Bipolar Junction Transistor (BJT)- Amplifiers

• Sinusoidal AC source causes base voltage to vary around the DC bias \(V_{BQ}\).

• The base current variation leads to a larger variation in collector current due to transistor current gain.

• Collector voltage decreases as the collector current increases, and vice versa.

• The collector-to-emitter voltage varies out of phase with the base voltage.

• Phase inversion: There is always a phase shift between base and collector voltage.

• The AC load line differs from the DC load line:

  • The effective AC collector resistance is \(R_L\) in parallel with \(R_C\), which is less than the DC collector resistance \(R_C\) alone.

• The distinction between the DC and AC load lines is explored further in the context of power amplifiers.

• Datasheets often provide only common-emitter \(h\) parameters, so \(r\)-parameters can be obtained: \[\begin{aligned}&{r'_e}=\frac{h_{re}}{h_{oe}}\\&r_{c}^{\prime}=\frac{h_{re}+1}{h_{oe}}\\&r_{b}^{\prime}=h_{ie}-\frac{h_{re}}{h_{oe}}(1+h_{fe})\end{aligned}\]

• Amplifiers operate in both ac and dc modes, but the common-emitter designation specifically refers to the ac operation.

• The output signal is 180° out of phase with the input signal.

How Phase Inversion Occurs:

• As \(V_{in}\) changes \(\Rightarrow\) \(I_B\) changes \(\Rightarrow\) \(I_C\) changes \(\Rightarrow\) deviates Q-point.

• If \(I_B\uparrow \Rightarrow~I_C\uparrow \Rightarrow\) voltage drop across the collector resistor \(R_C~\uparrow\)

• This leads to a decrease in the voltage at the collector.

• Thus, a change in \(V_{in}\) causes the opposite change in the collector voltage, resulting in phase inversion.

Conclusion

• The common-emitter amplifier provides voltage amplification and \(180^\circ\) phase inversion.

• Determine DC bias values by removing AC components (coupling and bypass capacitors) since they appear open for DC.

• Eliminate load resistor and signal source, not needed for DC analysis.

\[\begin{aligned} V_{\mathrm{B}} & =V_{\mathrm{E}}+0.7 \mathrm{~V}=2.7 \mathrm{~V} \\ V_{\mathrm{C}} & =V_{\mathrm{CC}}-I_{\mathrm{C}} R_{\mathrm{C}}=12 \mathrm{~V}-(3.58 \mathrm{~mA})(1.0 \mathrm{k} \Omega)=8.42 \mathrm{~V} \\ V_{\mathrm{CE}} & =V_{\mathrm{C}}-V_{\mathrm{E}}=8.42 \mathrm{~V}-2 \mathrm{~V}=6.42 \mathrm{~V} \end{aligned}\]

AC Signal Operation of an Amplifier

• Capacitors as Shorts:

  • Capacitors \(C_1\), \(C_2\), and \(C_3\) are replaced by shorts as their values make the capacitive reactance (\(X_C\)) negligible at the signal frequency.

• DC Source as Ground:

  • The DC source is replaced by ground in AC analysis because it has negligible internal resistance and does not produce an AC voltage, acting as an AC short.

• AC Equivalent Circuit:

  • Both \(R_C\) and \(R_1\) are connected to AC ground, which is linked to \(V_{CC}\) in the actual circuit.

  • The bypass capacitor \(C_2\), which keeps the emitter at AC ground, making it the circuit’s common point.

Signal (AC) Voltage at the Base

• AC Voltage Source Connection:

  • When an AC source \(V_s\) is connected to the base, the base voltage depends on various resistances.

• Factors Affecting Signal Voltage:

  • If the source’s internal resistance is \(0 \, \Omega\), the entire source voltage appears at the base.

  • With a nonzero internal resistance, three resistances must be considered:

    • Source Resistance (\(R_s\))

    • Bias Resistance (\(R_1 \parallel R_2\))

    • AC Input Resistance at the Base (\(R_{in(base)}\))

• Total Input Resistance:

  • The total input resistance, \(R_{in(tot)}\), is the parallel combination of \(R_1\), \(R_2\), and \(R_{in(base)}\): \[R_{in(tot)} = R_1 \parallel R_2 \parallel R_{in(base)}\]

• Voltage Divider Effect:

  • The source voltage, \(V_s\), is divided between \(R_s\) and \(R_{in(tot)}\), so the voltage at the base is given by: \[V_b = \left( \frac{R_{in(tot)}}{R_s + R_{in(tot)}} \right) V_s\]

• Special Case:

  • If \(R_s \ll R_{in(tot)}\), then \(V_b \approx V_s\).

