Comprehensive Study Notes on Power Amplifiers and Hybrid Equivalent Models

 # Introduction to Power Amplifiers and Operating Classes

An amplifier serves to receive a signal from a pickup transducer or other input source and provide a larger version of that signal to an output device, such as a speaker, or to another amplifier stage. Input signals are generally small, ranging from a few millivolts for cassette or CD inputs to a few microvolts for antennas. Small-signal amplifiers focus on linearity and gain magnitude, whereas power amplifiers (large-signal amplifiers) are designed to provide sufficient power to a load, typically from a few watts to tens of watts. The main features of large-signal amplifiers include circuit power efficiency, the maximum power handling capacity, and impedance matching to the output device.

Amplifiers are categorized into classes based on the duration of the output signal variation relative to a full input cycle of $360^{\circ}$. This is described as the operating cycle of the amplifier. A class A amplifier is biased such that the output signal varies for a full $360^{\circ}$ of the cycle. This requires the Q-point to be biased at a level (typically one-half the supply voltage) so the signal can swing up and down without reaching limiting supply levels. Class B amplifiers provide an output signal for only one-half of the input cycle ($180^{\circ}$). The DC bias for class B is set at $0\,V$, meaning the output is not a faithful reproduction of the input unless a push-pull connection is used, combining two class B operations to cover both positive and negative half-cycles. Class AB amplifiers are biased above the zero-base-current level of class B but below the one-half supply level of class A. Their output swing occurs between $180^{\circ}$ and $360^{\circ}$. Class C amplifiers are biased for operation at less than $180^{\circ}$ of the cycle, and they operate specifically with tuned or resonant circuits to provide a full cycle at a fixed frequency, common in radio applications. Class D amplifiers use digital techniques or pulse signals (chopping), recreation of the full cycle through sample-and-hold circuitry, achieving very high efficiency since the amplifier is only "on" for short intervals.

Power efficiency, defined as the ratio of AC output power ($P_o(ac)$) to DC input power ($P_i(dc)$), increases as we move from class A to class D. Class A has poor efficiency, with a maximum of $25\%$ for series-fed loads and $50\%$ for transformer-coupled loads. Class B can reach a maximum theoretical efficiency of $78.5\%$. Class AB efficiency falls between the ratings for class A and class B. Class D is the most efficient, often exceeding $90\%$.

The Series-Fed Class A Amplifier

The series-fed class A amplifier is a simple fixed-bias circuit where the load is connected directly to the collector. The DC bias point is fixed by the supply voltage $V_{CC}$ and the base resistor $R_B$. The DC base current is determined by the formula $I_B = \frac{V_{CC} - 0.7\,V}{R_B}$. The resulting collector current is $I_C = \beta I_B$, and the quiescent collector-emitter voltage is $V_{CE} = V_{CC} - I_C R_C$. To allow for the largest possible signal swing, the Q-point should be set such that the DC bias collector current is one-half of the maximum possible swing ($V_{CC}/R_C$) and the quiescent collector-emitter voltage is one-half of the supply voltage.

Power input from the DC supply is calculated as $P_i(dc) = V_{CC} I_{CQ}$. Even with an AC signal applied, the average current remains approximately the same in a class A series-fed amplifier. The AC power delivered to the load can be expressed in terms of RMS values as $P_o(ac) = V_{CE}(rms) I_C(rms)$, or alternatively as $P_o(ac) = I_C^2(rms) R_C = \frac{V_{CE}^2(rms)}{R_C}$. The percentage efficiency is calculated using \%\n = \frac{P_o(ac)}{P_i(dc)} \times 100\%. The maximum theoretical efficiency for this configuration is derived using maximum voltage ($V_{CC}$) and current ($V_{CC}/R_C$) swings. The maximum AC power is $P_o(ac)_{max} = \frac{V_{CC}^2}{8R_C}$, while the maximum DC input power is $P_i(dc)_{max} = \frac{V_{CC}^2}{2R_C}$, leading to a maximum efficiency of $25\%$.

Example 12.1 illustrates these concepts. For a circuit with $V_{CC} = 20\,V$, $R_B = 1\,k\Omega$, $R_C = 20\,\Omega$, and $\beta = 25$, the Q-point components are $I_{BQ} = 19.3\,mA$, $I_{CQ} = 482.5\,mA$, and $V_{CEQ} = 10.4\,V$. If an input signal results in a peak base current of $10\,mA$, the peak collector current is $I_c(p) = \beta I_b(p) = 25 \times 10\,mA = 250\,mA$. The output power is $P_o(ac) = \frac{I_c^2(p) R_C}{2} = \frac{(250 \times 10^{-3}\,A)^2 \times 20\,\Omega}{2} = 0.625\,W$. The input power is $P_i(dc) = 20\,V \times 0.48\,A = 9.6\,W$, resulting in an efficiency of $6.5\%$.

