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Kirchhoff''''s Current Law KCLKirchhoff''''s Current Law KCL states that the algebraic sum of currents leaving anynode or the algebraic sum of currents entering any node is zero.. Kirchhoff''''s

ĐẠI HỌC BÁCH KHOA HÀ NỘI TRƯỜNG ĐIỆN - ĐIỆN TỬ–––––––––––oOo–––––––––––

BÁO CÁO THÍ NGHIỆM

Lab #1: Kirchhoff's Current and Voltage Laws

I OBJECTIVES

To learn and apply Kirchhoff's Current Law

To learn and apply Kirchhoff's Voltage Law

To obtain further practice in electrical measurements

Compare experimental results with those using hand calculations

II BACKGROUND AND THEORY

1 Kirchhoff's Current Law (KCL)

Kirchhoff's Current Law (KCL) states that the algebraic sum of currents leaving anynode or the algebraic sum of currents entering any node is zero KCL can be stated asthe sum of the currents entering a node must equal the sum of the currents leaving anode

Kirchhoff's Current Law can also be expressed as follows:

a) The algebraic sum of the currents entering a junction (node) equals zero b) The algebraic sum of the currents leaving a junction (node) equals zero.c) The algebraic sum of currents entering a node equal to the algebraic sum ofcurrents leaving a node

2 Kirchhoff's Voltage Law (KVL)

Kirchhoff's Voltage Law (KVL) states that the algebraic sum of voltages around aclosed path is zero As you make a summation of voltages, it is suggested that youproceed around the closed path in a clockwise direction If you encounter a positive(+) sign as you first enter the circuit element, add that value On the contrary, if youfirst encounter a negative sign as you enter the circuit element, then subtract the value

of that voltage

We can state Kirchhoff's Voltage Law in three ways which are all equivalent:

The algebraic sum of the voltage drops around any closed path of an electriccircuit equals zero

The algebraic sum of the voltage rises around any closed path of an electriccircuit equal to zero

The algebraic sum of the voltage rises equals the algebraic sum of thevoltage drops around any closed path of an electric circuit

III EQUIPMENT AND PARTS LIST

2 Let R1=100Ω, R2 = 500Ω, R3= 50Ω, and R4= 150Ω in the equations of Step 1.Solve these equations by hand for V1 and V2

From V1 and V2 find Va, Vb, Vc, Ia, Ib, and lc

3 Measure the resistors Use these values to find V1, V2, Va, Vb, Vc, Ia, Ib, and Ic as

in Step 2

4 Construct the circuit shown in the Figure below

5 Use the multimeter to measure the indicated three currents and five voltages

6 Compare the results of Step 2 with those obtained from the theoretical calculations

of Steps 2 and 3 above

7 Using the measured values of the three currents, check KCL at node 2

8 Use your measured values of the source voltage, Va Vb, and Vc to checkKirchhoff's Voltage Law for the outer loop of the circuit

V CALCULATIONS AND COMPARISONS

Vb=Ia R 2=37.65 ×10 × 500 18.825= (V )Vc=Ic R 4=9.45× 10−3

Experimental results (mA) Theoretical results (mA) Differences (%)

=> The difference between experimental and theoretical results is less than 1%, which

is acceptable

VI CONCLUSIONS

The measured value are reasonably accurate and are roughly the same as the nominalvalues

Lab #2: Nodal Analysis

I OBJECTIVES

- To learn and apply Nodal analysis

- To obtain further practice in electrical measurements

- Compare experimental results with those using hand calculations

II BACKGROUND AND THEORY

Analysis of electrical networks involves the determination of node voltages and loop

or branch currents Nodal analysis refers to the technique of writing equations wherethe unknown quantities are the node voltages of the circuit Kirchhoff's current law isused to define the equations at each node in the circuit, using currents obtained byOhm's law

III EQUIPMENT AND PARTS LIST

- One useful property of the bridge circuit is the so-called balance condition thatoccurs when the relationship among the bridge resistors ("legs") is such that(R1/R2)-(R3/R4) Note that in the balance state the node voltages V2 and V3are equal, meaning that the current RS must be zero Also note that the balancecondition depends only upon the resistor ratios, not the applied voltage V

- The bridge circuit can be used to determine the value of an unknown resistor ifseveral known resistors are available For example, if the R3 leg was anunknown resistor we could use various combinations of known resistors for theR1, R2, and R4 legs until the balance condition was achieved We would thenknow that R3= R4*(R1/R2)

- Using the resistance and voltages given in the Figure, determine all nodevoltages and branch currents in the circuit Using the voltage and currentvalues, calculate the power dissipated by each resistor

- Determine a new value for R4, leaving the other resistors unchanged, that willbalance the bridge circuit Assume an adjustable resistor is used so that anyresistance value can be obtained

- Construct the circuit shown in the Figure

- Measure and record the resistance of each resistor you use.

