Click here to purchase the entire book in PDF format. Another way to consider this... We know that the voltage across the capacitor and the voltage across the resistor are always 90$^{\circ }$ apart at all frequencies, regardless of their phase relationships to the input voltage. Consider the Resistance and the Capacitive reactance as both providing components of the impedance, but 90$^{\circ }$ apart. Therefore, we can plot the relationship between these three using a right triangle as is shown in Figure 2.22. Figure 2.21: The triangle representing the relationship between the resistance, capacitive reactance and the impedance of the circuit. Note that, as frequency changes, only R remains constant. At this point, it should be easy to see why the impedance is the square root of the sum of the squares of R and $X_{C}$. In addition, it becomes intuitive that, as the frequency goes to $\infty$ Hz, $X_{C}$ goes to zero and the hypotenuse of the triangle, Z, becomes the same as R. If the frequency goes to 0 Hz (DC), $X_{C}$ goes to $\infty \Omega$ as does Z. Go back to the concept of a voltage divider using two resistors. Remember that the ratio of the two resistances is the same as the ratio of the voltages across the two resistors. $$\frac{R_{1}}{ R_{2}} = \frac{V_{1}}{V_{2}}$$ (3.87) If we consider the RC circuit in Figure 2.16, we can treat the two components in a similar manner, however the phase change must be taken into consideration. Figure 2.22 shows a triangle exactly the same as that in Figure 2.21 - now showing the relationship bewteen the input voltage, and the voltages across the resistor and the capacitor. Figure 2.22: The triangle representing the relationship bewteen the input voltage and the outputs of the high-pass and low-pass filters. Note that, as frequency changes, only $V_{IN}$ remains constant. So, once again, we can see that, as the frequency goes up, the voltage across the capacitor goes down until, at $\infty$ Hz, the voltage across the cap is 0 V and $V_{IN} = V_{R}$. Notice as well that this triangle gives us the phase relationships of the voltages. The voltage across the resistor and the capacitor are always 90$^{\circ }$ apart, but the phase of these two voltages in relation to the input voltage changes according to the value of the capacitive inductance which is, in turn, determined by the capacitance and the frequency. So, now we can see that, as the frequency goes down, the current goes down, the voltage across the resistor goes down, the voltage across the resistor approaches the input voltage, the phase of the low-pass filter approaches 90$^{\circ }$ and the phase of the high-pass filter approaches 0$^{\circ }$. As the frequency goes up, the voltage across the capacitor goes down, the voltage across the resistor appraoches the input voltage and the phase of the low-pass filter approaches 0$^{\circ }$ and the phase of the high-pass filter approaches 90$^{\circ }$.