Poles And Zeros

Trigonometric Explanation for falling gain at poles atom

Aim • Attempt to explain why the gain falls at poles, since it sounds. • First principle interpretation- Poles are frequencies of infinite gain. I would imagine encountering a pole should increase the gain.

sin2θ + cos2θ = 1

Understanding the sine wave

45°

Between 90° to 45° the SINE function falls from a 1 to a 0.707 and from 45° to 0° it falls from 0.707 to a 0. Which means first 45° takes the value down by ~30% and the next 45° takes it down by ~70%.

90°

sin2θ + cos2θ = 1

Understanding the sine wave Differentiating the SINE wave (meaning determining the slope of SINE) tells us that the slope is maximum near 0° and is least at 90°

45° 90°

sin2θ + cos2θ = 1

Stable System Pole

-ve S-Plane

Amplitude -σ1 + jω1

jω

X -σ

jω1 -σ1

-jω

σ sin2θ + cos2θ = 1

-ve S-Plane Amplitude -σ1 + jω1

jω

X -σ

jω1 -σ1

-jω

σ sin2θ + cos2θ = 1

-ve S-Plane Amplitude -σ1 + jω1

jω

X σ1

-σ

jω1 -σ1 When this distance is equal to this distance we say we encountered a pole

-jω

And here the gain starts to fall

σ sin2θ + cos2θ = 1

Amplitude -σ1 + jω1

jω

X θ

p

-σ

jω1

b

-σ1 When this (b) distance is equal to This (p) distance then this θ is equal to 45°

-jω

σ sin2θ + cos2θ = 1

this (h) is the distance of the pole from the frequency axis (freq at which we are bothered about the gain)

Amplitude h

jω

-σ1 + jω1 X θ

p

-σ

jω1

-σ1

-jω

σ sin2θ + cos2θ = 1

b

this (h) is the distance of the pole from the frequency axis (freq at which we are bothered about the gain)

Amplitude h -σ1 + jω1 X

θ1

jω

θ

p

-σ -σ1

-jω

jω1

b

h = p * cosec (θ)

Which means the ‘h’ starts increasing more rapidly after 45°, and therefore the distance from the pole start increasing exponentially and hence the effect of pole starts falling more rapidly. σ sin2θ + cos2θ = 1

h = p * cosec (θ) • Since the θ goes from 90° to ~0° going away from the pole the cosec function goes from 1 to ∞. • Which means the ‘h’ starts increasing more rapidly after 45°, hence the effect of pole falls more rapidly. • It is this rapidly reducing pole effect that causes the gain to fall when we encounter a pole. • Similarly encountering a zero has the reverse effect. It is the effect of zero reduces rapidly and hence the gain increases because of other poles now that the zero is having exponentially lesser and lesser effect. • The reason SINE slope was analyzed in the beginning is to use the fact that the slopes are changing before and after 45° sin2θ + cos2θ = 1