要約
Input and output noise in buck (stepdown) converters can concern the system designer. This application note provides a theoretical explanation of the individual contributions of conducted noise on the input and output sides of buck converters. These equations will allow the powersupply designer to optimize components for noise immunity.
Introduction
This application note gives a brief explanation of the individual contributions of conducted noise on the input and output sides of buck (stepdown) converters. Equations are listed to help calculate the peaktopeak noise contributions of each noise source. These equations will allow the designer to select components that meet their design objectives.
Assumptions
The equations listed in this application note are appropriate for any buck converter, independent of the control scheme, as long as the converter meets the following limitations:

The equations assume continuous conduction above 0A.

The load is a constant DC load.

The converter is 100% efficient.
This assumption describes the current flow in the inductor. Depending on the components used and on the load and the topology of the buck converter, the current flow through the inductor can look like that shown in Figure 1. The middle graph of Figure 1 (Graph B) shows discontinuous current flow. Discontinuous current flow occurs when the current through the inductor tries to go negative (i.e., current flows out of the output capacitor and into ground), but gets clamped at zero. Discontinuous conduction typically, but not always, occurs under lightload conditions.
The bottom graph of Figure 1 (Graph C) shows an example of continuous conduction where the current through the inductor drops below 0A. For the current to drop below 0A, the converter must have synchronous rectification and be designed to allow for that type of operation. With such a converter the current will typically, but not always, go below 0A under lightload conditions.
Whether the converter is synchronously rectified or not, continuous conduction above 0A will tend to happen at higher loads, as shown in the top graph in Figure 1. Thus, the equations listed in this application note should be appropriate when the converter is operating close to maximum loading.
Use Equation 1 below to test a particular design. If this inequality is met, the converter is operating in continuous conduction above 0A and the first assumption is satisfied.
V_{IN} = the buck regulator's input voltage
V_{OUT} = the buck regulator's output voltage
f = the frequency at which the buck regulator is switching
L = the value of the inductor
I_{OUT} = the output current
This assumption needs to be made for practical reasons. While assumption 1 is a reasonable assumption for most applications, this second assumption is most likely not true. In a typical buckconverter application, the current draw of the load will fluctuate. This fluctuation will cause noise, due to the finite response time of the converter and the equivalent series resistance (ESR) of the output capacitor (explained later), among other things. To calculate the output noise due to this fluctuation, the characteristics of the load need to be known. That is, of course, impossible for this application note.
This again is a practical consideration. The equations would be unwieldy if this assumption was not made. Most modern buck converters have excellent efficiency and the amount of error introduced by this assumption is minimal.
Background
Some of the noise sources in buck converters come from the nonideal properties of the components used. To better understand these noise sources, a brief discussion of the nonideal aspects of components follows. To keep this discussion simple, only the nonideal properties relative to buck converters are discussed.
Model for MOSFETs
When discussing a buck converter, a firstorder model for a MOSFET is a switch in series with a resistance, as shown in Figure 2. In this model, the MOSFET has a finite resistance between its drain and its source when the MOSFET is turned on. Typical R_{ON} (on resistances) can vary widely depending on the MOSFET and the application. As a point of reference, the IRF7303 used on the MAX1653EVKIT (evaluation kit) has a typical R_{ON} of 6.08Ω (V_{GS} = 4.5V).
A more complete model for a MOSFET is shown in Figure 3.
This model adds inductances and capacitances to the firstorder model in Figure 2. The three capacitors shown are:
 C_{GD} = Capacitance from the gate to the drain
 C_{GS} = Capacitance from the gate to the source
 C_{DS} = Capacitance from the drain to the source
Again, using the IRF7303 as an example, typical values for these capacitances are listed below (V_{DS} = 8V):
 C_{GD} = 120pF
 C_{GS} = 470pF
 C_{DS} = 180pF
In addition to these capacitances, both the drain and the source of the MOSFET have equivalent internal series inductances, L_{D} and L_{S}. The drain and source inductances are listed in the data sheet of the IRF7303 as 4nH and 6nH, respectively.
Model for Capacitors
The model for a capacitor is shown in Figure 4.
