Document Sample

Efficiency in Power Amplifiers Consider a power amplifier in which, at maximum signal amplitude, the transistor current exhibits a conduction angle where can take on values between 0 and 2 radians. For convenience, we take the time origin at the peak of the current so that we have conduction for 2 t 2 . For the moment, we assume for simplicity that . Then we can write, for the current at maximum signal, I m cos t cos 2 ; 2 t 2 iT t 0; o th e r w is e What we have done, essentially, is to subtract the constant cos 2 from the cosine so that the difference goes to 0 at t 2 . We simply define iT t to be zero for larger absolute values of t . For smaller input signals, the output current variations are smaller by a factor , 0 1, so that we can write I m c o s t c o s 2 ; cos t cos 2 iT t 0; o th e r w is e where ranges from 0 for no signal to 1 for the largest possible signal (that is, the signal such that if it were larger, the transistor collector to emitter voltage would try to change sign). For conduction angles , iT t is identically zero for sufficiently small signals (sufficiently small 's ). That is, for sufficiently small 's , cos t is never greater than cos 2 . In particular, iT t 0 if cos 2 . If cos 2 , then I m c o s t c o s 2 ; 2 t 2 iT t 0; o th e r w is e where 1 a rc c o s cos / 2 cos / 2 /2 0 cos / 2 Note that is the conduction angle for the largest possible signal, whereas is the (smaller) conduction angle for smaller signals. If cos / 2 , then 0 (no conduction). If 1 (largest signal), then , the usual conduction angle. We can easily generalize these results for 0 2 : I m c o s t c o s 2 ; 2 t 2 iT t 0; o th e r w is e where 1 a rc c o s cos / 2 cos / 2 /2 H cos / 2 where H is the Heaviside unit step function: 1 x 0 H (x) 0 x 0 These generalized results show that for sufficiently small signals cos / 2 , the current is either always zero 0 if , or never zero if . Sometimes it is important to relate Im to the permissible peak current through the transistor, I peak . Note from the results above that: I p e a k I m (1 c o s / 2 ) so that we can express Im as: I peak Im (1 c o s / 2 ) For calculating amplifier efficiency, however, we work with Im for simplicity in notation. With these results, we can calculate the average power drawn from the power supply. If the power supply voltage is V , then the instantaneous power delivered is V iT t so that the average power over a cycle is T /2 1 Pd c V iT t d t T T / 2 where T 2 . Using the expression for iT t , we find 1 Pd c V I m T cos t cos / 2 dt where, by definition, is chosen such that 2 . Let's change variables of integration: let t . Then / 2 1 Pd c V I m cos cos / 2 d 2 / 2 Because the integrand is even in and the limits of integration are symmetric about the origin, we can multiply the integral by 2 and integrate over only half the range: / 2 1 Pd c V I m c o s c o s / 2 d 0 Perform the integration: 1 / 2 Pd c V I m cos cos / 2 0 1 Pd c V I m s in cos 2 2 2 Here is a plot of the normalized DC power, p d c Pd c V I m : Notice that for 360 (class A amplifiers), the required power is constant, independent of the relative amplitude of the signal, while for smaller conduction angles, the required power increases with signal amplitude. For 180 , notice that no power is required for small signals because they fail to produce current in the transistor. We next calculate the power dissipated by the transistor. Suppose the voltage across the transistor is vT t . The instantaneous power dissipated by the transistor is v T t iT t . The average power dissipated over a cycle is 1 PT v T t iT t d t T The (ideal, undistorted) form of the voltage across the transistor that sees a resistive load is sinusoidal with a minimum corresponding to the current maximum: vT t V 1 cos t * where V * depends upon the circuit configuration. For the usual transformer coupled class A and B amplifiers, for typical tuned circuit class A, B, and C amplifiers and for direct-coupled split power supply configurations, V * is just the power supply voltage, V: V * V For RC-coupled class A amplifiers, typically V * V 2. In any case, the power dissipated is 1 PT V I m 1 c o s t c o s t c o s / 2 d t * T Change the variable of integration to t . 