Portable consumer systems' sizes are becoming increasingly smaller. As a result, the available space between components is also becoming tighter, making it more difficult to effectively separate digital and analog circuits.
Generally, system design engineers have to use many capacitors to filter digital noise, expecting to get a clean voltage source as the power supply for analog circuits. This article discusses the effect of filter capacitors.
In most voltage regulators, there are always two capacitors, Cin and Cout (Figure 1 below ).
|Figure 1: In most voltage regulators, there are always two capacitors, Cin and Cout.|
The main purpose of Cin is to filter AC noise, suppressing voltage transition applied to the regulator's input. Meanwhile, Cout's role is to form loop compensation (adding a zero, improving phase margin and inevitably generating a pole) and suppress AC voltage transition due to dynamic load or input. In the sense of attenuating high frequency components, filtering AC noise and suppressing voltage transition are equivalent essentially.
Some characteristics of capacitors manufactured by different dielectrics vary. Understanding the six basic parameters to describe electronic circuits, components or materials is necessary before studying capacitor features.
Resistance (R). Expressed in ohm, it is the ratio of DC voltage to DC current throw conductor.
Reactance (X) . It is introduced by storage energy components, like capacitors or inductors in AC circuitry, including capacitance and inductance. It is expressed in ohm.
Impedance (Z) . Expressed in ohm, it is a complex quantity as it consists of a real part, resistance, and an imaginary part, reactance. It is also expressed as Z = R + jX.
Conductance (G) . It is the ratio of DC current to DC voltage, the reciprocal of resistance. It is expressed in siemens.
Susceptance (B) . It is an imaginary part of admittance, including BC and BL, and is expressed in siemens.
Admittance (Y) . It is the reciprocal of impedance and is expressed in siemens. Also a complex quality, its real part is conductance and its imaginary part is susceptance. It can also be expressed as Y = G + jB. Y is used to describe combinations of components in parallel.
|Figure 2: Impedance is usually used to describe components in series. Admittance is used to describe combinations of components in parallel..|
Impedance is usually used to describe components in series (Figure 2, above ). For a combination of components in series, if Phi is larger than 0 degrees, it means that it's inductive to device terminals – the closer to 90 degrees, the more inductive it is.
|Table 1: Each capacitor type has advantages and disadvantages.|
If Phi is 90 degrees, it is a pure inductor. If Phi is less than 0 degrees, it means that it's capacitive – the closer to -90 degrees, the more capacitive it is. If Phi is -90 degrees, it's a pure capacitor.
Table 1 above lists the features of the different types of capacitors according to dielectric classification.
|Figure 3: All devices have parasitic components – unwanted inductance in capacitors and resistors, unwanted resistance in capacitors etc.|
There is neither pure resistance nor reactance in the real world, but a combination of these impedance elements. All devices have parasitic components such as unwanted inductance in capacitors and resistors, and unwanted resistance in capacitors. The equivalent circuit for real electrolytic capacitor is shown in Figure 3, above .
For multiple layer ceramic capacitors, its equivalent circuit is shown in Figure 4 below. For filter capacitors, it is good to keep them capacitive even at high frequency – never inductive. This means that Phi is equal or close to -90 degrees.
|Figure 4: For filter capacitors, it is good to keep them capacitive even at high frequency – never inductive. This means that Phi is equal or close to -90 degrees.|
The capacitor's performance can be determined comprehensively if serial resistance (RS), serial capacitance (CS), serial inductance (LS), Z and Phi across two leads are obtained at different frequencies.
For a 50V/10 microfarad aluminum electrolytic capacitor, frequency exceeding 800kHz represents inductance instead of capacitance. Both ceramic capacitors have high Q and are still quite capacitive, even when frequency is close to 1MHz.In parallel
If there is always a difference between the real and ideal capacitors, one might wonder: What are the effects of Cin and Cout?
It is recommended to use two capacitors in some application guideline manual -a 10 microFarad aluminum electrolytic capacitor and a smaller one, a 0.1 microFarad ceramic capacitor. To study the effect of filtering AC noise with two capacitors here, R0 is introduced. However, it does not affect AC performance, as shown in Figure 5, below .
|Figure 5: R0 expresses inner resistance of voltage source.|
Here, R0 expresses inner resistance of voltage source. R1 is the effective series resistance (ESR) of C1; R2 is the effective series resistance of C2. The equation below shows the transfer function formula.
