--- theme: default title: 'Lecture 35: Emitter follower circuit' info: | ## DI02011011: Electronics Circuit and Application (ECA) Feedback in amplifiers --- # DI02011011: Electronics Circuit and Application (ECA) ## Lecture 35: Emitter follower circuit ### Feedback in amplifiers --- # Lecture Agenda: Emitter Follower Circuit Breakdown - Circuit Topology & DC Biasing Analysis ($V_B, V_E, I_E, r_e$) - Identification of 100% Voltage-Series Feedback Mechanism ($\beta = 1.0$) - Small-Signal Hybrid-\pi$ AC Equivalent Circuit Modeling - Exact Mathematical Derivation of Voltage Gain ($A_v = \frac{R_E'}{r_e + R_E'}$) - Exact Mathematical Derivation of Input Impedance ($R_{in} = R_B \parallel [r_\pi + (1+\beta_0)R_E']$) - Exact Mathematical Derivation of Output Impedance ($R_{out} = R_E \parallel [r_e + \frac{R_S'}{1+\beta_0}]$) - Current Gain ($A_i \approx \beta_0$) and Power Gain ($A_p \approx \beta_0$) Analysis - High-Frequency Response & Absence of Miller Effect Multiplication - Solved Numerical Problem: Complete DC & AC Emitter Follower Analysis - Practical Application: High-Impedance Sensor Buffer Design - Large-Signal Drive Limitations & Asymmetrical Cutoff Clipping - Summary & Key Engineering Takeaways --- # Circuit Topology & DC Operating Point Analysis - Circuit Configuration: Collector connected directly to $V_{CC}$ (AC ground), input signal $V_i$ applied to Base, output signal $V_o$ taken from Emitter across resistor $R_E$. - Base Voltage DC Bias: Voltage divider biasing gives DC Base voltage $V_B = V_{CC} \frac{R_2}{R_1 + R_2}$. - Emitter DC Voltage & Current: Emitter DC voltage $V_E = V_B - V_{BE} = V_B - 0.7\text{ V}$. DC Emitter current $I_E = \frac{V_E}{R_E}$. - Dynamic Emitter Resistance ($r_e$): $r_e = \frac{V_T}{I_E} = \frac{26\text{ mV}}{I_E}$. Base-emitter dynamic resistance $r_\pi = (1 + \beta_0) r_e$. --- # Feedback Topology: 100% Voltage-Series Feedback - Feedback Signal Sampling: Output voltage $V_o = V_E$ is sampled directly at the emitter. - Series Feedback Injection: 100% of output voltage $V_o$ is fed back in series opposition to base input $V_i$: $v_{be} = v_i - v_o = v_b - v_e$. - Feedback Ratio: $\beta = 1.0$ ($100\%$ voltage-series negative feedback). - Closed-Loop Gain Formulation: Open-loop gain $A_v = g_m (R_E \parallel r_o) \gg 1$. Closed-loop voltage gain $A_{vf} = \frac{A_v}{1 + 1 \cdot A_v} = \frac{A_v}{1 + A_v} \approx 1.0$ V/V. --- # Small-Signal Hybrid-\pi$ AC Equivalent Circuit Model - AC Model Representation: Replace BJT with hybrid-\pi$ model featuring base-emitter resistance $r_\pi$, dependent current source $g_m v_\pi = \beta_0 i_b$, and output load $R_E' = R_E \parallel R_L$. - Input Voltage Loop Equation: $v_i = v_\pi + v_o = i_b r_\pi + i_e R_E'$. - Emitter AC Current Relationship: $i_e = i_b + \beta_0 i_b = (1 + \ensuremath{\beta}_0) i_b$. - Output Voltage Equation: $v_o = i_e R_E' = (1 + \beta_0) i_b R_E'$. --- # Exact Derivation of Voltage Gain ($A_v$) - Gain Definition: $A_v = \frac{v_o}{v_i} = \frac{(1 + \beta_0) i_b R_E'}{i_b r_\pi + (1 + \beta_0) i_b R_E'} = \frac{(1 + \beta_0) R_E'}{r_\pi + (1 + \beta_0) R_E'}$. - Simplification with $r_e$: Divide numerator and denominator by $(1 + \beta_0)$, substituting $r_e = \frac{r_\pi}{1 + \beta_0}$: - Final Voltage Gain Formula: $A_v = \frac{R_E'}{r_e + R_E'}$, where $R_E' = R_E \parallel R_L$. - Gain Characteristics: Since $R_E' \gg r_e$ (typically $R_E' \approx 1\text{ k}\Omega - 10\text{ k}\Omega$ vs $r_e \approx 5 - 25\ \Omega$), $A_v$ is positive (in-phase output, $0^\circ$ phase shift) and slightly less than unity ($0.98 - 0.999$). --- # Exact Derivation of Input Impedance ($R_{in}$) - Base Input Impedance ($R_{in,base}$): $R_{in,base} = \frac{v_i}{i_b} = \frac{i_b r_\pi + (1 + \beta_0) i_b R_E'}{i_b} = r_\pi + (1 + \beta_0) R_E'$. - Resistance Reflection Rule: Emitter resistance $R_E'$ reflected back into the base loop is multiplied by factor $(1 + \beta_0)$. - Total Stage Input Impedance ($R_{in}$): Including bias resistors $R_B = R_1 \parallel R_2$: - Final Input Impedance Formula: $R_{in} = R_B \parallel R_{in,base} = R_B \parallel \left[ r_\pi + (1 + \beta_0)(R_E \parallel R_L) \right]$. Reaches hundreds of $k\Omega$ to $M\Omega$. --- # Exact Derivation of Output Impedance ($R_{out}$) - Test Source Setup: Deactivate source ($v_s = 0\text{ V}$), apply test voltage $v_x$ at emitter port, measure test current $i_x$. - Base Loop Equation: $i_b = -\frac{v_x}{r_\pi + (R_S \parallel R_B)}$, where $R_S' = R_S \parallel R_B$. - Emitter Test Current: $i_x = -i_b - \beta_0 i_b + \frac{v_x}{R_E} = v_x \left[ \frac{1 + \beta_0}{r_\pi + R_S'} + \frac{1}{R_E} \right]$. - Final Output Impedance Formula: $R_{out} = \frac{v_x}{i_x} = R_E \parallel \left[ \frac{r_\pi + R_S'}{1 + \beta_0} \right] = R_E \parallel \left[ r_e + \frac{R_S \parallel R_B}{1 + \beta_0} \right]$. Yields extremely low output impedance ($10\ \Omega - 50\ \Omega$). --- # Current Gain ($A_i$) & Power Gain ($A_p$) Derivation - Transistor Current Gain: $A_{i,transistor} = \frac{i_e}{i_b} = 1 + \beta_0 \approx \beta_0$ (typically $100 - 300$). - Stage Current Gain ($A_{is}$): Accounting for bias divider splitting and load splitting: - Stage Current Gain Formula: $A_{is} = \frac{i_L}{i_s} = A_{i,transistor} \cdot \frac{R_B}{R_B + R_{in,base}} \cdot \frac{R_E}{R_E + R_L}$. - Power Gain ($A_p$): $A_p = A_v \cdot A_{is} \approx 1.0 \cdot A_{is} \approx \beta_0$. Although voltage gain is unity, the Emitter Follower provides high power gain driven entirely by current amplification! --- # High-Frequency Response & Absence of Miller Effect - Miller Effect Review: In Common-Emitter stages, base-collector capacitance $C_\mu$ is multiplied by $(1 + |A_v|)$, creating large input Miller capacitance $C_{in,M} = C_\mu (1 + A_v)$. - Common Collector Immunity: In Emitter Follower, Collector is AC grounded. Base-to-Collector voltage equals base voltage $v_b$, so $C_\mu$ is NOT multiplied by voltage gain ($A_{v,BC} = 0$). - Base-Emitter Capacitance Bootstrapping: Base-to-emitter voltage is $v_{be} = v_i(1 - A_v) \approx 0$. Effective $C_\pi$ input capacitance is bootstrapped down to $C_{\pi,in} = C_\pi (1 - A_v) \ll C_\pi$. - Ultra-Wide Bandwidth: Complete absence of Miller multiplication grants the Emitter Follower exceptionally high bandwidth, ideal for high-speed line drivers. --- # Numerical Problem: Complete AC & DC Emitter Follower Analysis - Given Parameters: $V_{CC} = 15\text{ V}$, $R_1 = 33\text{ k}\Omega$, $R_2 = 33\text{ k}\Omega$, $R_E = 3.3\text{ k}\Omega$, load $R_L = 1\text{ k}\Omega$, source $R_S = 600\ \Omega$, transistor $\beta_0 = 150$, $V_{BE} = 0.7\text{ V}$. - Step 1 - DC Analysis: $V_B = 15 \times \frac{33k}{66k} = 7.5\text{ V}$. $V_E = 7.5 - 0.7 = 6.8\text{ V}$. $I_E = \frac{6.8\text{ V}}{3.3\text{ k}\Omega} = 2.06\text{ mA}$. Dynamic resistance $r_e = \frac{26\text{ mV}}{2.06\text{ mA}} = 12.62\ \Omega$. $r_\pi = (151) \times 12.62 = 1.905\text{ k}\Omega$. - Step 2 - Voltage Gain ($A_v$): $R_E' = 3.3k \parallel 1k = 767.4\ \Omega$. $A_v = \frac{767.4}{12.62 + 767.4} = 0.9838\text{ V/V}$. - Step 3 - Input Impedance ($R_{in}$): $R_{in,base} = 1.905k + (151 \times 767.4) = 117.78\text{ k}\Omega$. $R_B = 33k \parallel 33k = 16.5\text{ k}\Omega$. $R_{in} = 16.5k \parallel 117.78k = 14.47\text{ k}\Omega$. - Step 4 - Output Impedance ($R_{out}$): $R_S' = R_S \parallel R_B = 600 \parallel 16.5k = 578.9\ \Omega$. $R_{out} = 3.3k \parallel \left[ 12.62 + \frac{578.9}{151} \right] = 3.3k \parallel [12.62 + 3.83] = 3.3k \parallel 16.45 = 16.37\ \Omega$! --- # Practical Application: High-Impedance Sensor Buffer Design - Problem Scenario: A piezoelectric transducer with high source impedance $R_S = 100\text{ k}\Omega$ drives a low impedance load $R_L = 100\ \Omega$. Compare direct connection versus inserting an Emitter Follower buffer ($R_{in} = 500\text{ k}\Omega$, $R_{out} = 10\ \Omega$, $A_v = 0.99$). - Direct Connection Signal Voltage: $V_L = V_S \frac{R_L}{R_S + R_L} = V_S \frac{100}{100,000 + 100} = 0.000999 V_S$ ($99.9\%$ of signal LOST due to loading!). - Buffered Connection Signal Voltage: - Input voltage $V_{in} = V_S \frac{R_{in}}{R_S + R_{in}} = V_S \frac{500k}{600k} = 0.8333 V_S$. - Output voltage $V_L = A_v V_{in} \frac{R_L}{R_{out} + R_L} = (0.99)(0.8333 V_S) \frac{100}{10 + 100} = 0.750 V_S$. - Performance Gain: Signal voltage transfer restored from $0.1\%$ to $75.0\%$ — a $750$-fold improvement in voltage delivered to the load! --- # Limitations of Emitter Follower: Cutoff & Clipping - Asymmetrical Drive Capability: Emitter follower can source large transient currents (transistor turns ON hard), but sinking current is strictly limited by DC bias current $I_E = \frac{V_E - V_{EE}}{R_E}$. - Negative Swing Cutoff Clipping: On large negative input voltage swings, if load current demand $\frac{\Delta V}{R_L} > I_E$, transistor cuts off completely ($I_C = 0$), clipping the negative waveform. - Complementary Push-Pull Solution: Replace single transistor and resistor $R_E$ with NPN-PNP complementary pair (Class-AB Emitter Follower). - Push-Pull Advantage: NPN transistor sources positive load current; PNP transistor sinks negative load current, achieving symmetrical drive capability. --- # Summary & Key Takeaways: Emitter Follower Circuit - Voltage Gain & Phase: Near-unity voltage gain ($A_v = \frac{R_E'}{r_e + R_E'} \approx 0.98 - 0.999$), non-inverting ($0^\circ$ phase shift). - Impedance Transformation: High input impedance ($R_{in} = R_B \parallel [r_\pi + (1+\beta_0)R_E']$) and low output impedance ($R_{out} = R_E \parallel [r_e + \frac{R_S'}{1+\beta_0}]$). - Feedback & Bandwidth: Operates as 100% voltage-series feedback amplifier ($\beta = 1.0$), immune to Miller multiplication, yielding ultra-wide bandwidth. - Buffer Application: Ideal impedance matching buffer bridging high-impedance sensors and low-impedance loads.