  • This means the voltage at the base is nearly equal to the input voltage if the source resistance is much smaller than the total input resistance.

• Design Trade-Offs:

  • High input resistance is desired to minimize signal loading.

  • Small resistances are necessary for stable biasing.

  • Achieving both high input resistance and stable biasing involves trade-offs in component selection.

Input Resistance at the Base

• Use the simplified \(r\)-parameter model of the transistor to derive the AC input resistance at the base.

Output Resistance of Common-Emitter Amplifier

• The output resistance of a common-emitter amplifier : \[R_{\text{out}} \approx R_C\]

• The more detailed expression is:

  \[R_{\text{out}} = R_C \parallel r_c'\] where \(r_c'\) is the internal AC collector resistance of the transistor.

• Since \(r_c' >> R_C\), the approximation is usually valid.

Voltage Gain

• Equation represents Voltage gain (\(A_\nu\)) from base to collector .

• Overall Amplifier Gain: from the source voltage to the collector, we must include the attenuation of the input circuit.

• Attenuation: The reduction in signal voltage as it passes through a circuit, corresponding to a gain of less than 1.

• Attenuation Example:

  • If the signal amplitude is reduced by half:

    • Attenuation = 2.

    • Gain = \(\frac{1}{\text{Attenuation}} = 0.5\).

  • Case Study:

    • Input signal: 10 mV.

    • Output signal: 2 mV.

    • Attenuation = \(\frac{10\,\text{mV}}{2\,\text{mV}} = 5\).

    • Gain = \(\frac{1}{5} = 0.2\).

• Key Takeaways

  1. Attenuation reduces the signal voltage as it passes through a circuit.

  2. Attenuation and gain are inversely related: \[\text{Gain} = \frac{1}{\text{Attenuation}}\]

  3. The overall gain of the amplifier depends on both the voltage gain (\(A_\nu\)) and the attenuation of the input circuit.

\[\begin{aligned} \text{Attenuation} & = \frac{V_s}{V_b} = \frac{R_s + R_{in(tot)}}{R_{in(tot)}} \\ A_v^{\prime} & =\left(\frac{V_c}{V_b}\right)\left(\frac{V_b}{V_s}\right)=\frac{V_c}{V_s} \end{aligned}\]

Effect of Emitter Bypass Capacitor on Voltage Gain

• \(C_2\) shorts AC signal around \(R_E\), maintaining emitter at AC ground.

• Maximum gain achieved: \[\text{Gain} = \frac{R_C}{r_e^{\prime}}\]

• \(X_C\) must be very small compared to \(R_E\) across operating frequencies.

• Rule-of-thumb: \[10X_C \leq R_E \quad \text{at minimum operating frequency.}\]

• Ensure \(X_C \approx 0\,\Omega\) for optimal performance.

Without Bypass Capacitor

• Without a bypass capacitor, the emitter isn’t at AC ground.

• \(R_E\) appears in the AC path, increasing, reducing AC voltage gain. \[A_v=\frac{R_{\mathrm{C}}}{r_e^{\prime}+R_{\mathrm{E}}}\]

Effect of a Load on the Voltage Gain

• The total ac collector resistance \[R_c=\frac{R_\mathrm{C}R_L}{R_\mathrm{C}+R_L}\]

• Replacing \(R_C\) with \(R_c\) \[A_v=\frac{R_c}{r_e^{\prime}}\]

• If \[\begin{aligned} &R_c < R_C ~(\text{because of}~R_L) \Rightarrow A_v \downarrow \\ &R_L >> R_C \Rightarrow R_c \cong R_C \Rightarrow \text{very little effect of load} \end{aligned}\]

Stability in Amplifiers:

• Stability measures how well an amplifier maintains its design values over changes in temperature or transistor parameters (e.g., \(\beta\)).