Transformer-Coupled Class A Amplifiers

Transformer-coupled class A amplifiers achieve a higher maximum efficiency of $50\%$. A transformer allows for voltage and current levels to be stepped up or down and transforms the load impedance as seen at the primary. The voltage transformation ratio is $\frac{V_2}{V_1} = \frac{N_2}{N_1}$, and the current transformation ratio is $\frac{I_2}{I_1} = \frac{N_1}{N_2}$. The impedance transformation depends on the square of the turns ratio $a = \frac{N_1}{N_2}$. The reflected impedance $R'_L$ is given by $R'_L = a^2 R_L$. For example, a $15:1$ transformer with an $8\,\Omega$ load reflects an effective resistance of $1800\,\Omega$ ($1.8\,k\Omega$). To match a $16\,\Omega$ speaker to a $10\,k\Omega$ primary load, the required turns ratio is $a = \sqrt{\frac{10000}{16}} = 25:1$.

In transformer-coupled circuits, the DC resistance of the winding is typically very small. This results in a nearly vertical 0-ohm DC load line on collector characteristics, where $V_{CEQ} = V_{CC}$. The AC load line depends on the reflected resistance $R'_L$ and has a slope of $\frac{-1}{R'_L}$. The peak-to-peak signal swings are defined as $V_{CE(p-p)} = V_{CEmax} - V_{CEmin}$ and $I_{C(p-p)} = I_{Cmax} - I_{Cmin}$. AC power is calculated as $P_o(ac) = \frac{(V_{CEmax} - V_{CEmin})(I_{Cmax} - I_{Cmin})}{8}$.

Example 12.4 demonstrates this for a circuit with $V_{CC} = 10\,V$, a $3:1$ transformer, and an $8\,\Omega$ load. With a reflected resistance of $R'_L = 3^2 \times 8 = 72\,\Omega$, the Q-point is $V_{CEQ} = 10\,V$ and $I_{CQ} = 140\,mA$. Using graphical analysis, $V_{CEmax} = 18.3\,V$, $V_{CEmin} = 1.7\,V$, $I_{Cmax} = 255\,mA$, and $I_{Cmin} = 25\,mA$. The output power is $P_o(ac) = \frac{(18.3 - 1.7)(255 \times 10^{-3} - 25 \times 10^{-3})}{8} = 0.477\,W$. In Example 12.5, with input power $P_i(dc) = 10\,V \times 140\,mA = 1.4\,W$, the efficiency is $34.1\%$. Example 12.6 verifies that efficiency drops significantly for smaller voltage swings; for instance, a swing of $V(p) = 2\,V$ on a $12\,V$ supply yields only $1.39\%$ efficiency compared to $50\%$ for a full $12\,V$ swing.

Class B Amplifier Operation and Efficiency

Class B operation occurs when the transistor is biased at cutoff ($0\,V$ bias). Two transistors are required to handle opposite half-cycles in a configuration known as a push-pull circuit. This configuration provides greater efficiency than class A. The average current ($I_{dc}$) drawn from the supply for a peak output current $I(p)$ is given by $I_{dc} = \frac{2I(p)}{\pi}$, making total input power $P_i(dc) = V_{CC} \frac{2I(p)}{\pi}$. Output AC power is $P_o(ac) = \frac{V_L^2(p)}{2R_L}$. The efficiency equation is \%\n = \frac{\pi V_L(p)}{4 V_{CC}} \times 100\%. Maximum circuit efficiency occurs when $V_L(p) = V_{CC}$, reaching $78.54\%$.

Transistor power dissipation is the difference between input and output power: $P_{2Q} = P_i(dc) - P_o(ac)$. The total power dissipated by two transistors is split equally: $P_Q = \frac{P_{2Q}}{2}$. Maximum dissipation does not occur at maximum output, but rather when the peak load voltage is $V_L(p) = 0.636 V_{CC}$. At this point, $P_{2Qmax} = \frac{2 V_{CC}^2}{\pi^2 R_L}$.

