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audio_engineering:class_b_power_calculations

# Power Calculations for Class B Amplifiers

For the calculations below, the following are assumed: v_o is the amplifier output voltage, i_o is the output current, and R_L is the load resistance. It's also assumed the amplifier has a bipolar power supply with rails V_(C C) and V_(E E), where V_(E E) = -V_(C C). We further assume that transistor base currents are negligible (so collector currents equal output currents) and that the zero-conduction dead zone likewise is negligible.

These analyses are based on sinusoidal signals as this is the accepted standard for power and thermal design in audio. The formulae will be valid for other signals with a crest factor of sqrt(2). Alternative analyses based on worst-case signals (i.e., those having a crest factor of one, e.g, square waves), may be instructive but are not presented here.

A majority of the following is derived from material presented by Dr. Cristina Crespo1) 2) 3).

## Power into the load

The average output power bar(P_L) into load R_L for a sine wave with amplitude v_(op) is calculated as:

bar(P_L) = v_(o RMS) * i_(o RMS) = v_(o RMS) * v_(o RMS)/R_L = (v_(op)/sqrt(2))^2/R_L = (v_(op))^2/(2R_L)

Power into the load

bar(P_L) = (v_(op))^2/(2R_L)

The maximum power into the load happens when v_(op) is at its maximum possible value of V_(C C).

## Required supply power for given output

We start by calculating bar(P_S)_(V+), the average power over one cycle delivered by the power supply to the transistor connected to V_(C C) (i.e., the “positive” transistor). Letting i_(C+) be the collector current flowing through that transistor, this is calculated as:

bar(P_S)_(V+) = 1/T int_(0)^(T) V_(C C) * i_(C+) dt

 = 1/T int_(0)^(T/2) V_(C C) * v_o/R_L dt
 = 1/T int_(0)^(T/2) V_(C C) * v_(op)/R_L*sin(omega t) dt

Note the change in the upper integral limit from from T to T/2. This is because i_(C+) from T/2 to T is zero.

Expanding the above yields:

bar(P_S)_(V+) = 1/T int_(0)^(T/2) V_(C C) * v_(op)/R_L*sin(omega t) dt

 = 1/T int_(0)^(T/2) V_(C C) * v_(op)/R_L*sin(2pi f t) dt
 = 1/T int_(0)^(T/2) V_(C C) * v_(op)/R_L*sin((2pi)/T t) dt

Solving the integral using:

int sin(ax) dx = -1/a cos(ax)

yields:

bar(P_S)_(V+) = 1/T[V_(C C) * v_(op)/R_L * (-T)/(2pi) * cos((2pi)/T t)]_(t=T/2) - 1/T[V_(C C) * v_(op)/R_L * (-T)/(2pi) * cos((2pi)/T t)]_(t=0)

 = 1/T * V_(C C) * v_(op)/R_L * (-T)/(2pi) * [cos((2pi)/T t)_(t=T/2) - cos((2pi)/T t)_(t=0)]
 = -(V_(C C) * v_(op))/(2pi R_L) * [cos((2pi)/T t)_(t=T/2) - cos((2pi)/T t)_(t=0)]
 = -(V_(C C) * v_(op))/(2pi R_L) * [cos((2pi)/T T/2) - cos((2pi)/T 0)]
 = -(V_(C C) * v_(op))/(2pi R_L) * [cos(pi) - cos(0)]
 = -(V_(C C) * v_(op))/(2pi R_L) * (-1 - 1)
 = -(V_(C C) * v_(op))/(2pi R_L) * (-2) = (V_(C C) * v_(op))/(pi R_L) = 1/pi * v_(op)/R_L * V_(C C)

Owing to symmetry, the transistor connected to V_(E E) will consume the same average power over one cycle as the one connected to V_(C C), therefore the total average power delivered by the power supply is:

bar(P_S) = bar(P_S)_(V+) + bar(P_S)_(V-) = bar(P_S)_(V+) + bar(P_S)_(V+) = 2 bar(P_S)_(V+)

Substituting yields:

bar(P_S) = 2/pi * v_(op)/R_L * V_(C C)

Required supply power for given output bar(P_S) = 2/pi * v_(op)/R_L * V_(C C)

## Maximum required supply power

From the above we can see that bar(P_S) increases linearly with v_(op). Therefore, the maximum power delivered by the power supply bar(P_S)_max will happen when v_(op) = V_(C C). Thus:

bar(P_S)_max = 2/pi * V_(C C)/R_L * V_(C C) = 2/pi * (V_(C C))^2/R_L

 = 2/pi * 2bar(P_L)_max = 4/pi * bar(P_L)_max

Maximum required supply power

bar(P_S)_max = 2/pi * (V_(C C))^2/R_L
or
bar(P_S)_max = 4/pi * bar(P_L)_max

## Power dissipation

The power dissipated as heat in both transistors P_D is the difference between the power consumed and the power delivered to the load:

P_D = bar(P_S) - bar(P_L)

 = (2/pi * v_(op)/R_L * V_(C C)) - [(v_(op))^2/(2R_L)]
 = ((2 V_(C C))/(pi R_L) * v_(op)) - [(v_(op))^2/(2R_L)]

## Maximum power dissipation

The plot of P_D vs. v_(op) describes a broad curve that intersects the origin when v_(op) = 0 and has a peak somewhat before its final value when v_(op) = V_(C C). To find the maxima of this curve, we solve for (d P_D)/(d v_(op)) = 0. Thus:

(d P_D)/(d v_(op)) = (2 V_(C C))/(pi R_L) - 2(v_(op)/(2R_L))  = (2 V_(C C))/(pi R_L) - v_(op)/(R_L)

The above is zero when:

v_(op) = 2/pi V_(C C)

Substituting this into the relation for P_D above yields the maximum (i.e., worst-case) power dissipation P_(Dmax):

P_(Dmax) = ((2 V_(C C))/(pi R_L) * 2/pi V_(C C)) - [(2/pi V_(C C))^2/(2R_L)]

 = (2/pi V_(C C))^2 *1/R_L - [(2/pi V_(C C))^2/(2R_L)]
 = (2/pi V_(C C))^2 *1/(2R_L)
 = 2/pi^2 * (V_(C C))^2/R_L

Alternatively,

P_(Dmax) = (2/pi V_(C C))^2 *1/(2R_L)

 = (2/pi)^2 * (V_(C C))^2/(2R_L)
 = (2/pi)^2 P_(L max)  = 4/pi^2 P_(L max)

Maximum power dissipation

P_(Dmax) = 2/pi^2 * (V_(C C))^2/R_L
or
P_(Dmax) = 4/pi^2 P_(L max)

occurs when

v_(op) = 2/pi V_(C C)

Remember that this is the total dissipated power. It will be spread uniformly across all output devices.

## Power efficiency

The power efficiency eta is defined as the ratio of the power delivered to the load to the power consumed:

eta = bar(P_L) / bar(P_S)

Substituting values for sinusoidal signals yields:

eta = [(v_(op))^2/(2R_L)] / [2/pi * v_(op)/R_L * V_(C C)] = pi/4 * v_(op)/V_(C C)

The power efficiency when the amplifier is at maximum power dissipation is interesting though not terribly useful:

eta_("at max power dissipation") = pi/4 * (2/pi V_(C C))/V_(C C)  = 1/2 = 50%

More useful is the maximum efficiency, which happens when v_(op) = V_(C C) :

eta_max = pi/4 * V_(C C)/V_(C C) = pi/4 ~~ 78.5%

Maximum power efficiency

eta_max ~~ 78.5%

1)
Cristina Crespo. “Class B Output Stage: Push-Pull Network.” Video. YouTube, May 23, 2017. https://www.youtube.com/watch?v=xXIZOg0BKag.
2)
———. “Class B Push-Pull Stage: Power Calculations.” Video. YouTube, May 23, 2017. https://www.youtube.com/watch?v=YltVJ3KzaR4.
3)
———. “Example: Class B Output Stage.” Video. YouTube, May 23, 2017. https://www.youtube.com/watch?v=et5wzj2Xh7E.
audio_engineering/class_b_power_calculations.txt · Last modified: 2023/08/01 23:47 by mithat