transformer equation which uses “4.44” because the Royer oscillator generates a square wave and not a sinusoidal signal. The output voltage is dependent upon the turns ratio of the number of turns on the primary winding, NP, to the number of turns on the secondary winding, N_S:



From these relationships, we can see that both the output voltage and operating frequency are not fixed and are dependent on the input voltage. Therefore unregulated DC/DC converters should ideally only be used with regulated input voltages.

In practice, there are “hidden” feedback mechanisms that improve the performance of Royer oscillators above what the theory predicts. The primary, secondary and feedback windings all exhibit an interaction with each other due to leakage inductances and coupling capacitances. The windings can be arranged on the core to increase or decrease these interactions or even to shield one winding from the influence of another. For example, unregulated converters can be made short-circuit proof by winding the secondary between the primary and feedback windings so that the short-circuited output turns form a shield which reduces the coupling from primary to secondary. The converter continues to oscillate when the output is shorted, but at a much reduced power that the switching components can tolerate. The unregulated converter will run hotter into a dead short, but it will survive. As soon as the short circuit is removed, the converter will revert to its normal operating mode with full power.

Closed Loops

The dependence of the output on the input voltage can be removed by using a feedback loop. Essentially, a feedback path is provided to an error amplifier which compares the actual output with the desired output and corrects the output to bring it into line. As the correction is always in the opposite direction to the error (if the output is too high, reduce it, - if the output is too low, increase it), the feedback is said to be “Negative”. If the feedback loop is “Positive” then any errors will be amplified and the output will either oscillate or rapidly go to the minimum or maximum level. Ensuring that positive feedback conditions do not arise under transient operating conditions is one of the most challenging aspects of the loop design.

The beauty of feedback is that changes of input voltage will be compensated for as well as any changes in the output voltage due to changes in the load. The same feedback loop corrects for both situations. Another advantage of closed feedback loops is that the input and output do not have to have the same units. A feedback loop can be used to make a constant current output from a variable input voltage supply. The error amplifier simply adjusts the output according to a feedback signal based on the output current rather than on the output voltage (in effect, it becomes a transconductance amplifier instead of a voltage amplifier). To analyse the feedback design, let us take a simple non-isolated buck regulator. A typical circuit diagram could be:



In terms of functional blocks, Figure 2.1 can be reduced to:



Each functional block will have a gain, K. The power switching elements (FETs) will have gain of K_PWR, the output filter formed from L₁ and C₁ with K_FILT(S), the feedback element (the resistor divider formed from R₁ and R₂ ) will have gain K_FB. The resulting feedback signal is compared with the reference voltage, V_REF at the summing point and the error amplified by the error amplifier A₁ with gain K_EA(S) to control the P_WM modulator which has a gain K_MOD. Some of these gain blocks will have a high amplification and some will attenuate the signal, but overall the open loop gain (the sum of all of the gains) will be positive and typically be around 1000.



The simple circuit shown in Fig 2.1 will have resonances (poles) caused by the LC output filter at the frequency:



And an additional resonance (zero) caused by the capacitor’s ESR:



At frequencies above ƒ_PO, the gain decreases at a rate of -40dB/decade due to the second order LC characteristic of the output filter. The point at which it reaches unity (0dB gain) is the crossover frequency, ƒ_C. At the frequency ƒ_ZO, the effect of the first order RC filter due to the ESR of the filter capacitor changes the gain slope to -20dB/decade. A plot of the normalised gain against frequency shows that the slope and phase change with frequency:



The phase plot is the phase change additional to the 180° caused by the inverting input f the error amplifier, A1.

As we can see from the phase plot, the circuit is unstable at the crossover frequency as the phase change is -180° or -360° in total. This will cause the converter to veer into the positive feedback region and the output will start to ring or break into oscillation.

By increasing the gain in the error amplifier stage, the frequency where the overall gain equals unity can be moved to a safer region. The phase margin (the difference between the overall phase and -180° at the system ƒ_C) and the gain margin (the system gain at -180° phase) define how stable the feedback loop is (Fig. 2.4).

Feedback Loop Compensation

The further away the chosen system crossover frequency from the power stage cross-over frequency is, the more stable the output (it has better gain and phase margins), but the slower the transient response. A phase margin of approximately 45° provides for good response with a little overshoot, but no ringing.

