Download Example 1.5.

1.4. The Source-Free Parallel RLC Circuits
 Applying KCL at the top node gives;
 takig derivative with respect to t and dividing by C results in;
1.4. The Source-Free Parallel RLC Circuits
𝑑2
𝑑𝑡 2
𝑠2
𝑑
𝑑𝑡
s
1.4. The Source-Free Parallel RLC Circuits
Overdamped Case (∝> 𝝎𝟎 )
𝑠1 𝑎𝑛𝑑 𝑠2 negative and real
Critically Damped Case (∝= 𝝎𝟎 )
𝑠1 𝑎𝑛𝑑 𝑠2 real and equal
1.4. The Source-Free Parallel RLC Circuits
Under Damped Case (∝< 𝝎𝟎 )
𝑠1 𝑎𝑛𝑑 𝑠2 are complex
𝐴1 𝑎𝑛𝑑 𝐴2 can be determined from initial conditions;
Example 1.5.
In the parallel circuit of fig. , find 𝑣(𝑡) for t>0,
assuming
𝑣 0 = 5 𝑉, 𝑖 0 = 0, 𝐿 = 1𝐻 𝑎𝑛𝑑 𝐶 = 10 𝑚𝐹.
Consider these cases;
R=1.923 Ω, R=5 Ω, R=6.25 Ω.
İf R=1.923 Ω
Example 1.5.
Since (∝> 𝝎𝟎 ) in this case, overdamped.
The roots of the characteristic equation are;
Apply initial conditions to get 𝑨𝟏 and 𝑨𝟐
At t =0
Example 1.5.
Must be differeantiated
At t =0
With 𝑨𝟏 and 𝑨𝟐 the solution gets…
Example 1.5.
When R=5 Ω
𝜔0 remains 10;
Since α= 𝜔0 , the responce is
Critically damped…
Apply initial conditions to get 𝑨𝟏 and 𝑨𝟐
Example 1.5.
Must be differeantiated
With 𝑨𝟏 and 𝑨𝟐 the solution gets…
Example 1.5.
İn the last case R=6.25 Ω…
Solution is…
Response for three
degrees of damping
1.5. Step Response of Series RLC Circuits
This equation has two compenants;
Natural response;
𝒗𝒏 (𝒕)
Forced response;
𝒗𝒇 (𝒕)
1.5. Step Response of Series RLC Circuits
 İf we set 𝑉𝑠 = 0, 𝑣(𝑡) contains only natural response (𝑣𝑛 )
 𝑣𝑛 (t) can be expressed as three conditions;
 The forced response is the steady-state or final value of 𝑣(𝑡)
 The final value of the capacitor voltage is the same as the source
voltage 𝑉𝑠 .
1.5. Step Response of Series RLC Circuits
 Thus the complate solution…
Example 1.6.
For the circuit in Fig., find
𝑣 𝑡 𝑎𝑛𝑑 𝑖(𝑡) for t>0. Consider these
cases:
R=5Ω, R=4Ω,R=1Ω.
Example 1.6.
Example 1.6.
 𝑣𝑓 is the forced response or steady-state
response.
 It is final value of the capacitor voltage.
 𝑣𝑓 =24 V.
For t>0 , current i,
Example 1.6.
 Finally,
 but we have to solve i(t)…
1.6. Step Response of Parallel RLC
Circuits
Example 1.7.
Solution:
 For t<0, the switch is open
 The circuit partitioned into two independent subcircuits.
Capacitor voltage equal the
voltage of 20Ω resistor.
Example 1.7.
 For t>0, the switch is closed
 We have parallel RLC circuit.
 The voltage source is off or short-circuited.
Example 1.7.
The final value of I…
Using initial conditions we get 𝑨𝟏 and 𝑨𝟐
Example 1.7.
From i(t) we obtain v(t)…