• \(A_\nu\) depends on \(r_e'\), which is sensitive to temperature and \(I_E\). \[A_\nu = \frac{R_C}{r_e'} \Leftarrow r_e^\prime~ \text{changes, gain becomes unstable.}\]

• No Bypass Capacitor: \[\begin{aligned} A_\nu & = \frac{R_C}{r_e' + R_E} \\ A_\nu & \approx \frac{R_C}{R_E} \qquad R_E \gg r_e' ~~ \text{(stable but lower gain)} \end{aligned}\]

Swamping Technique

• A compromise between fully bypassed and unbypassed \(R_E\).

• Reduces the effect of \(r_e'\) on \(A_\nu\).

• \(R_E\) is split into two resistors: \(R_{E1}\) (unbypassed) and \(R_{E2}\) (bypassed).

• Both \(R_{E1}\) and \(R_{E2}\) affect DC bias , but only \(R_{E1}\) affects AC gain.

  \[A_\nu = \frac{R_C}{r_e' + R_{E1}} \approx \frac{R_C}{R_{E1}} ~~ (R_{E1} \gg r_e')\]

• Achieves reasonable gain while maintaining stability.

• Swamping increase the input resistance \[R_{in(base)}=\beta_{ac}(r_{e}^{\prime}+R_{\mathrm{E}1})\]

• Current Gain

  \[\begin{aligned} \text{base-to-collector~} \beta_{ac} &= \dfrac{I_c}{I_b} \\ \text{overall~} A_i &= \dfrac{I_c}{I_s} \\ I_s &= \dfrac{V_s}{R_s + R_{\text{in(tot)}}} \end{aligned}\]

• Power Gain \[\begin{aligned} A_p & = A_v^\prime \cdot A_i \qquad (A_v^\prime=V_c/V_s) \end{aligned}\]

• Input Signal: Applied to the base via a coupling capacitor

• Output Signal: Taken from the emitter, also coupled via a capacitor

• Collector: AC ground (no effect on AC signal)

• Voltage Gain

  \[\begin{aligned} V_{out} & =I_eR_e \\ V_{in} &=I_e(r_e^{\prime}+R_e) \\ A_\nu & =\dfrac{V_{out}}{V_{in}} =\frac{I_eR_e}{I_e(r_e^{\prime}+R_e)} =\frac{R_e}{r_e^{\prime}+R_e} <1 \quad (R_e = R_E || R_L) \\ A_{v}&\cong1 \quad (R_{e}>r_{e}^{\prime})\\ \end{aligned}\]

  

• Input Resistance: High \(R_{in}\), making it a useful buffer circuit.

  \[\begin{aligned} R_{in(base)} & =\frac{V_{in}}{I_{in}}=\frac{V_b}{I_b}=\frac{I_e(r_e^{\prime}+R_e)}{I_b} \\ &\cong\frac{\beta_{ac}I_b(r_e^{\prime}+R_e)}{I_b} \quad (I_e\cong I_c=\beta_{ac}I_b) \\ &\cong\beta_{ac}(r_{e}^{\prime}+R_{e}) \\ &\cong\beta_{ac}R_{e} \quad (\text{if}~ R_{e}>r_{e}^{\prime}) \\ R_{in(tot)} & =R_1\parallel R_2\parallel R_{in(base)} \end{aligned}\]

• Output Resistance: With the load removed, \[R_{out}\cong\left(\frac{R_s}{\beta_{ac}}\right)\parallel R_\mathrm{E}\]

• Current and Power Gain \[\begin{aligned} & A_i=\frac{I_e}{I_{i n}} \quad (I_{i n}=V_{i n} / R_{\text {in }(t o t)} .)\\ & A_p=A_v A_i \cong A_i \quad (A_v \cong 1) \end{aligned}\]

• Application:

  • A common-emitter amplifier with a 1.0 k\(\Omega\) collector resistor (R\(_C\)) must drive a low-resistance load (8\(\Omega\) speaker).

  • The speaker is capacitively coupled to the amplifier’s output, causing the load to appear in parallel with the collector resistor.

