Frequency Behavior of RC Circuits and Phase Shifter (E17)

Frequency Behavior of RC Elements (E17)
Aim of the Experiment

  • Investigate the properties of filters formed by series connection of ohmic resistors and capacitors (RC elements).

  • Understand a circuit that allows two sinusoidal voltages to be phase-shifted relative to each other by arbitrary angles, without altering their magnitudes.

Theoretical Background Part A: Filters

Basic RC Circuits and Complex AC Calculation

  • High-pass and low-pass filter circuits are derived from a basic RC series circuit, connected to an AC voltage source (Figure 1a).

  • The behavior of this circuit (current and partial voltages across R and C as a function of input AC voltage U(t)) can be determined by solving a differential equation.

  • For sinusoidal currents and voltages of constant frequency, a simplified complex AC calculation is used.

  • AC currents and voltages are represented by a complex number describing their amplitude and phase.

  • The time function e^{i
    ho t} is omitted because it is the same for all circuit elements.

  • Circuit analysis: Kirchhoff's rules, which involve current and voltage addition, simplify to addition of time-independent complex numbers (vector addition in the complex plane).

    • For the circuit in Figure 1a, the total voltage U is:

      U = UR + UC = R I + Z_C I \quad (1)

    • Definitions:

      • Source voltage: U = U_0 e^{i
        ho t}

      • Current: I = I_0 e^{i(\rho t - \phi)}

      • Capacitive impedance: Z_C = 1 / (i\rho C)

    • Note on phase: The negative sign of \phi is a generally observed convention.

    • Current calculation (omitting e^{i\rho t}):

      I0 e^{-i\phi} = U0 / (R - i / (\rho C)) \quad (2)

    • For real I0 and U0, \phi corresponds to the phase of the complex number U0 / (R - i / (\rho C)) and I0 is its magnitude.

    • Magnitude of current:

      I0 = U0 / \sqrt{R^2 + 1 / (\rho^2 C^2)} \quad (3)

    • Phase of current:

      \tan \phi = 1 / (\rho R C) \quad (4)

High-Pass Filter (Figure 1b)

  • The output voltage UA is taken across the resistor R (UR).

  • Transfer function: The ratio of output voltage to input voltage (U{A,0} / U{E,0} or U{R,0} / U0). The complex transfer function is given by:

    H{HP}(i\rho) = \frac{UR(i\rho)}{U_{in}(i\rho)} = \frac{R}{R + \frac{1}{i\rho C}} = \frac{i\rho RC}{1 + i\rho RC} \quad (5)

  • Output voltage expression: UR = R I = R I0 e^{-i\phi} = U{R,0} e^{-i\phiR} \quad (6)

    • Magnitude: U{R,0} = U0 / \sqrt{1 + 1 / (\rho R C)^2} \quad (7)

    • Phase: \phi_R = \phi \quad (8)

  • Frequency dependence: The partial voltage across the resistor depends on the generator frequency:

    • For high frequencies: U{R,0} \approx U0 (almost undamped).

    • For low frequencies: U{R,0} \ll U0.

  • This property gives the circuit its name: High-pass.

Low-Pass Filter (Figure 1c)

  • The output voltage UA is taken across the capacitor C (UC).

  • Transfer function: The ratio of output voltage to input voltage (U{A,0} / U{E,0} or U{C,0} / U0). The complex transfer function is given by:

    H{LP}(i\rho) = \frac{UC(i\rho)}{U_{in}(i\rho)} = \frac{\frac{1}{i\rho C}}{R + \frac{1}{i\rho C}} = \frac{1}{1 + i\rho RC} \quad (9)

  • Output voltage expression: UC = ZC I = (-i / (\rho C)) U0 / (R - i / (\rho C)) = U0 / (1 + i\rho R C) \quad (10)

    • Represented by magnitude and phase: UC = U{C,0} e^{-i\phi_C}

    • Magnitude: U{C,0} = U0 / \sqrt{1 + (\rho R C)^2} \quad (11)

  • Frequency dependence: The low-pass filter transmits DC and low-frequency AC signals, but attenuates high-frequency signals.