Besides simply increasing the error amplifier gain at all operating frequencies to move the system corner frequency into a safe area, the error amplifier phase shift can be made frequency dependent by adding compensation to the op-amp feedback:



The compensation component values can be chosen so that the phase reverses and adds to the phase margin at the critical crossover frequency, thus increasing the stability. This allows the output filter to be less heavily damped, thus accelerating the DC/DC converter’s reaction to transients without risking excessive overshoot or oscillation.

Right Half Plane Instability

In topologies that drive the output inductor with a continuous current via a diode, such as boost, buck/boost, flyback and forward converters, the conduction time of the diode adds a delay into the feedback loop. If the load suddenly increases, the duty cycle has to be temporarily increased to transfer more energy into the inductor. However, a high duty cycle give less time (t_OFF) for the diode to conduct, so the average diode current during t_OFF actually decreases (right Fig. 2.8). As the output current is supplied through the diode, the output current also decreases. This condition remains until the average inductor current slowly rises and the diode current reaches its correct value.



This phenomenon, where the diode current must first decrease before it can increase, is known as the Right Half Plane (RHP) instability, because the output current is temporarily 180° out of phase with the duty cycle. For example in a simple boost converter (Fig. 1.13), a temporary additional zero occurs according to:



The RHP instability is almost impossible to compensate for, especially as the zero changes with the load current. The solution is to design the feedback loop with a cross-over frequency well below the lowest frequency where RHP zeroes arise (this has the disadvantage of reducing the DC/DC converter’s reaction time to step load changes) to use a buck/boost converter in discontinuous mode (DCM) to eliminate the problem altogether.

Slope Compensation

One further potential cause of feedback loop instability is sub-harmonic or bifurcate instability. The root cause is the PWM comparator that compares the feedback voltage level with the timing saw-tooth voltage ramp (refer back to the block diagram 1.40). The problem can occur because the energy in the inductors are not completely discharged with each switching cycle so that current flows back into the feedback circuit at the wrong time or simply due to switching noise on the comparators inputs. The effect is the same: the PWM modulator generates a bifurcated or double-beat.



The solution to the sub-harmonic instability problem is called slope compensation. An artificial ramp waveform (usually derived from the slope of the inductor current or sometimes directly from the timing capacitor voltage) is added to the feedback voltage to avoid false triggering or re-triggering of the PWM comparator.

Analyzing Loop Stability in Analogue and Digital Feedback Systems

Finding Analogue Loop Stability Experimentally

It is possible to determine the feedback loop stability experimentally using a Bode Plot apparatus. A sine wave generator can be used with an audio transformer to inject a disturbance signal into the control loop. The frequency of the sine wave is ramped up until the disturbance on the output is as large as the disturbing signal. The gain is then unity and thus the disturbing frequency must be ƒC, the corner frequency of the feedback loop. The phase difference between the disturbing signal and the output is the phase margin. By further increasing the frequency until the phase difference is -180°, the gain margin can be found.

Finding Analogue Loop Stability using the Laplace Transform

An alternative to the experimental method is to derive the poles and zeros mathematically. To do this, we need to know the transfer function of the converter.

For the simple buck converter shown in Fig.2.1, the transfer function is:



The letter ‘s’ indicates that the variable has a frequency dependence. The transfer function can be solved using the Laplace Transform (LT), but to understand the LT, first we need to consider the Fourier Transform.

The Fourier Transform (FT) is a special form of the LT. Fourier determined that any periodic signal is the sum of sinusoidal signals of various frequency, phase and amplitude (the Fourier Series). The transform shifts from the time domain to the frequency domain (and vice versa). The result of a Fourier Transform on a periodic signal is the equivalent Fourier Series or spectrum. Figure 2.12 shows graphically the first six harmonics of a square wave:



The Fourier transform is an intergral function from minus infinity to plus infinity, which can be written as:



Mapped onto the S-domain, the variable of the FT is s = jω. The results are imaginary variables only.

The Laplace Transform is a superset of the FT. The variable of LT is in the complex plane. The integration starts at zero instead of -∞. This means that it can also be used to analyse step or semi-infinite signals, such as a pulse or exponential decaying series. The Laplace transform can be written as:



Mapped onto the S-domain, the variable of the LT is s = σ + jω. Using the LT, it is possible to simulate the feedback loop mathematically and generate a pole-zero plot in the s-plane. The vertical axis is imaginary, the horizontal axis real. The higher up or down the imaginary axis one travels, the faster the oscillations occur. The further in the negative real axis one travels, the faster the decay and the further in the real positive axis, the faster the growth.



