*A practical measurement approach with the use of a network analyser*

*Knowing the stability criteria’s of a switching power supply and measuring its close loop gain and phase over a certain frequency range helps the designer to understand potential risks of instability of the analysed design. *

*By Werner Berns, Application Design Centre Manager Europe, National Semiconductor, Michele Sclocchi, Principal Application Engineer Power Management Europe, National Semiconductor, Gianpaolo Lisi, University of Salerno, Italy*

This article shows how to use a network analyser to measure the loop stability and gives guidance on how to interpret the results.

### Mathematical model enough?

Beside the fact, that mathematical models of switching power supplies are getting better, they still have some limitations about the accuracy. The main reasons for those limitations are unknown details about the system. For example: component parasitics, PCB layout, temperature effects, propagation delays, non-linearity of all semiconductors. A real measurement almost always differs from the mathematical prediction and the efforts to get good models can be extremely high and time consuming. Therefore, it’s always a good idea to actually measure the transfer function. Of course a mathematic model is very useful to calculate the compensation network (R-C values) before one actually build it and then cross check by doing a real loop measurement and do a fine tune. Just to clarify, it’s not a question of whether to use a mathematical model OR a real loop stability measurement. An expert in power supply designs will always do both.

### The feedback loop

Figure 1 shows a simplified block diagram of a typical switching power supply.

• The power stage is typically the pulse width modulation (PWM) controller with a power switch transistor, output filter (inductor/transformer and 1 or more capacitor connected in parallel).

• The error amplifier controls the output voltage of the power supply by sensing the output voltage and comparing it with a fixed voltage reference.

• The compensation network is G(s) and H(s), which typically is part of the circuitry which the designer can adjust in order to make the power supply stable.

The gain of the error amplifier network is the ratio of the feedback impedance to the input impedance. The voltage adjust resistor (Rfb1) does not effect the AC gain calculation since it does not carry AC current. The feedback loop is the path through the input impedance, the error amplifier, the power stage and back to the input impedance.

### Phase shift and loop gain

A feedback is used in all voltage regulators to keep the output voltage constant. The output voltage is sampled through a resistor divider and that signal is fed into the negative input of the error amplifier. Since the other input of the error amplifier is tied to a reference voltage, the error amplifier will supply current as required to the pass transistor to keep the regulated output at the correct DC voltage.

It is important to note that for a stable loop, negative feedback must be used. The response of the loop will opposes any change at the output. This means that if the output voltage tries to rise (or fall), the loop will respond to force it back to the nominal value.

If a sinusoidal waveform (or noise) is injected into the loop, the signal will go through the loop and it will come back multiplied by the loop gain with a certain lag phase respect to the injected signal. The phase shift is defined as the total amount of phase lag, referred to the starting point of -180° (negative feedback loop) that is introduced into the feedback signal as it goes around the loop. The loop gain is defined as the ratio of the amplitude of the signal that goes through the loop divided by the amplitude of the injected signal: Loop Gain [dB] = 20*log(Va/Vb).

Let's assume we inject a sinusoidal signal into the loop across a wide frequency range. At low frequencies, the signal comes back with a larger amplitude and at high frequencies it is attenuated. All measured values, the gain and the phase, will be recorded. The result of the measurement is the so called bode plot (see example in Figure 2). The shown graphs say a lot about the loop stability and certain points are of special interest.

**Crossover Frequency (f _{c}**)

The point at where the injected signal comes back at the same amplitude (0dB), is called cross over frequency or unity gain frequency.

**Phase Margin (ϕ _{m})**

The phase margin is defined as the difference (in degrees) between the total phase shift of the feedback signal and the -180° measured at the cross over frequency.

**Gain Margin (G _{m})**

The gain margin is the amount of negative gain (attenuation) where the total phase shift is 180°.

### What would we like to achieve?

The compensation network needs to be optimised in order to meet the static and dynamic performance requirements while maintaining stability.

An ideal loop gain should have the following attributes:

- Fast loop response, achieved by a high bandwidth (high cross zero frequency).

- Loop gain slope of 20dB/decade from low frequency to half the switching frequency.

- Large DC gain to achieve high DC accuracy over load and line variations.

- Good noise immunity, with low gain at high frequencies, close to the switching frequency.

- Flat phase curve near crossover frequency.

- Good phase margin in order to have a good stability with minimum overshoot.

The bandwidth of the control loop determines the speed of the loop in responding to a transient condition. Higher crossovers are preferred but there are practical limitations.

As a rule of thumb, 1/10 of the switching frequency is a good starting point. But more than 1/5 is not recommended. The higher the bandwidth is, the faster the load transient response speed will potentially be. However, if the duty cycle saturates during load transient, further increasing the small signal bandwidth will not help. There are also other practical limitations that depends on the type of control loop used, and topology, for example, in voltage mode control, the LC filter will ring at the resonance frequency, the control must eliminate this by having a reasonable gain at the resonance frequency.

In flyback and boost topology the main limitation is the RHP zero, where the cross over frequency has to be 1/3 lower than the RHP zero frequency. Also, if the main performance is step load, there are no practical benefits in raising the cross over frequency above the output capacitor ESR cross over limits ( fc < 1/ 2 πCR_{c}).

Sufficient phase margin is required to prevent oscillations. The step response can be seen in a second order system where the damping factor is ζ≈ϕ_{m} /100. Optimal phase margin is at 52° (blue graph). Lower phase margin leads to under damped system response (red graph) and higher phase margin leads to over damped system response (green graph).

As mentioned before, there are two main parameters that give a figure of merit of how stable the system is: Phase margin and Gain Margin. In theory, even 20° of phase margin in the worst case scenario could be enough for a stable design, however more degrees of margin will ensure a stable loop in any conditions. When we specify and measure the phase margin we also have to consider how much it will degrade in the worst conditions, since line load and temperature changes will tend to degrade the phase margin from nominal value.

It is also important to monitor the minimum phase shift at frequency below the cross over frequency. If the phase shift gets close to 0, the system can oscillate when the gain decreases, for example with an increase of load or decrease of line voltage.

### Implementation of a close loop measurement

A network analyser is used for this measurement, by injecting the sinusoidal output signal into the control loop, with a sweep frequency from few tens of hertz to above the operating switching frequency and measuring two signals A and B as shown in Figure 4.

The sinusoidal signal is injected through an isolation transformer. In order to not distort the close loop system too much, the injected signal has an amplitude of few tens to hundred of mV. A resistor is connected in parallel with the output of the transformer. The two output channels of the network analyzer are connected at the connection points of the transformer to measure the loop input (ch B) and loop output (ch A). Gain and phase of the function chA/chB is plotted in a logarithm scale over frequency.

The isolation transformer is needed to ensure a floating sinusoidal voltage injected to the feedback loop across an inserted resistor of few ohms. The resistor connected in parallel with the output of the transformer is in series with the feedback resistor divider and should have a resistance value much lower than the feedback resistors in order to not change the DC output voltage. The transformer should have low primary to secondary capacitance and flat frequency response. Transformers designed for this purpose are available in the market and are typically sold for few hundred Euros. However, a simple transformer can be self-made by winding two strands of wires in a toroid core. Figure 5 shows an example of a network analyser screen shot.

Yellow graph represents the gain and the green graph represents the phase. The red line marks 0dB and 0° and the scale it 10dB respectively 45° per division. The switching frequency in this example was 1.5MHz.

### Reading the results, we get the following values:

The crossover frequency is 94,8kHz (marker 1) and the phase margin is about 60°.

The gain margin is about 16dB and the gain at the switching frequency is below -40dB (good noise immunity).