  • Phase for low-pass:

    \tan \phi_C = -\rho R C \quad (12)

Band-Pass Filter (Figure 1d)

  • Formed by connecting a high-pass and a low-pass filter in series.

  • Transfer function: Approximately the product of the individual high-pass and low-pass transfer functions.

  • Phase: Calculated as the sum of the individual phase shifts: \phi{R1} + \phi{C2}, where \phi{R1} is for the high-pass and \phi{C2} is for the low-pass.

Gain and Cut-off Frequency

  • Gain (Attenuation):

    • Measured in Bel or Decibel (dB).

    • Defined as the logarithm of the ratio of output power (PA) to input power (PE):

      V = \log(PA / PE) \cdot 1 \text{ Bel} = 10 \cdot \log(PA / PE) \cdot 1 \text{ dB} \quad (13)

    • If output resistance RA equals input resistance RE:

      V = 10 \text{ dB} \cdot \log(|UA / UE|^2) = 20 \text{ dB} \cdot \log(|UA / UE|) \quad (14)

  • Cut-off Frequency (\rhog or \nug):

    • Defined as the frequency where the output power has dropped to half of the input power.

    • This implies the output voltage magnitude is UA = UE / \sqrt{2}.

    • For both high-pass and low-pass filters, the cut-off frequency is:

      \rhog = 2\pi \nug = 1 / (R C)

  • Gain and Phase for High-Pass (expressed using \nu_g):

    V(\nu) = -10 \text{ dB} \cdot \log(1 + (\nug / \nu)^2) \quad \text{and} \quad \tan \phiR = \nu_g / \nu \quad (15)

  • Three cases for High-Pass Gain behavior (refer to Figure 2):

    1. Low Frequencies (\nu \ll \nu_g):

      • The '1' in the logarithm argument of Eq. (15) can be neglected.

      • Approximation: V(\nu) \approx 20 \text{ dB} \cdot (\log \nu - \log \nu_g).

      • This is a linear equation for V(\nu) as a function of \log \nu.

      • Slope: +20 \text{ dB per frequency decade} (factor of 10 increase in frequency) or approximately +6 \text{ dB per octave} (factor of 2 increase in frequency).

      • This represents an asymptote for amplitude increase at low frequencies.

    2. Cut-off Frequency (\nu = \nu_g):

      • 1 + (\nu_g / \nu)^2 = 2

      • Gain: V(\nu_g) = -10 \text{ dB} \cdot \log 2 \approx -3 \text{ dB}.

    3. High Frequencies (\nu \gg \nu_g):

      • The '1' in the logarithm argument of Eq. (15) dominates.

      • Gain V is practically independent of \nu.

      • Transfer function is approximately 1, so V \approx 0 \text{ dB}.

  • Low-Pass Analog: An analogous analysis for the low-pass filter yields comparable results. The gain curve needs to be mirrored along the y-axis at \nu = \nu_g. Additionally, the sign of the phase function changes.

Phasor Diagram Analysis (Figure 3)

  • Graphical Representation: Voltages U0, UR, and U_C are represented as vectors (phasors) in the complex plane.

    • The length of the arrow corresponds to the magnitude of the voltage.

    • The angle to the real axis indicates the phase shift relative to the source voltage U.

  • Triangle Formation: According to Kirchhoff's voltage law, the vectorial sum UR + UC must equal U. Thus, the three voltages form a triangle.

  • Right-Angled Triangle: For a capacitor, current and voltage are always shifted by \pi/2 relative to each other. For a resistor, voltage is in phase with the current. This means UR and UC must be shifted by \pi/2 relative to each other, forming a right-angled triangle that lies on a Thales' circle.