- Use the multimeter to adjust the power supply Measure and record all nodevoltages and branch currents

Lab #4: Thévenin and Norton Equivalent Circuits

I OBJECTIVES

- To understand the Thévenin's and Norton equivalent of a circuit

- To check the experimental values versus calculated values

- To find the conditions for maximum power delivered to a load

- To build a Thévenin equivalent of the original circuit and check to see if itreally is equivalent

II BACKGROUND & THEORY

Voc and R , are computed as follows: V = Vt t oc

Rt=vt

isc

=voc

iscThévenin's and Norton circuits are equivalent and if one is known, the other is alsoeasily determined Actually, measuring the short circuit current in a real circuit is oftennot recommended (the circuit may not be designed to handle the high current) and maydamage the circuit Circuits with no independent sources require a different technique;

a source must be connected to the output and the current or voltage measured

2 Maximum Power Transfer

The load resistance that absorbs the maximum power from a two-terminal circuit isequal to the Thévenin resistance

The maximum voltage at the output of a linear source is:

inThe maximum current at the output of a linear source is:

imax=isc=in=vt

RtThe maximum power delivered by a linear source to matched load resistance R = RL T

= R is:n

pmax= vt 2

Take an additional measurement that will allow you to construct the Theveninequivalent

2 Place the following "loads" across the output and record voltage across and currentthrough.

V EXPERIMENT RESULTS

Vopen = 10.05 (V)

Ishort = 26.02 (mA)

Equivalent Thevenin’s resistor: 380.64 (Ω)

When loads are placed:

With this additional measurement data, we can construct the Thevenin equivalentcircuit, which consists of a voltage source (Vth) in series with a resistor (Rth) TheThevenin voltage will be the open-circuit voltage measured earlier, and the Theveninresistance will be calculated based on the load measurements

Lab #5: Power Relationship

II BACKGROUND AND THEORY

An arbitrary connection between a source and a load can be depicted in Fig 1 Notethat the voltage source and source resistance Rs could actually be the Theveninequivalent circuit for a more complicated network

Case 1: Maximum voltage transfer

To maximize the load voltage Vload the relationship between Rs and Rload can beexpressed as:

Vload= RlRs+Rl×Vsource

Which means that Vload is always less than Vsource, and is maximized for Rload >>Rs

So in order to maximize the voltage at the load, we must design the circuit so that theload resistance is much greater than the source resistance

Case 2: Maximum current transfer

If we need to supply the maximum available current from the source to the load, weneed a different relationship between RL and Rs Specifically:

Iload=Vsource/(Rs+RL)

Assuming the source resistance is fixed, the load current is maximized for RL << RS

Case 3: Maximum power transfer

To maximize the power delivered to the load, assuming the source resistance is fixed,

we need to maximize the power expression with respect to RL

Pload=Vload 2

III EQUIPMENT AND PARTS LIST

Consider the circuit in Fig 2 Then use specific resistance values to calculate thepower dissipated in each of the resistors with the optimum load resistor attached Use aprogram to solve for the branch currents if you wish.

4.2 Laboratory experiment

1 Assemble the circuit in Fig 2 using the supply for Vsource, R =330Ω, and RS L=1Ω.Use the multimeter to set Vsource to 12V (DC), Measure and record the voltage acrossRL

2 Now replace the 1Ω load resistors in turn with the nominal resistor values (resistorper value) in your lab kit (10Ω, 100Ω, 200Ω, etc.) and record the load voltage.Remember to record the actual values of the resistors you use 3 Construct the circuitshown in Fig 2 using the nominal resistor indicated (record the values used) Set thepower supply to 12V Using the symbolic expression you derived in the pre-lab andthe measured values of your resistors, calculate the value of R for the maximum powertransfer Construct a resistor of this value using a carefully adjusted 1kΩ potentiometerand/or resistors from your lab kit and attach it to your circuits Measure the voltageacross your load resistor and across each of the other resistors in the circuit Alsomeasure the current flowing in the power supply

4 Now replace the load resistor in tum with each of the nominal resistor values in yourlab kit (1kΩ, 100Ω, 200Ω, etc.) and record the load voltage in each case

V CALCULATIONS AND COMPARISONS

We have

50 Ω 50 Ω 150 Ω 500 Ω 100 Ω

For this experiment, first we found the equivalent resistor of the circuit by Thevenin’stheorem and confirmed it with the actual measurement of open voltage and shortcircuit current The equivalent resistor is:

Req❑=145.74 Ω

Now with the circuit following Figure 2, we replace the load resistor with someavailable resistors in the lab and record the voltage and current at the load We came

up with the table:

R load (Ω) V load (V) V load/V source I load (A) P load (W)

0 50 100 150 200 250 0

0 50 100 150 200 250 0

VI CONCLUSIONS

- I load decreases with R load

- V load/V source increases with R load

- P load reaches a maximum at the nearest value to the equivalent resistor

Lab #8: Second Order Circuit

II BACKGROUND AND THEORY

The characteristic equation is derived as

ζ is called the damping factor: ω is called the undamped natural resonant frequency If

ζ < 1 the system response is underdamped

ζ = 1 the system response is critically

ζ > 1 the system response is overdamped

With (1) and (2), solving the equation we have:

ωn=√ 1LC

2 ζωn=RtotalL

III EQUIPMENT AND PARTS LIST

- Digital multimeter (DMM)

- DC power supply

IV EXPERIMENT RESULTS

1 Circuit for experiment

- Actual experiment data: