EEG 326 Frequency Response Lecture Flashcards

General Frequency Response Characteristics of Amplifiers

• Mid-Frequency Range Assumptions:

  • In previous treatments of linear amplifiers, coupling and bypass capacitors were treated as ideal short circuits.

  • Transistor internal capacitances (parasitic, load, and transistor-specific capacitances) were treated as ideal open circuits with respect to the AC input signal.

  • The assumption was that the signal frequency is high enough for coupling/bypass capacitors to be shorts, yet low enough for parasitic/transistor capacitances to be opens. This is only true in the mid-frequency range.

• Frequency Bands and Gain Variation:

  • Low Frequency Band (f < f_L): Gain increases as frequency increases. This roll-off at low frequencies is due to the effects of the coupling and bypass capacitors.

  • Mid-Band: Gain is almost constant with frequency. In this range, coupling and bypass capacitors act as short circuits, while stray and transistor capacitances act as open circuits.

  • High Frequency Band (f > f_H): Gain decreases as frequency increases. This roll-off at high frequencies is caused by stray capacitance and transistor capacitance effects.

• Decibel (dB) Measurement:

  • The decibel is a unit of logarithmic gain measurement used to express amplifier response.

  • At the lower corner frequency ($f = f_L$) and the upper corner frequency ($f = f_H$), the gain is $3\,dB$ less than the maximum (mid-band) gain.

• Amplifier Bandwidth ($f_{BW}$):

  • Bandwidth is defined as the difference between the high and low corner frequencies: $f_{BW} = f_H - f_L$

  • Example: For an audio amplifier with a frequency range of $20\,Hz$ to $20\,kHz$, the design must ensure f_L < 20\,Hz and f_H > 20\,kHz.

• Equivalent Circuits by Frequency Range:

  • Midband Range: Coupling/bypass capacitors = Shorted; Stray/transistor capacitances = Open.

  • Low-Frequency Range: Coupling and bypass capacitors must be included in the equivalent circuit and amplification equations. Stray and transistor capacitances are treated as open circuits.

  • High-Frequency Range: Transistor, parasitic, and load capacitances must be taken into account. Coupling and bypass capacitors are treated as shorted.

System Transfer Functions and Bode Plots

• s-Domain Analysis:

  • Frequency transfer functions are obtained using the complex frequency $s$.

  • Capacitor complex impedance: $Z_C = \frac{1}{sC}$.

  • Inductor complex impedance: $Z_L = sL$.

  • System transfer functions can represent current gain, voltage gain, transimpedance, or transconductance.

  • To obtain the frequency response from an s-domain function, set $s = j\omega$.

• Poles and Zeros:

  • A transfer function in the s-domain is generally expressed as:     $H(s) = K \frac{(s - z_1)(s - z_2)\dots}{(s - p_1)(s - p_2)\dots}$

  • $K$ is a constant.

  • $z_1, z_2, \dots$ are "zeros": at these frequencies ($s = z_n$), the transfer function becomes zero.

  • $p_1, p_2, \dots$ are "poles": at these frequencies ($s = p_n$), the transfer function diverges toward infinity.

Analysis of First-Order (RC) Circuits

• Series Coupling Capacitor Circuit (High-Pass Network):

  • Circuit configuration involves input voltage $V_i$, resistor $R_s$ in series with capacitor $C_s$, and the output taken across parallel resistor $R_p$.

  • Transfer function: $\frac{V_o(s)}{V_i(s)} = \frac{R_p}{R_s + R_p + \frac{1}{sC_s}} = \frac{s R_p C_s}{1 + s(R_s + R_p)C_s}$.

  • Rearranged: $\frac{V_o(s)}{V_i(s)} = \left( \frac{R_p}{R_s + R_p} \right) \left( \frac{s\tau_s}{1 + s\tau_s} \right)$, where the time constant $\tau_s = (R_s + R_p)C_s$.

  • Setting $s = j\omega$: $T(j\omega) = \left( \frac{R_p}{R_s + R_p} \right) \frac{j\omega\tau_s}{1 + j\omega\tau_s}$.

  • Magnitude: $|T(jf)| = \left( \frac{R_p}{R_s + R_p} \right) \frac{2\pi f\tau_s}{\sqrt{1 + (2\pi f\tau_s)^2}}$.

  • Phase: $\theta = 90^{\circ} - \tan^{-1}(2\pi f\tau_s)$.

  • As $f \rightarrow 0$, $\theta = 90^{\circ}$. At the corner frequency where $f = \frac{1}{2\pi\tau_s}$, $\theta = 45^{\circ}$. As $f \rightarrow \infty$, $\theta = 0^{\circ}$.

• Parallel Capacitor Load Circuit (Low-Pass Network):

  • Circuit involves $V_i$, source resistance $R_s$, and a load consisting of $R_p$ and $C_p$ in parallel.

  • Transfer function: $T(jf) = \left( \frac{R_p}{R_s + R_p} \right) \left( \frac{1}{1 + j2\pi f\tau_p} \right)$.

  • Time constant: $\tau_p = (R_s || R_p)C_p$.

  • Magnitude: $|T(jf)| = \left( \frac{R_p}{R_s + R_p} \right) \frac{1}{\sqrt{1 + (2\pi f\tau_p)^2}}$.

  • Phase: $\theta = -\tan^{-1}(2\pi f\tau_p)$.

Bode Plot Mechanics

• Definition: A technique for approximate plots of magnitude and phase.

• Magnitude Components (Logarithmic sums of three terms for high-pass RC):

  1. Constant Term: $\frac{R_p}{R_s + R_p}$. Since this ratio is less than unity, its value in dB is less than zero and is independent of frequency.

  2. Rising Slope Term: Corresponding to the numerator $s\tau$. This term increases linearly with frequency at a rate of $20\,dB/decade$ (or $6\,dB/octave$).

  3. Falling Slope Term: Corresponding to $\frac{1}{1 + s\tau}$. At low frequencies, it is $0\,dB$. At the corner frequency ($f = 1/(2\pi\tau)$), it is $-3\,dB$. At very high frequencies, it decreases linearly at $-20\,dB/decade$.

• Combined Plot: The rising slope and constant term dominate at low frequencies. After the corner frequency (break-point), the falling slope term cancels the rising slope, resulting in a flat response (mid-band gain).

Combined Capacitor Effects (SCTC and OCTC)

• If a circuit contains both a series coupling capacitor ($C_s$) and a parallel load capacitor ($C_p$), we can analyze them individually if their values differ significantly.

• Low Frequencies (Open-Circuit Time Constant - OCTC):

  • Treat $C_p$ as an open circuit.

  • $\tau_s = (R_s + R_p)C_s$.

• High Frequencies (Short-Circuit Time Constant - SCTC):

  • Treat $C_s$ as a short circuit.

  • $\tau_p = (R_s || R_p)C_p$.

• Corner Frequencies:

  • Lower corner: $f_L = \frac{1}{2\pi\tau_s}$.

  • Upper corner: $f_H = \frac{1}{2\pi\tau_p}$.

Transistor Frequency Response: External Capacitors

• Common-Emitter with Input Coupling Capacitor:

  • Assumptions: Output resistance $r_o = \infty$ (valid if r_o >> R_C and r_o >> R_E).

  • Small-signal voltage gain: $A_v(s) = \frac{V_o(s)}{V_i(s)} = -g_m r_{\pi} R_C \left( \frac{R_B}{R_B + R_{ib}} \right) \frac{1}{(R_{si} + R_i) + \frac{1}{s C_C}}$.

  • Simplified form: $A_v(s) = \left( \frac{-g_m r_{\pi} R_C}{R_{si} + R_i} \right) \left( \frac{R_B}{R_B + R_{ib}} \right) \left( \frac{s\tau_s}{1 + s\tau_s} \right)$.

  • Time constant: $\tau_s = (R_{si} + R_i)C_C$, where $R_i = R_B || [r_{\pi} + (1 + \beta)R_E]$.

• Common-Source with Output Coupling Capacitor:

  • Assumptions: Signal generator resistance is negligible compared to $R_G$. $r_o = \infty$.

  • Even with the capacitor at the output, the response is a high-pass network.

  • The effective resistance seen by $C_C$ is found by setting independent sources to zero ($g_m V_{gs} = 0$), making it an open circuit.

  • Effective resistance: $R_{eff} = R_D + R_L$.

  • Time constant: $\tau = (R_D + R_L)C_C$.

• Emitter-Follower with Output Coupling Capacitor:

  • Determine the $3\,dB$ frequency using the resistance seen by the output capacitor $C_{C2}$.

  • Parameters for analysis typically include $\beta = 100$, $V_{BE(on)} = 0.7\,V$, and $V_A = 120\,V$.

Load Capacitor Effects

• This occurs in a parallel capacitive load configuration, acting as a low-pass network.

• At high frequencies, the impedance of $C_L$ decreases, shunting the output to ground and driving the output voltage toward zero.

• The corner frequency is found via the time constant approach involving $\tau_p$.

Combined Coupling and Load Capacitors Example

• Circuit Parameters:

  • $R_s = 0.1\,k\Omega$, $R_1 = 40\,k\Omega$, $R_2 = 5.7\,k\Omega$, $R_C = 5\,k\Omega$, $R_E = 0.5\,k\Omega$, $R_L = 10\,k\Omega$.

  • $C_C = 10\,\mu F$, $C_L = 15\,pF$.

  • Transistor: $V_{BE(on)} = 0.7\,V$, $\beta = 100$, $V_A = \infty$.

• DC/Small-Signal Analysis:

  • $I_{CQ} = 0.99\,mA$.

  • $g_m = 38.08\,mA/V$, $r_{\pi} = 2.626\,k\Omega$.

  • Input Resistance $R_i = r_{\pi} + (1 + \beta)R_E = 2.63 + (101)(0.5) = 53.1\,k\Omega$.

• Results:

  • Midband Gain $|A_v|_{max} = 6.14$.

  • $\tau_s = (R_s + R_1 || R_2 || R_i)C_C = 46.6\,ms$.

  • $\tau_p = (R_C || R_L)C_L = 50\,ns$.

  • $f_L = \frac{1}{2\pi(46.6 \times 10^{-3})} = 3.42\,Hz$.

  • $f_H = \frac{1}{2\pi(50 \times 10^{-9})} = 3.18\,MHz$.

  • $f_{BW} \approx 3.18\,MHz$.

Gain-Bandwidth Product (GB)

• Assuming corner frequencies are far apart, $f_{BW} \approx f_H$.

• The Gain-Bandwidth Product is essentially constant for a given load capacitance: $GB = |A_v|_{max} \times f_H$.

• Amplifier design involves a trade-off between higher gain and wider bandwidth.

Bypass Capacitor Effects

• Bypass capacitors ($C_E$ or $C_S$) are used to allow stabilizing resistors ($R_E$ or $R_S$) to be part of the DC bias without reducing small-signal gain (as they are shorted at mid-band signal frequencies).

• Transfer Function:     $A_v(s) = \left( \frac{-g_m r_{\pi} R_C}{R_s + r_{\pi} + (1 + \beta)R_E} \right) \frac{1 + s R_E C_E}{1 + s \frac{R_E(R_s + r_{\pi})C_E}{R_s + r_{\pi} + (1 + \beta)R_E}}$.

• This has the form: $A_v(s) = A_{v0} \frac{1 + s\tau_A}{1 + s\tau_B}$.

• Asymptotes:

  • As $\omega \rightarrow 0$ (Open circuit): $|A_v| = \frac{g_m r_{\pi} R_C}{R_s + r_{\pi} + (1 + \beta)R_E}$.

  • As $\omega \rightarrow \infty$ (Short circuit): $|A_v| = \frac{g_m r_{\pi} R_C}{R_s + r_{\pi}}$.

• The Bode plot features two horizontal asymptotes connected by a sloped region defined by corner frequencies $f_A = \frac{1}{2\pi\tau_A}$ and $f_B = \frac{1}{2\pi\tau_B}$.

Questions & Discussion

• Exercise 1 (Series Coupling): $R_S = 2\,k$, $R_P = 8\,k$, $f_L = 50\,Hz$.

  • Calculate $C_s$: $\tau_s = \frac{1}{2\pi(50)} = 3.18\,ms$. $C_s = \frac{3.18\,ms}{2\,k + 8\,k} = 0.318\,\mu F$.

• Exercise 2 (Parallel Load): $R_S = 4.7\,k$, $R_P = 25\,k$, $C_P = 120\,pF$.

  • Calculate $f_H$: $\tau_p = (4.7\,k || 25\,k) \times 120\,pF = 3.956\,k \times 120\,pF = 0.4747\,\mu s$. $f_H \approx 335\,kHz$.

• PSpice Results Discussion: PSpice simulations for combined effects of multiple coupling capacitors ($C_{C1} = 1\,\mu F$, $C_{C2} = 2\,\mu F$) and a bypass capacitor ($C_E = 50\,\mu F$) show that the low-frequency roll-off slope increases with each additional capacitor ($6\,dB/octave$ for one, $12\,dB/octave$ for two, etc.).