# SINGLE-TIME-CONSTANT NETWORK

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```SINGLE-TIME-CONSTANT NETWORK
STC Circuits
Time constant τ = CR for an RC network; τ = L/R for an LR network.
Example: Reduce the network to an STC network, and find its time constant.
Thevenin
Rth=(R1||R2+R3)||R4
Rth=
Vth
∴τ = Rth ⋅ C = [ R4 || (( R1 || R2 ) + R3 )] ⋅ C
Example: Reduce to a STC, and obtain τ.
C1
VI
+
C2
R
VO
-
Cth
C th = C1 + C 2
∴τ = R(C1 + C 2 )
Example: COMPENSATED ATTENUATOR
USING SUPERPOSITION
LPF
HPF
vo = vo1 + v o 2
τ = ( R1 || R2 )(C1 + C 2 )
Classification of STC Networks
STC: HP “High-Pass”
LP “Low-Pass”
Examples:
Low-Pass Type:
High-Pass Type:
FREQUENCY RESPONSE OF STC NETWORKS
Low-Pass Networks
T (S ) =
Or
output
K
=
input 1 + ( S / ω 0 )
T ( jω ) =
ω0 is cut-off frequency, which is constant.
K
1 + j (ω / ω 0 )
ω0 =
1
τ
τ = time constant
Example:
VO ( s)
1 / SC
1
1
1
=
=
=
=
VI ( s) R + 1 / SC 1 + SRC 1 + Sτ 1 + ( S / ω 0 )
| T ( jω ) |=
K
1+ (
ω 2
)
ω0
φ (ω ) = − tan −1 (ω / ω 0 )
Let
ω
&gt;&gt; 1
ω0
∴ | T ( jω ) |≈ K
ω0
ω
i.e. if ω doubles in value (an octave), |T(jω)| is halved.
20 log10 0.5 = −6dB / octave
if ω increases by a factor of 10 (a decade), |T(jω)| drops to 1/10th of its value.
10 log10 0.1 = −20dB / decade
High-Pass Network
Vo ( s )
s /ω0
R
sCR
s
=
=
=
=
Vi ( s ) R + 1 1 + sCR 1 + ( s / ω 0 ) s + ω 0
sC
In general:
T ( s) =
Ks
s + ω0
| T ( jω ) |=
φ = tan −1 (
Kjω
=
jω + ω 0
ω0
)
ω
K
1− j
ω0
ω
=
K
1 + (ω 0 / ω ) 2
```