Obviously, there are two poles, two zeros in the system expressed by the formula. When Z1 = 1/(2Pi x R1C1), Z2 = 1/(2Pi x R2C2), C1>>C2 and R1>>R2, zeros can be expressed as P1 = ½ Pi x (RO + R1)C1 and P2 = 1/2Pi x [(R0 || R1) + R2]C2.
Assuming a frequency of 100kHz, the ESR of the 50V/10 microFarad electrolytic capacitor is 774 m-ohms, and the 16V/1 microfarad ceramic capacitor is 190 milli-ohms. Moreover, R0 = 1 ohm, C1 = 8.21 microFarad, C2 = 0.997 microFarad, R1 = 774 milli-ohms and R2 = 190 milliohms (Figure 6 below ).
|Figure 6: Assuming a frequency of 100kHz, the ESR of the 50V/10F electrolytic capacitor is 774 milli-ohms and the 16V/1 microfarads ceramic capacitor is 190 milli-ohms.|
In Figure 6a, there are two zeros, Z1 and Z2; and two poles, P1 and P2. Z1 is formed by R1 and C1. Attenuation will end at Z1 without C2.
Only C2 keeps attenuation to continue. Meanwhile, Z2 is formed by R2 and C2. So if you would like to keep attenuation, make sure that C2 < c1="" and="" r2="">< r1.="">
The plots in Figure 6 are calculated by MathCAD under the assumptions that C and ESR are independent from frequency.
|Figure 7: Both C and ESR vary with frequency and represent inductiveness at high frequencies.|
However both C and ESR vary with frequency and represent inductiveness at high frequencies.
Figure 7 above shows the plots with these considerations in mind. Figure 8 below shows the plots obtained from network analyzer (Agilent 4395A).
When frequency is less than 100kHz, there is almost no difference between Figure 7 and Figure 8a below. Gain plot goes up when frequency exceeds 700kHz due to equivalent series inductance's (ESL) effect.
|Figure 8: Gain plot goes up when frequency exceeds 700kHz due to the effect of ESL.|
When C1 and C2 meet the condition of R1C1 = R2C2, it can be simplified to one pole and one zero according to Equation 1. There are two cases that meet the rule: 1. C1 is the same as C2. 2. For specific capacitor types, ESR is generally the reciprocal of capacitance.
|Figure 9: The equivalent capacitance Ce = C1 || C2, and the equivalent ESR, Re = R1 || R2.|
The equivalent capacitance Ce = C1 || C2, and the equivalent ESR, Re = R1 || R2. Figure 9 above shows the circuit of three capacitors in parallel. The transfer function is expressed in the equation below:
It is easy to find that there are three poles and three zeros in the system.
Assuming that R0 = 1 ohm, C1 = 10 microFarad, C2 = 1 microFarad, C3 = 0.1 microFarad, R1 = 2 ohms, R2 = 100 milli-ohms and R3 = 50 milli-ohms, frequency plots are shown in Figure 10 below , as calculated by MathCAD.
|Figure 10: Attenuation starts from P1 formed by C1, R0 and R1, and ends at Z3 formed by C3 and R3.|
Attenuation starts from first pole (P1) formed by C1, R0 and R1, and ends at third zero (Z3) formed by C3 and R3. Likewise, if they meet the condition of R1C1 = R2C2 = R3C3, these three capacitors can be one equivalent capacitor, with a Ce = C1 || C2 || C3 and Re = R1 || R2 || R3. In linear voltage regulator applications, users are most concerned about AC noise at the output terminal. The AC noise comes from only two sources – input and inner circuits of the regulators itself. Fortunately, new-generation regulators can solve this problem.
Good power supply rejection ratio (PSRR) performance can suppress AC noise from input, especially at low frequency. Also, output noise generated by regulators' inner circuits is very low that it can be ignored.
For example, in AP2121, PSRR can reach 70dB from DC to 1kHz, and output noise from 10Hz to 100kHz is only 30 microVrms. Users can easily obtain clean regulated voltage without a bigger capacitor, thus saving system cost and PCB space.
To achieve a significant attenuation of AC noise for output of linear regulators, it is good to use multiple capacitors in parallel with different capacitance and ESR. Attenuation will start from the first pole, of which the frequency depends on the capacitor with the bigger size, and end at the last zero of which the frequency depends on the capacitor with the smaller size.
It is very important to keep a short distance between capacitor leads and chip pins to prevent ESL distribution through the lead and long wires. However, when using new-generation linear regulators with high PSRR and low noise, it is not necessary to use multiple capacitors in parallel. One 1 microFarad ceramic cap is enough and is recommended.
Peter Wang is System Engineer at BCD Semiconductor Manufacturing Co. Ltd.