RC Bode Phase and Group Delay

Predict corner frequency responses

Let's verify the response at the corner frequency for an RC circuit with R = 10 kΩ and C = 1 nF. At the corner frequency, the normalized frequency is u = 1.

1. First, calculate the corner frequency: fc = 1 / (2πRC). With R = 10,000 Ω and C = 10^-9 F, RC = 10^-5 s. Thus, fc ≈ 15915.494 Hz.
2. For u = 1, the transfer function becomes H(i) = 1 / (1 + i) = 0.5 - 0.5i.
3. The amplitude gain is 1 / √(1 + 1²) = 1 / √2 ≈ 0.707. In decibels, this is -10 log₁₀(2) ≈ -3.01 dB.
4. The phase shift is -atan(1) = -π/4 ≈ -0.785 radians.
5. The group delay is RC / (1 + 1²) = 10^-5 / 2 = 5 × 10^-6 seconds.

Examine the Bode responses

Open the RC Bode Phase and Group Delay module and use its controls to see how the frequency affects the transfer function.

1. Set Resistance kiloohms to 10 and Capacitance nanofarads to 1.
2. Set Frequency over corner to 1. Verify that the amplitude gain is approximately 0.707 and the gain in decibels is -3.01.
3. Observe that the phase is -0.785 radians (-45 degrees), and the group delay is 0.000005 seconds. The indicator on the Bode plot will be exactly at the -3 dB point on the gain curve.
4. Change Frequency over corner to 10. The gain decibels will drop to approximately -20 dB, indicating a -20 dB/decade roll-off for the lowpass filter.
5. Set Frequency over corner to 0. This simulates a DC signal where the transfer real part is 1, imaginary is 0, gain is 0 dB, phase is 0, and group delay is at its maximum value of RC = 10 μs.

Model limitations

This model assumes ideal components where the resistor has no parasitic inductance or capacitance, and the capacitor has no equivalent series resistance (ESR) or inductance (ESL). In a real high-frequency circuit, these parasitics become significant and alter the frequency response from the ideal 1 / (1 + iwRC) behavior. Additionally, this model is specifically for an unloaded RC filter; connecting a load impedance across the capacitor will change the transfer function. Note that group delay differs from phase delay (-phase/w) and represents the time delay of the amplitude envelope, not the signal propagation speed.

Further reading

For a comprehensive guide on RC filters, including theoretical background and interactive animations, refer to the UNSW Physics: RC Filters resource. It provides deeper insights into the behavior of lowpass and highpass configurations.