Question

Problem 1 Hi-fi audio amplifiers are usually built as discrete operational amplifiers with a relatively large gain (g500) and with feedback that reduces the closed-loop gain to a smaller value k-32. Since loudspeakers predominatly act like a resistor (nominal impedance: 8) with an inductor in series, the feedback network can be used to improve the frequency re sponse in the audio range up to 20 kHz. We are primarily concerned with gain and frequency response of the amplifier Amp Vour(s) C1 Feedback GrB(S) Discrete operational our(s) V(s) Isp(S) Ls R GfB(S) VeB(s) Figure Simplified sche matic (A) and block diagram representation (B) of a hi-fi ampli fier output stage with feedback network and loudspeaker impedance. The magnetic force that drives the membrane is proportional to the coil current Is, and the coil current is therefore the output variable that we are interested in.

Hi-fi audio amplifiers are usually built as discrete operational amplifiers with a
relatively large gain (g ? 500) and with feedback that reduces the closed-loop gain
to a smaller value k = 32. Since loudspeakers predominatly act like a resistor
(nominal impedance: 8
) with an inductor in series, the feedback network can be
used to improve the frequency response in the audio range up to 20 kHz. We are
primarily concerned with gain and frequency response of the amplifier. One of the key features of amplifiers with feedback is that the gain is dominated
by the external feedback network, which allows larger variations of the
internal gain g. Presently, the feedback network is a 1:32 resistor divider with
the capacitor C1 omitted. Therefore, VFB(t) = VOUT (t)/32. What is the actual gain of the amplifier k = VOUT (t)/V(t)? [5 points]
Note: Do not use the ideal op-amp model, because g = 500 does not allow the
approximation g ? ?. With manufacturer tolerances, the internal gain g can vary between 300 and
700. What is the range of variation for k? A loudspeaker is connected to the amplifier. Its resistive (DC-) impedance
is 8
, but the coil and crossover add an inductive impedance of L=0.8mH.
To obtain the coil current IS PK(s), Ohm’s Law can be applied, but with the
combined resistive and inductive impedance: Plot the frequency response (Bode diagram) of the combined transfer function
IS PK(s)/V(s). Clearly indicate the asymptotes and the cutoff frequency. How
strong is the attenuation of the signal at the higher end of the audio range at 15
kHz, compared to low frequencies? [10 points]
Note: Please do not confuse the linear frequency f and the angular frequency
! = 2 f . Since a frequency response as seen in C. is clearly unacceptable, the capacitor
C1 is used to decrease the feedback signal at higher frequencies. Components
are chosen as follows: R1=320k
, R2=10k
, C1=10nF. The transfer function
of the feedback network is therefore Determine the combined transfer function IS PK(s)/V(s) when the frequency
compensation capacitor C1 is installed and determine the poles and zeros (if
any) of the new transfer function. Plot the Bode diagram of the new transfer function. Clearly indicate the asymptotes
and the new cutoff frequency. How strong is the attenuation of the signal
at the higher end of the audio range at 15 kHz now? [10 points]
Note: One zero and one pole have very similar locations on the s-plane. For
the Bode plot, you may assume that they have the same frequency and cancel
out.

Answer 1