All capacitors have what looks like a resistor in series with its capacitance. This resistor is referred to as the equivalent series resistance (ESR) of the capacitor. ESR varies significantly with the type of capacitor used. Ceramic capacitors typically have very low ESR; the average aluminum electrolytic capacitor can have high values of ESR; and tantalums capacitor EST values fall somewhere in between. In recent years, there have been a number of specialty capacitors designed to minimize ESR. Below is a listing of some examples:
 AVX TPS series. Example: 10V, 100µF capacitor with an ESR as low as 0.065Ω at 100kHz
 Kemet® A700 series. Example: 6.3V, 100µF capacitor with an ESR as low as 0.03Ω (max) at 100kHz
 Panasonic® ECJ series. Example: 16V, 4.7µF capacitor with an ESR as low as 0.017Ω (max) at 100kHz
Note that the ESR of a capacitor varies with frequency.
Also in this model is the equivalent series inductance (ESL) of the capacitor. Some typical inductances for ceramic chip capacitors are listed below:
 0.1µF at 1.7nH
 0.01µF at .84nH
 1000pF at 1.7nH
Values for tantalum and aluminum capacitors are more difficult to specify, but are probably on the order of 5nH to 10nH for a good highvalue, surfacemount device.
Model for a ReversedBias Diode
The catch diode used on some buck converters can be modeled as an inductance in series with a capacitor when reversed biased. The MBR030 used on the MAX1653 evaluation board has a capacitance of approximately 55pF when reversed biased with 8V. This capacitance varies significantly with bias voltage. The inductance is not listed in the data sheet, but is probably on the order of 1nH to 2nH.
Model for CircuitBoard Traces
Even the traces on circuit boards are not perfect. For our discussion, we will model the traces on the circuit boards as inductances.
Output Noise
Figure 6 shows a typical buck converter, with Figure 7 showing some typical waveforms for that converter. Graph D in Figure 7, labeled I(C_{OUT}), shows the current flowing into and out of the output capacitor. Note that the net DC current is zero (because it is a capacitor), but that there is AC current flowing.
This AC current working against the output capacitor's finite capacitance and ESR generates the output noise.
The first noise source we will discuss is finite output capacitance. Assuming for the moment that the output capacitor is ideal and that there is no ESR, the voltage on the capacitor is given by the ageold capacitor equation, V = I/C∫idt. With a little math, this formula can be represented as Equation 2. Equation 2 calculates the peaktopeak output noise due to a finite output capacitance. It is interesting to note that the output noise is independent of the load current.
V_{IN} = the buck regulator's input voltage
V_{OUT} = the buck regulator's output voltage
f = the frequency at which the buck regulator is switching
C_{OUT} = the capcitance of the output capacitor
L = the value of the inductor
R_{COUTESR} = the ESR of the output capacitor
V_{NOUTESR} = the peaktopeak noise voltage due to the finite capacitance of the output capacitor
Figure 8 shows the output waveform measured using a MAX1653EVKIT with highquality ceramic capacitors that have negligible ESR. The output ripple that looks like a sin wave is due mostly to finite output capacitance.
Minimizing this source of noise is academically simple: just increase the value of the output capacitance. If there were no ESR, this would truly be the case. However, choosing a larger capacitor can, under some circumstances, actually increase the output noise if the ESR of the capacitor is larger.
Output Capacitor ESR
The contribution to the noise from the ESR of the output capacitor is easy to calculate. The peaktopeak noise is given by V_{PP} = di × R_{ESR}, where di is the peaktopeak AC current going into the capacitor. Again, with a little mathematics, this equation can be expressed as shown in Equation 3. This equation gives the peaktopeak noise due solely to the output capacitor's ESR.
V_{IN} = the buck regulator's input voltage
V_{OUT} = the buck regulator's output voltage
f = the frequency at which the buck regulator is switching
L = the value of the inductor
R_{COUTESR} = the ESR of the output capacitor
V_{NOUTESR} = the peaktopeak noise voltage due to the ESR of the output capacitor
Choosing the correct output capacitor is critical for a lownoise design. The correct capacitor balances cost, low ESR, and high capacitance.
