Capacitors

Capacitors are a key component in a circuit designer’s tool kit, thus a short discussion on evaluating their affect on circuit performance is in order. Capacitors have an impedance of XC = 1/2πfC. Note that when the frequency is zero the capacitive impedance (also known as reactance) is infinite, and that when the frequency is infinite the capacitive impedance is zero. These end-points are derived from the final value theorem, and they are used to get a rough idea of the effect of a capacitor. When a capacitor is used with a resistor, they form what is called a break-point. Without going into complicated math, just accept that the break frequency occurs at f = 1/(2π RC) and the gain is –3 dB at the break frequency.

The low pass filter circuit shown in Figure 3–10 has a capacitor in parallel with the feedback resistor. The gain for the low pass filter is given in Equation 3–20.

 

Figure 3–10. Low-Pass Filter

 

At very low frequencies XC ⇒ ∞, so RF dominates the parallel combination in Equation 20, and the capacitor has no effect. The gain at low frequencies is –RF/RG. At very high frequencies XC ⇒ 0, so the feedback resistor is shorted out, thus reducing the circuit gain to zero. At the frequency where XC = RF the gain is reduced by √2 because complex impedances in parallel equal half the vector sum of both impedances.

Connecting the capacitor in parallel with RG where it has the opposite effect makes a high pass filter (Figure 3–11). Equation 3–21 gives the equation for the high pass filter.

Figure 3–11.High-Pass Filter

 

At very low frequencies XC ⇒ ∞, so RG dominates the parallel combination in Equation 3–21, and the capacitor has no effect. The gain at low frequencies is 1+RF/RG. At very high frequencies XC ⇒ 0, so the gain setting resistor is shorted out thus increasing the circuit gain to maximum.

This simple technique is used to predict the form of a circuit transfer function rapidly. Better analysis techniques are presented in later chapters for those applications requiring more precision.