ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Control Systems

Tutorial 1

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

April 15, 2023 Dr Wazhar - FKM Kuktem 2

What is the meaning of single loop and multi-loop electrical network?

Single loop network(one mesh or current)

Multi-loop electrical network(more than one mesh or current)

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Creating transfer function of multi-loop electrical network

Steps:1. Decide input and output2. Obtain relationship between

voltage and current (Ohm’s law)

3. Convert to frequency domain4. Obtain transfer function

between output and input

Example 1

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Step 1: Decide input and output

Input:Supply voltage v(t)

Output:Capacitor voltage vc(t)

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Step 2: Write equations

(𝑖¿¿1 (𝑡 )−𝑖3 (𝑡 ))+( 𝑑𝑖1 (𝑡 )𝑑𝑡

−𝑑 𝑖2 (𝑡 )𝑑𝑡 )2+( 𝑖1 (𝑡 )−𝑖2 (𝑡 ))=𝑣 (𝑡)¿

Taking Laplace transform

)()1()21()22( 321 sVIIsIs

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Step 2: Write equations

4( 𝑑𝑖2 (𝑡 )𝑑𝑡

−𝑑𝑖3 (𝑡 )𝑑𝑡 )+3( 𝑑𝑖2 (𝑡 )

𝑑𝑡 )+ (𝑖2 (𝑡 )−𝑖1 (𝑡 ) )+2( 𝑑𝑖2 (𝑡 )𝑑𝑡

−𝑑𝑖1 (𝑡 )𝑑𝑡 )=0

Taking Laplace Transform

0)4()91()21( 321 IsIsIs

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Step 2: Write equations

4( 𝑑𝑖3 (𝑡 )𝑑𝑡

−𝑑𝑖2 (𝑡 )𝑑𝑡 )+1∫ 𝑖3 (𝑡 )𝑑𝑡+(𝑖3 (𝑡 )− 𝑖1 (𝑡 ) )=0

Taking Laplace Transform

0)11

4()4()1( 321 Is

sIsI

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Alternatively the circuit can be written in s-domain directly (OPTIONAL)

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Step 3: Obtain relationship between voltage and current (Ohm’s Law)

0)11

4()4()1( 321 Is

sIsI

0)4()91()21( 321 IsIsIs

)()1()21()22( 321 sVIIsIs

0

0

)(

)14(41

4)19()12(

1)12()22(

3

2

1

2

sV

I

I

I

s

sss

sss

ss

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Step 4: Obtain transfer function between output and input

)(

)(

sVOutput

sVInput

c

)(1

)( 3 sIssVc

0

0

)(

)14(41

4)19()12(

1)12()22(

3

2

1

2

sV

I

I

I

s

sss

sss

ss

Need to solve this simultaneous equations for I3

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Step 4: Obtain transfer function between output and input

0

0

)(

)14(41

4)19()12(

1)12()22(

3

2

1

2

sV

I

I

I

s

sss

sss

ss

sss

s

sss

ss

s

ss

sVss

sI

)14(41

4)19()12(

1)12()22(

041

0)19()12(

)()12()22(

)(

2

3

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Step 4: Obtain transfer function between output and input

132207722

)1138()(

423

2

3

ssss

sssVsI

)(1

)( 3 sIssVc

132207722

1138

)(

)(

)(

)()(

423

2

ssss

ss

sV

sV

sR

sCsG C

Therefore,

Since,

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Example 2

Find the transfer function VL(s)/V(s)

Redrawing the circuit in s-domain

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Step: 2 Writing equations

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Example 3

Find the transfer function X1(s)/U(s) and X2(s)/U(s).

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Example 3

Taking Laplace transform

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

Example 3

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Example 4

Determine the transfer function Y2(s)/F(s)

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Example 4