Example 12.7 shows that for a class B amplifier with $V_{CC} = 30\,V$ driving a $16\,\Omega$ load at $20\,V$ peak, $I_{dc} = 0.796\,A$ and $P_i(dc) = 23.9\,W$. Output power is $P_o(ac) = 12.5\,W$, and efficiency is $52.3\%$. Example 12.8 calculates the maximums for the same circuit: maximum $P_o(ac) = 28.125\,W$, maximum $P_i(dc) = 35.81\,W$, and maximum power dissipated by each transistor is $5.7\,W$.

Practical Class B Push-Pull Circuit Configurations

Push-pull operation can be realized via several circuit topologies. Generating phase-inverted signals for the two amplifier stages can be achieved using a center-tapped transformer, a BJT phase-splitter stage with outputs from both collector and emitter, or op-amp stages (one inverting, one non-inverting). Complementary-symmetry push-pull circuits use complementary transistors (one NPN and one PNP). These transistors conduct on opposite half-cycles: the NPN on the positive and the PNP on the negative. A major disadvantage is crossover distortion, which occurs because transistors do not switch instantly at the zero-voltage crossover point. Biasing in class AB can mitigate this. High-power practical versions often use Darlington-connected transistors for higher output current and lower output resistance. Quasi-complementary push-pull amplifiers allow for the use of matched NPN power transistors for both output devices. This is achieved by using a NPN-NPN Darlington pair for one half and a PNP-NPN feedback pair for the other, maintaining complementary operation while using identical high-power output transistors.

Class C and Class D Amplifiers

Class C amplifiers are biased to operate for less than $180^{\circ}$ of the input signal. They utilize a tuned (LC tank) circuit at the output to maintain a full cycle at the resonant frequency. This specialization makes them unsuitable for audio, but ideal for fixed-frequency communications. Class D amplifiers are designed for high efficiency, typically exceeding $90\%$. They use digital or pulse-type signals to drive power transistors in an "on/off" manner, minimizing power loss since current flows primarily when voltage drop across the device is very low. Input sinusoidal signals are converted to pulse-width modulated (PWM) waveforms using a chopping waveform (sawtooth generator) and a comparator. The resulting digital waveform is then amplified and converted back to a sinusoidal signal using a low-pass filter. Power MOSFETs are popular driver devices for these circuits.

The Hybrid Equivalent Model and h-Parameters

The hybrid equivalent model (h-parameters) is a two-port system used for transistor analysis, often preferred because its parameters are standard listings on device specification sheets. There are four basic h-parameters: $h_{ie}$ (input impedance), $h_{re}$ (reverse transfer voltage ratio), $h_{fe}$ (forward transfer current gain), and $h_{oe}$ (output admittance). For the common-emitter configuration, the small-signal variables are related by the equations $V_1 = h_{ie} I_1 + h_{re} V_2$ and $I_2 = h_{fe} I_1 + h_{oe} V_2$. These parameters vary with operating conditions like collector current ($I_C$) and supply voltage ($V_{CE}$). In approximate hybrid models, $h_{re}$ and $h_{oe}$ are often ignored if their values are sufficiently small relative to circuit resistors, simplifying the circuit to $h_{ie}$ and a current source $h_{fe} I_B$.

Example 5.20 demonstrates analysis for a fixed-bias configuration with $R_B = 330\,k\Omega$, $R_C = 2.7\,k\Omega$, $h_{fe} = 120$, $h_{ie} = 1.175\,k\Omega$, and $h_{oe} = 20\,\mu A/V$. The input impedance is $Z_i = R_B \parallel h_{ie} \approx 1.171\,k\Omega$. The output impedance is $Z_o = \frac{1}{h_{oe}} \parallel R_C \approx 2.56\,k\Omega$. The voltage gain is $A_v = \frac{-h_{fe}(R_C \parallel 1/h_{oe})}{h_{ie}} \approx -262.34$. Example 5.21 performs calculations for a common-base configuration. With $R_E = 2.2\,k\Omega$, $R_C = 3.3\,k\Omega$, $h_{fb} = -0.99$, and $h_{ib} = 14.3\,\Omega$, the results are $Z_i = R_E \parallel h_{ib} \approx 14.21\,\Omega$, $Z_o \approx 3.3\,k\Omega$, and $A_v = \frac{-h_{fb} R_C}{h_{ib}} \approx 229.91$. The hybrid model parameters can be converted to other models (like the $r_e$ model) using relationships such as $h_{ie} = \beta r_e$ and $h_{fe} = \beta$.