Input Noise
Many times engineers calculate the output noise, while ignoring the input noise. Buck converters will also cause conducted noise to be "injected" onto the input supply. For example, in the case of a 5V to 3.3V converter, the buck converter will cause noise on the 5V supply as well as generate noise on the 3.3V output. If the 5V supply has noisesensitive components powered from it, this injected noise could be important. The section below describes what causes this noise. There are basically three major contributors: finite input capacitance, the ESR of the input capacitor, and ringing caused by the stray inductance and stray capacitance in the circuit.
Input Capacitor Capacitance
The current waveform that the input capacitor "sees" is shown in Graph B of Figure 7. This AC current works against the input capacitor's finite capacitance to create noise on the input supply. This noise is calculated in the same way as the noise on the output due to finite capacitance. Again, we start with the basic capacitor equation, V = I/C∫idt. Putting this equation into a form we can use yields the equations listed below (Equation 4). When making this calculation, be sure to include all the capacitances on the input supply, and not just the capacitance physically located next to the buck converter.
V_{IN} = the buck regulator's input voltage
V_{OUT} = the buck regulator's output voltage
f = the frequency at which the buck regulator is switching
L = the value of the inductor
I_{OUT} = the output current
V_{NINCIN }= the peaktopeak noise voltage due to the finite capacitance of the input capacitor
C_{IN} = the capacitance of the input capacitor
Figure 9 shows a typical waveform of the noise from finite input capacitance. This plot was taken with highquality ceramic capacitors that minimize any effect due to capacitor ESR. The large, highfrequency noise is unrelated to the finite input capacitance and is caused by the capacitor's ESL. This is discussed further under the section Input Capacitor's ESL.
Input Capacitor ESR
The contribution to the noise by the input capacitor's ESR is calculated using the same method used for the output capacitor. This peaktopeak noise is given by V_{PP}  di × R_{ESR}, where di is the peaktopeak AC current going into the capacitor. Equation 5 is the same basic formula, but with variables that are directly related to the circuit. Figure 10 shows a typical waveform generated from an input capacitor's ESR. Again, the large highfrequency noise is due to the stray inductance and capacitance in the circuit; it should be ignored in the discussion of ESR noise. Noise from stray inductance will be discussed later.
V_{IN} = the buck regulator's input voltage
V_{OUT} = the buck regulator's output voltage
f = the frequency at which the buck regulator is switching
L = the value of the inductor
I_{OUT} = the DC output current
R_{CINESR} = the input capacitor's ESR
V_{NINESR} = the peaktopeak noise voltage from the ESR of the input capacitor
Input Capacitance ESL
Figure 10 shows the noise on the input of a MAX1653EVKIT. Notice that there is some highfrequency noise at the points where the MOSFET switches. This noise is shown on an expanded time scale in Figure 11.
While the noise contributions from the input capacitance and ESR are fairly intuitive, this noise source is much less obvious. We can redraw the typical synchronous buck converter shown in Figure 6 to include all the nonideal components, e.g., the MOSFETS, catch diode, input capacitor and traces (Figure 12).
Figure 12 can be simplified into Figure 13 by combining components and discarding parts that have negligible contributions. By taking some aggressive liberties, a further simplification of the circuit can be made to give Figure 14. While these aggressive liberties are not technically "accurate," they are necessary to simplify understanding and the equations. With our newly simplified circuit, it now becomes obvious that there is a series RLC tank circuit. When the switch closes, this tank circuit resonates in a classic manner. A series RLC circuit has a response as shown in Figure 15.
The equations that govern this RLC circuit with respect to our simplified model are given below:
When the buck converter uses an nchannel MOSFET on the high side, the variables given in the above equations can be defined as:
V_{PP} = the maximum peaktopeak noise voltage
L_{S} = L_{t1} + L_{q1d} + L_{q1s} + L_{t2} + L_{t3} + L_{q2s} + L_{q2s} + L_{t4}
L_{C} = the equivalent series inductance of the input capacitor
V_{IN} = the buck regulator's input voltage
τ = the ringdown time constant
R_{ESR} = the ESR of the input capacitor
R_{ON} = the onresistance of the highside MOSFET
C_{S} = C_{q2DS} + C_{q2GD}
The case where a pchannel MOSFET is used on the high side is not as simple. When an nchannel MOSFET is used, the capacitance of the highside switch can be ignored. This is because a floating gate driver is used, and it basically shorts out most of the highside MOSFET's capacitances. In the case of a pchannel MOSFET, a floating gate driver is not used, and the capacitances cannot be ignored. Therefore, our aggressive oversimplification of the circuit is not as nearly as accurate in the case of a pchannel supply as it is for an nchannel supply. However, the basic concepts and the solutions are the same.