2 1 PT V I m 1 cos cos cos / 2 d * 2 2 As before, the integrand is even and the limits of the integral are symmetric so we can simplify the limits of integration: 2 1 PT V I m 1 cos cos cos / 2 d * 0 / 2 1 c o s cos cos / 2 cos cos / 2 d PT V I m * 2 2 0 / 2 1 1 c o s / 2 c o s c o s / 2 cos d PT V I m * 2 2 0 / 2 V Im * 1 PT 1 c o s s in c o s s in c o s 2 2 2 2 0 1 * V Im PT 1 c o s s in s in 2 cos cos 2 2 2 2 2 2 2 2 This expression gives the power dissipated in the transistor in terms of the conduction angle, , and the signal size (normalized to the maximum signal), . For class A amplifiers 360 , notice that the greatest dissipation occurs for the smallest signals. For class B amplifiers 180 , close examination of the plot shows that the transistors work harder for intermediate sized signals than for large signals. Let's specialize to the cases for which V * V . Such a restriction excludes mainly RC coupled amplifiers. In these amplifiers, a resistor in series with the transistor drops the voltage across the transistor to a value significantly less than the power supply voltage and in the process drops the efficiency by dissipating power in addition to that necessary in other configurations. As a consequence, RC coupled amplifiers are not very useful for power amplifiers, anyway. For the common case V * V , we can calculate the power in the signal as Ps ig Pd c PT because there is no resistor in the circuit to dissipate any power. Thus V Im 1 Ps ig s in 1 c o s s in s in 2 cos cos cos 2 2 2 2 2 2 2 2 2 2 2 V Im 1 Ps ig s in 1 c o s s in s in 2 cos cos cos 2 2 2 2 2 2 2 2 2 2 2 V Im 1 Ps ig s in s in cos s in 2 cos s in cos cos 2 2 2 2 2 2 2 2 2 2 2 2 V Im 1 Ps ig c o s s in s in 2 cos 2 2 2 2 2 2 1 Ps ig V I m s in c o s c o s s in 2 2 2 2 2 2 Here is a plot of the normalized signal power, p s ig Ps ig V I m : Not surprisingly, the signal power increases with signal amplitude. For 180 , note again that the signal amplitude must exceed some threshold before any signal power is produced. The decrease of the signal power with decreasing conduction angle may seem to be inconsistent with the use of conduction angles of 180 (class B and C) in most high power amplifiers. For high power amplifiers, however, the efficiency Ps ig Pd c is a primary concern. s in cos c o s s in Ps ig 2 2 2 2 2 2 , Pd c s in cos 2 2 2 Here is a plot of the efficiency: This plot clearly shows the high efficiencies available at small conduction angles and large amplitudes. Operating too close to the maximum, however, can mean that a small change in signal amplitude can result in no power output at all. A typical compromise is to choose large amplitude operation 1 with 120 . To examine the efficiency at large amplitudes in more detail, we specialize to the case 1 . In that case, and becomes: 1 s in s in c o s s in Ps ig 2 2 2 2 2 2 m ax Pd c s in cos 2 2 2 s in s in 1 2 2 2 m ax 2 s in cos 2 2 2 2 s in s in 1 2 2 m ax 4 s in cos 2 2 2 If we employ the trigonometric identity s in 2 x 2 s in x c o s x we can write 1 s in m ax 4 s in cos 2 2 2 It is easy to see that for class A amplifiers 2 , m ax 0 .5 . (With RC coupling, it is lower by the factor V * V 0 .5 .) For class B amplifiers , m ax 4 7 8 % . For class C amplifiers ( small), it is not so easily shown (with L’Hospital’s rule) that m a x approaches 1 . Here is a plot of maximum efficiency vs. conduction angle: Power Amplifier Efficiency 1.2 maximum efficiency, etamax 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 conduction angle, theta

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 5 |

posted: | 12/6/2011 |

language: | English |

pages: | 11 |

OTHER DOCS BY nuhman10

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.