  

• Parallel Resistance Calculation: \[R_c = R_C \parallel R_L = 1.0\ \text{k}\Omega \parallel 8\ \Omega = 7.94\ \Omega\]

• Impact on Voltage Gain:

  • Without load: \[A_v = \frac{R_C}{r_e'} = \frac{1.0\ \text{k}\Omega}{5\ \Omega} = 200\]

  • With 8\(\Omega\) speaker load: \[A_v = \frac{R_c}{r_e'} = \frac{7.94\ \Omega}{5\ \Omega} = 1.59\]

  • Conclusion: The voltage gain is significantly reduced due to the low resistance of the speaker.

• Solution: Use an emitter-follower configuration with a Darlington pair to buffer the amplifier and speaker, restoring voltage gain and preventing excessive loading.

\[\begin{aligned} \text{Voltage Gain~} A_{\nu} & =\frac{V_{out}}{V_{in}}=\frac{V_{c}}{V_{e}}=\frac{I_{c}R_{c}}{I_{e}(r_{e}^{^{\prime}}\parallel R_{\mathrm{E}})}\cong\frac{I_{e}R_{c}}{I_{e}(r_{e}^{^{\prime}}\parallel R_{\mathrm{E}})} \\ &\cong\frac{R_c}{r_e^{^{\prime}}} ~~(\text{if~}R_E>>r_e^{\prime}) \end{aligned}\]

Note: \(A_\nu\) same as for the common-emitter amplifier, but no phase inversion from emitter to collector.

\[\begin{aligned} \text{Input resistance~}R_{in(emitter)} & =\frac{V_{in}}{I_{in}}=\frac{V_{e}}{I_{e}}=\frac{I_{e}(r_{e}^{\prime}\|R_{\mathrm{E}})}{I_{e}} \\ & \cong r_{e}^{\prime} ~~ (\mathrm{If~}R_\mathrm{E}>r_e^{\prime}) \\ \text{Output resistance~} R_{out}&\cong R_{\mathrm{C}} \\ \text{Current gain~} A_i \cong 1 \\ \text{Power gain~} A_p & = A_v \end{aligned}\]

Basic operation of a differential amplifier:

• Both inputs grounded:

• Bias voltage on input 1 with input 2 grounded:

• Bias voltage on input 2 with input 1 grounded

Final Outcome:

• The increase in \(I_{C2}\) results in a decrease in \(V_{C2}\).

• The decrease in \(I_{C1}\) results in an increase in \(V_{C1}\).

• Modes of Signal Operation:

  • Single-Ended Differential Input

  • Double-Ended Differential Input

  • Common-Mode Input

• Key role: Amplify desired signals while rejecting unwanted noise.

Single-Ended Differential Input

• One input grounded, signal applied to the other.

• Two cases:

  • Signal applied to Input 1:

    • Inverted, amplified signal at Output 1.

    • Noninverted, amplified signal at Output 2.

  • Signal applied to Input 2:

    • Inverted, amplified signal at Output 2.

    • Noninverted, amplified signal at Output 1.

Double-Ended Differential Inputs

• Two opposite-polarity (out-of-phase) signals applied to both inputs.

• Each input affects the output:

  • Input 1 alone produces certain output signals.

  • Input 2 alone produces another set of output signals.

  • Superimposing both results in final output signal.

Common-Mode Inputs

• Identical signals (same phase, frequency, and amplitude) applied to both inputs.

• Individual input effects:

  • Input 1 alone produces signals of opposite polarity at outputs.

  • Input 2 alone does the same.

  • When both signals are applied, they cancel each other out.

Importance of Common-Mode Rejection

• Eliminates unwanted signals (noise) appearing on both inputs.

• Noise sources:

  • Radiated energy from adjacent lines.

  • 60 Hz power line interference.

  • Other electromagnetic sources.

• Enhances signal integrity in communication and instrumentation applications.

Common-Mode Rejection Ratio (CMRR)

• Desired signals appear on only one input or with opposite polarities on both inputs.

• These desired signals are amplified and appear on the outputs.

• Unwanted signals (noise) appearing with the same polarity on both inputs are canceled.

• The measure of an amplifier’s ability to reject common-mode signals is the Common-Mode Rejection Ratio (CMRR): \[\mathrm{CMRR} = \frac{A_{v(d)}}{A_{c m}}\]

• A high CMRR means high differential gain and low common-mode gain.

• CMRR is often expressed in decibels (dB): \[\mathrm{CMRR} = 20 \log \left(\frac{A_{v(d)}}{A_{c m}}\right)\]