To check the relative accuracy of our simplified circuit in Figure 14, we compare the response of the MAX1653EVKIT (Figure 11) with the ideal waveform (Figure 15). The ideal waveform was generated using values obtained from the typical components used on the MAX1653 EV board. The MAX1653EVKIT is ideal for comparison purposes, because it uses a highside nMOSFET.
The first thing to notice is that the first lowgoing cycle on the EV kit is not as big as that predicted by our simple model. This is primarily because of the finite turnon time of the MOSFET: the MOSFET does not instantaneously go from high impedance to low impedance. This transition takes time. In the case of the MAX1653EVKIT, this transition is on the order of 5ns to 15ns. Because this transition time is roughly on the same order as the period of our ringing, the first cycle gets attenuated. This works in our favor to reduce the peaktopeak noise. Consequently, except for the first lowgoing cycle, our simplified model matches the lab results as well as can be expected.
Now that we understand the source of this ringing, how do we minimize it? Looking at Equation 6, we find that the noise is based on three variables: the input voltage (over which we have little control), the input capacitor's series inductance, and the sum of the inductances in the circuit. It becomes obvious (both mathematically and intuitively) to use input capacitors that have small series inductances. In addition, placing a small, lowinductance ceramic chip capacitor in parallel with the bulk capacitance can reduce the size of this noise. Keep these capacitors as close as possible to the MOSFETs (and/or catch diode). See Figure 16. A .01µF ceramic chip capacitor tends to be ideal in terms of capacitance and low inductance. In addition to placing the .01µF capacitors at the input of the buck converter, placing these capacitors close to sensitive circuits that use the V_{IN} supply will greatly reduce the noise at these circuits. One might also be tempted to reduce the slew rate of the gate drive to the MOSFET. Although this will reduce the noise, it could have negative effects on other aspects of the buck converter's performance, specifically its efficiency.
At this point it is appropriate to mention measuring techniques. Good measurement techniques are a good idea in general, and are critical in trying to measure inputcapacitance ESL noise. When using a scope probe, keep the ground lead as short as possible. Connect both the signal portion of the probe and the ground lead directly across what you are trying to measure. Remember that this noise source is due to inductances on the order of nHs. It does not take much trace length to introduce nHs of inductance and ruin your measurement.
If further reduction of this noise source is necessary, some inputsupply filtering can virtually eliminate the noise. The simplest and most economical way is to separate the V_{IN} supply from the rest of your circuitry as much as possible. This approach adds a little inductance between the V_{IN} supply and the rest of the circuitry. This extra inductance provides some isolation, thus effectively filtering this noise source. See Figure 17. Placing a small inductor in series with this supply line will virtually eliminate this noise source altogether. An inductor on the order of 200nH to 1µH is probably best. Too large an inductor can have negative effects on circuit behavior. See Figure 18.
Summary of Equations
Noise due to Finite Capacitance  
Input Noise  
Output Noise  
Noise due to ESR of Capacitor  
Input Noise  
Output Noise 
V_{IN} = the buck regulator's input voltage
V_{OUT} = the buck regulator's output voltage
f = the frequency at which the buck regulator is switching
C_{IN} = the capacitance of the input capacitor
C_{OUT} = the capacitance of the output capacitor
L = the value of the inductor
I_{OUT} = the output current
R_{CINESR} = the input capacitors
R_{COUTESR} = the ESR of the output capacitor
V_{NOUTESR} = the peaktopeak noise voltage due to the ESR of the output capacitor
V_{NOUTC} = the peaktopeak noise voltage due to the finite capacitance of the output capacitor
V_{NINESR} = the peaktopeak noise voltage due to the ESR of the input capacitor
V_{NINCIN} = the peaktopeak noise voltage due to the finite capacitance of the input capacitor
Note: The noise from the different sources is not in phase and, therefore, cannot be directly added.