  • Frequency Dependence on the Thales' Circle: As the frequency \nu of the source voltage changes, UR changes in both magnitude and phase. However, since the vector sum of UR and UC is constant (U) and the phase between them is always \pi/2, the tip of the UR phasor must move along the Thales' circle.

    • Low Frequency (\nu \ll \nug): UR \ll UC. From the diagram, this implies \phiR \approx \pi/2 and \phi_C \approx 0.

    • Cut-off Frequency (\nu = \nug): UR = U_C. The phase angles are \pm \pi/4.

    • High Frequency (\nu \gg \nug): UR \gg UC. This results in \phiR \approx 0 and \phi_C \approx -\pi/2.

Theoretical Background Part B: Phase Shifter

RC Phase Shifter Circuit (Figure 4)

  • A circuit designed to shift the phase of two sinusoidal voltages relative to each other by arbitrary angles without changing their magnitudes.

  • Transformer: The transformer in Figure 4 is primarily for measurement purposes, allowing decoupling from ground (explained in procedure).

  • Kirchhoff's Loop Rule: For the voltages in this circuit, at any given time:

    U(t) = U1 + U2 = UR + UC \quad (16)

  • Phasor Diagram (Figure 4):

    • The voltages U, UR, and UC again form a triangle.

    • The sum U1 + U2 also equals the source voltage.

    • Since ohmic resistors R1 and R2 do not cause any phase shift of the current, their voltage phasors lie parallel to the real axis (Re) and precisely cover the hypotenuse of the triangle.

  • Right-Angled Triangle on Thales' Circle: Similar to Part A, the triangle is right-angled and lies on a Thales' circle.

  • Condition R1 = R2: If R1 = R2, point A in the phasor diagram is exactly at the center of the hypotenuse. The voltage U_{AB} between points A and B then forms the radius of the circle.

  • Key Property: Constant Magnitude, Variable Phase: If R, C, or the source frequency \rho are changed, UR changes in magnitude and phase. However, the tip of the UR phasor (point B in Figure 4) can only move along the Thales' circle. This means the voltage U{AB} only changes its phase angle \phi{AB} relative to the source voltage, while its magnitude remains constant.

Experimental Setup and Procedure Part A

Circuit Construction and Measurement

  • Build the filter circuits shown in Figure 1 in sequence.

  • The AC input voltage U_E is supplied by a sinus generator, which allows stepless adjustment of frequency \nu over a wide range. The generator voltage is variable but maintains a constant (frequency-independent) value once set.

  • The input (UE) and output (UA) voltages are measured using a two-channel oscilloscope.

Data Collection and Plotting

  • Measure UE, UA, and the phase angles \phi of the filters over a sufficiently wide frequency range to capture both linear regions of the gain factor V (refer to Figure 2).

  • Initially, record two measurement points per decade (e.g., 10 \text{ Hz}, 30 \text{ Hz}, 100 \text{ Hz}, 300 \text{ Hz}, etc.).

  • Plot UA / UE (or gain V in dB) and \phi as functions of \log \nu on a double-logarithmic diagram concurrently with the measurements.

  • Subsequently, take additional measurement points near the filter's cut-off frequencies to precisely locate the curve's 'knee' or bend.

Grounding Consideration

  • Important: The ground connections of the oscilloscope and the signal generator must be carefully handled to avoid creating ground loops, which can interfere with measurements. Ensure that all ground connections are at a common reference potential to prevent unwanted current flows and measurement errors. When using multiple instruments, their grounds should ideally be common or isolated if necessary, especially with phase-sensitive measurements. For example, the outer conductor of the BNC cable connected to the oscilloscope input is usually connected to the ground of the device. If both input and output channels are used from the same oscilloscope, their grounds are internally connected. This must be considered when connecting the circuit to avoid short circuits. The transformer in Figure 4 is used to provide galvanic isolation for specific measurements, decoupling the circuit from the generator's ground.