High Pass Filter Example: RC Circuit Exam Problem & Algebra Walkthrough

For a standard passive RC high pass filter example targeting audio crossovers or AC-coupling for microcontroller ADCs, a 10 kΩ resistor paired with a 47 nF capacitor yields a theoretical cutoff frequency ($f_c$) of roughly 338 Hz. When fed a 2.0 Vrms sine wave at 60 Hz, this specific filter attenuates the output to 0.35 Vrms while introducing an +80° phase lead. This walkthrough breaks down the exact algebra, the complex voltage divider theorem, and the common calculation traps that cost points on electrical engineering exams and bench tests.

The Practice Problem Statement & Frequency Response Table

Which Theorem Applies and Why?

To solve this, we use the Complex Voltage Divider Theorem. In an RC high-pass filter, the capacitor is in series with the signal path and the resistor is tied to ground. Because the capacitor introduces frequency-dependent reactance, we cannot use simple DC resistance ratios. We must treat the capacitor as a complex impedance ($Z_C = -jX_C$) and the resistor as a real impedance ($Z_R = R$). The output is taken across the resistor, meaning the transfer function relies on the ratio of the resistive impedance to the total circuit impedance.

Before diving into the algebra, here is the complete frequency response profile for the 10 kΩ / 47 nF combination we will derive. Notice how the gain and phase shift dramatically as we cross the cutoff threshold.

Table 1: Frequency Response of 10kΩ / 47nF High-Pass Filter (Vin = 2.0 Vrms)

Frequency (Hz)	Capacitive Reactance ($X_C$)	Total Impedance ($Z$)	Voltage Gain ($V_{out}/V_{in}$)	$V_{out}$ (Vrms)	Phase Lead ($\phi$)
10	338,627 Ω	338,774 Ω	0.0295	0.059 V	+88.3°
60 (Test)	56,438 Ω	57,318 Ω	0.1745	0.349 V	+80.0°
338 ($f_c$)	10,018 Ω	14,155 Ω	0.7065 (-3dB)	1.413 V	+45.1°
1,000	3,386 Ω	10,558 Ω	0.9471	1.894 V	+18.7°
10,000	338 Ω	10,006 Ω	0.9994	1.999 V	+1.9°

Step-by-Step Algebraic Solution

Part A: Component Selection for $f_c \approx 300$ Hz

The foundational equation for the -3dB cutoff frequency of an RC filter is:

$$f_c = \frac{1}{2 \pi R C}$$

We need to select standard E12 values. Let's arbitrarily fix the resistor to a standard 10 kΩ ($10,000 \, \Omega$) to keep thermal noise reasonable and avoid loading the source. Rearranging the formula to solve for $C$:

$$C = \frac{1}{2 \pi R f_c}$$

Substitute the knowns:

$$C = \frac{1}{2 \cdot \pi \cdot 10,000 \cdot 300}$$

$$C = \frac{1}{18,849,555} \approx 53.05 \times 10^{-9} \text{ F} = 53.05 \text{ nF}$$

The closest standard E12 capacitor values are 47 nF and 56 nF. We will select 47 nF (a highly common, cheap ceramic value). Let's verify the actual cutoff frequency this creates:

$$f_{c(actual)} = \frac{1}{2 \cdot \pi \cdot 10,000 \cdot 47 \times 10^{-9}} = \frac{1}{0.002953} \approx \mathbf{338.6 \text{ Hz}}$$

Part B: Output Voltage and Phase at 60 Hz

Now we analyze the circuit at the 60 Hz test frequency. First, calculate the capacitive reactance ($X_C$):

$$X_C = \frac{1}{2 \pi f C} = \frac{1}{2 \cdot \pi \cdot 60 \cdot 47 \times 10^{-9}}$$

$$X_C = \frac{1}{1.7719 \times 10^{-5}} \approx \mathbf{56,438 \, \Omega}$$

Next, find the total magnitude of the circuit impedance ($Z$). Because $R$ and $X_C$ are 90° out of phase, we use the Pythagorean theorem:

$$Z = \sqrt{R^2 + X_C^2} = \sqrt{10,000^2 + 56,438^2}$$

$$Z = \sqrt{100,000,000 + 3,185,247,844} = \sqrt{3,285,247,844} \approx \mathbf{57,317 \, \Omega}$$

Apply the complex voltage divider formula for a high-pass configuration (where output is taken across the resistor):

$$V_{out} = V_{in} \times \frac{R}{Z} = 2.0 \times \frac{10,000}{57,317}$$

$$V_{out} = 2.0 \times 0.17447 \approx \mathbf{0.349 \text{ Vrms}}$$

Finally, calculate the phase angle ($\phi$). In a high-pass filter, the output voltage across the resistor leads the input voltage:

$$\phi = \arctan\left(\frac{X_C}{R}\right) = \arctan\left(\frac{56,438}{10,000}\right) = \arctan(5.6438)$$

$$\phi \approx \mathbf{+80.0^\circ}$$

Sanity Checks, Common Traps, and Verification

Answer Sanity Check

• Order of Magnitude: 60 Hz is nearly a full decade below our 338 Hz cutoff. A first-order filter attenuates at 20 dB/decade. We should expect heavy attenuation. Our result of 0.349 V (roughly 17% of the input) aligns perfectly with this expectation.
• Phase Check: As frequency approaches 0 Hz, a high-pass filter's phase lead approaches +90°. At 60 Hz (below $f_c$), a phase lead of +80° is logically sound.
• Units: Reactance and Impedance are in Ohms (Ω), Gain is unitless (V/V), and final output is in Vrms. All units cancel and resolve correctly.

How to Verify the Answer Independently

Never trust hand-calculated AC theory on the bench without a simulation sanity check. Open LTspice and build the schematic: place a 2V AC sine source, a 47nF capacitor in series, and a 10k resistor to ground. Run an AC analysis command (.ac dec 100 1 10k). Plot the Bode diagram. You will visually confirm the -3dB point sits exactly at 338 Hz, and probing the 60 Hz marker on the trace will yield -15.1 dB (which translates to a 0.175 linear gain multiplier) and a phase trace crossing +80°. For physical verification, build it on a breadboard using a C0G/NP0 dielectric capacitor to avoid the microphonic and distortion artifacts common in X7R ceramics, and measure with a true-RMS multimeter or oscilloscope.

FAQ: High Pass Filter Edge Cases

What if my application requires an exact 300.0 Hz cutoff?

Standard 5% or 10% passive components make an exact 300 Hz cutoff impossible. To achieve high precision, you must either use a trimmer potentiometer for $R$, or switch to an active filter topology (like a Sallen-Key high-pass) using an op-amp like the TL072, which allows you to tune the Q-factor and cutoff independently using precision 1% resistors.

Does the physical capacitor type matter for this circuit?

Absolutely. If this high pass filter example is being used for audio AC-coupling, do not use standard X7R or Y5V ceramic capacitors. These exhibit severe voltage coefficients (capacitance drops as voltage increases) and piezoelectric microphonics, which will inject distortion into your audio signal. Always specify C0G/NP0 ceramics or metallized polypropylene film capacitors for audio paths. For simple DC-blocking on a microcontroller GPIO pin, a cheap X7R is perfectly acceptable.