Zeroes always lie on the real axis. Complex conjugate pole pairs in the left-half of the s-plane combine to generate a response that is a decaying sinusoid of the form Ae^-σtsin (ωt + Ø), where A and Ø are initial conditions, σ is the rate of decay and ω is the frequency in radians/ seconds.

A pole pair that lies on the imaginary axis, ±jω (i.e, with no real component) generates an oscillation with constant amplitude.

The distance of a Pole from the origin, 0, indicates how damped the response is: the closer to the origin, the slower the rate of decay. If a Pole lies on the origin, it means that the system is operating at DC.

If a Pole lies on the right hand plane, the system is unstable (this is the origin of the term RHP instability described in section 2.4.1)

Finding Digital Loop Stability using the Bilinear Transform

If a digital signal processor is used to generate the feedback loop compensation, the stability of the digital loop can be derived from the Laplace Transform using a further transform to correct for the sampling rate.

As the input signal to a digital system is no longer time-continuous, the s-plane values need to be transformed into the z-plane discrete-time values using a bilinear transform (Tustin’s Method).

The result of this mapping is that the stable region of z-plane becomes a circle with radius = 1 (unit circle).



The far right edge of the circle (ω = 0) represents DC. The far left edge of the circle represents the aliasing frequency. Any poles that lie outside the circle will be unstable. The poles of the feedback loop can now be plotted in the z-plane, the positions of poles represent responses normalised to the sample rate rather than to continuous time as in the s-plane.

Digital compensation firstly assumes the DSP’s sampling frequency is much greater than the system crossover frequency, so that any simulations are accurate. Then there are two common approaches to find the compensation values: re-design and direct design. With digital re-design, a linear model of a switching converter is established and the feedback loop compensation is simulated conventionally in the s-domain. The results of the analogue compensation are mapped into z-domain to complete the digital compensation design. For the direct design approach, a discrete model of switching power converter is wholly simulated and the compensation design is calculated directly in the z-domain. This requires accurate models of the analogue parts using such programs as Spice^TM or Matlab^TM.

The result of either method is the same: a matrix of values stored in a look-up table. The DSP or μC will then take the digitised input signal, enter it into the computational matrix and output the resultant value either as an analogue control signal or, more often, as a direct PWM drive output. In the latter case, the comparator and PWM circuits will also be synthesised digitally. This eliminates the analogue control loop errors arising from slope compensation and RHP instability. If a different feedback compensation response is required to handle a different mode of operation, the digital controller can smoothly switch between look-up tables without resetting any of the output values, an ability that analogue controllers cannot match. Thus fewer compromises need be made in selecting the compensation characteristics.

It is this lack of compromise and ability to switch rapidly between fast transient response or a stable output that makes digital feedback loop so attractive. As the cost of micro- controllers continues to fall, more and more DC/DC converters will migrate towards fully digital or hybrid feedback loop controllers.

Digital Feedback Loop

The circuit above (Fig. 2.15) shows a simplified microcontroller-based DC/DC converter. All timing is under digital control - both the full-bridge power stage and the synchronous rectification on the output.

The microcontroller has integrated op-amp elements on-board. This means that the sense inputs can be connected directly to the microcontroller. As the microcontroller has information regarding input voltage, output voltage and output current, there is no need for external circuits to monitor short circuits or overload conditions. The input voltage monitoring allows both a controlled start-up and a programmable under-voltage lockout with adaptive hysteresis. The fourth op-amp input is used to monitor overtemperature conditions, either within the DC/ DC converter or at the remote load. The consequences of an over-temperature condition are programmable according to the required performance in the application, for example, shutdown and latch-off, shut-down and auto-restart after cooling down or power-limiting to reduce heat dissipation.

The external data connection allows operating conditions to be updated on-the-fly or various pre-programmed performance options to be selected. In addition, the bidirectional communication bus allows fault reporting and status updates.



Figure 2.16 above shows the internal state machine diagrammatically. The various controller subroutines use matrix look-up tables to calculate the correct response in real-time.



According to the operational conditions, the system controller routine can switch in or out different matrix tables.The advantage of a digital controller is also a much reduced BOM as well as intelligent control of output voltage and current.