Chapter 7 Review

Key Terms

Cramer’s rule
An explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.


linear circuit
An electronic circuit in which, for a sinusoidal input voltage of frequency

any steady-state output of the circuit (the current through any component, or the voltage between any two points) is also sinusoidal with frequency



load resistor
An electrical component or portion of a circuit that consumes electric power. Normally, the term “load” is used to refer specifically to the active component for which power consumption is mainly intended, as opposed to internal resistance or resistors used in conjunction with capacitors or inductors for timing.



maximum power transfer theorem
Maximum power is transferred to a load when the internal resistance equals the load resistance; for a Thévenin-equivalent circuit, this is true when the load resistance equals the Thévenin resistance.

mesh analysis
A method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in an electrical circuit.

mesh current
The currents defined in mesh analysis as flowing around each loop of a planar circuit. The actual current through each branch of the circuit is found as a combination of the mesh currents in that branch.

planar circuit (mesh)
An electrical circuit that can be drawn on a plane surface with no wires crossing each other.

superposition theorem
The response (voltage or current) in any branch of a linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by short circuits.

Thévenin-equivalent circuit
A simple circuit, equivalent to any more complex linear circuit, involving the Thévenin voltage and Thévenin resistance relative to a particular load. In the Thévenin-equivalent circuit, the Thévenin voltage and resistance are connected in series with the load.

Thévenin resistance
One component of a linear circuit’s Thévenin-equivalent, found by removing the load resistor from the original circuit and calculating the total equivalent resistance between the two load connection points.

Thévenin’s theorem
An electrical circuit theorem through which any complex linear circuit may be replaced by its Thévenin-equivalent with respect to a given load.

Thévenin voltage
One component of a linear circuit’s Thévenin-equivalent, found by removing the load resistor from the original circuit and calculating the potential difference from one load connection point to the other.



Key Equations


Cramer’s rule
Current through a load resistor
Voltage across a load resistor
Power dissipated in a load resistor


Summary

7.1 Mesh Analysis

  • Steps in mesh analysis method:
    1. Draw mesh current loops, ensuring:
      1. each loop is unique; and
      2. all circuit elements—voltage sources, resistors, capacitors, inductors, etc. and short circuits—are covered by at least one loop.
    2. Apply loop rule as described in Kirchhoff’s Rules (particularly with reference to Figure 6.3.5) and solve simultaneous equations.
    3. Add or subtract mesh currents in branches that are covered by multiple loops, depending on the direction of each loop and the sign of each current calculated in step 2.
  • When and are known, the elements of are given by Cramer’s rule:


where

is the

-th element of

and

is the matrix formed by replacing the

-th column of

with

.

  • Steps to read off and directly from a planar circuit:
    1. Draw one mesh current loop inside each loop of the circuit.
    2. Work your way around each loop , reading off terms and as:
      1. is the sum for each voltage source that passes from negative to positive, and for each voltage source that passes through from positive to negative,
      2. is the sum for each resistor that passes, and
      3. is the sum for each resistor passed by a mesh current adjacent to .

7.2 Superposition Theorem

  • Linear circuits can be analysed by calculating contributions to current for each voltage source independently
  • Steps in superposition method:
    1. Replace all potential sources but one with a short circuit; find the voltage/current through each branch of the network.
    2. Repeat for each potential source.
    3. Add up all the separate voltages/currents in each branch.

7.3 Thévenin’s Theorem

  • Any linear circuit containing several voltage sources and resistors can be simplified to an equivalent circuit with a single voltage source and resistance connected in series with a load. Specifically, the three components connected in series are:
    1. Load resistor, ;
    2. Thévenin voltage , found by removing from the original circuit and calculating the potential difference from one load connection point to the other;
    3. Thévenin resistance , found by removing from the original circuit and calculating the total equivalent resistance between the two load connection points.
  • Once the Thévenin voltage and resistance are determined, the Thévenin-equivalent circuit is redrawn with the Thévenin voltage attached to the Thévenin resistance in series, and any load resistance attached between the two connection points
  • Maximum power transfer theorem: maximum power is transferred to a load when the internal resistance equals the load resistance; for a Thévenin-equivalent circuit, this is true when the load resistance equals the Thévenin resistance

Answers to Check Your Understanding

7.1 a.

b.

c.

7.2 Replacing the

6.00~\mathrm{V}

source with a short means current from the

9.00~\mathrm{V}

source passes through the short. With the

9.00~\mathrm{V}

source replaced with a short, the current through the

7.00~\Omega

resistor due to the

6.00~\mathrm{V}

source is

Therefore, the current through the

7.00~\Omega

resistor is

I_{7\Omega}=\frac{6.00}{7.00}~\mathrm{A},

and the power dissipated is

P_{7\Omega}=\left(\frac{6.00}{7.00}~\mathrm{A}\right)^2(7.00~\Omega)=5.14~\mathrm{W}.

In fact, since the

7~\Omega

-resistor is connected directly to the terminals of the

6.00~\mathrm{V}

source, the current is

I_{7\Omega}=\frac{6.00}{7.00}~\mathrm{A}

regardless of the value of the

9.00~\mathrm{V}

battery.

7.3.

V_{\mathrm{th}}=V_1

,

R_{\mathrm{th}}=R_3

.


Conceptual Questions

7.1 Mesh Analysis

1. Sally and Frank are asked to find the

\mathbf{R}

and

\mathbf{V}

matrices for the circuit below. They both follow the procedure for reading off matrix elements described in Mesh Analysis. When comparing results, they find that they have the same elements in the first rows of both

\mathbf{R}

and

\mathbf{V}

, but the elements in the second and third rows of their matrices are swapped. How did this happen? Did one of them make an error? If given values for the voltages and resistances, should they expect to find the same currents through each component? Explain.

7.2 Superposition Theorem

2. How much power is used by the

9~\Omega

-resistor in the circuit below? How much power is used by the

3~\Omega

-resistor? Explain why the answers are obvious, with no need to do any work. (Hint: this problem is best approached using superposition method).

3. (a) Use superposition theorem to determine the potential difference across the resistor in the circuit below. (b) Explain why mesh analysis is a bad approach to use for this problem (e.g. you could demonstrate this by applying the mesh strategy and Cramer’s rule to calculate mesh currents and interpret the result).

7.3 Thévenin’s Theorem

4. Efficiency. The efficiency of a circuit is defined as the ratio between the power used by a load and the total power used. (a) Show that the general expression for efficiency can be written as

For any given

V_{\mathrm{th}}

and

R_{\mathrm{th}}

, show that: (b) the condition for minimum efficiency (

R_L\gg R_{\mathrm{th}}

) corresponds to maximum current, but the voltage across and power dissipated at the load are both small; (c) the condition for maximum efficiency (

R_L\ll R_{\mathrm{th}}

) corresponds to maximum voltage across the load, but current through and power dissipated at the load are both small; (d) the condition of maximum power transfer (

R_L=R_{\mathrm{th}}

), while only moderately efficient, corresponds to both moderate voltage across and moderate current through the load. (Hint: You may find it useful to refer to Equations 7.3.1-7.3.3).

Problems

7.1 Mesh Analysis

5. Apply Cramer’s rule to find expressions for

I_1

and

I_2

in terms of the given resistances and voltages.

6. Refer to the circuit in Problem 5, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

R_1=3~\Omega

,

R_2=5~\Omega

, and

R_3=7~\Omega

. Apply Cramer’s rule to calculate the values of

I_1

and

I_2

.


7. Apply Cramer’s rule to find expressions for

I_1

and

I_2

in terms of the given resistances and voltages.

8. Refer to the circuit in Problem 7, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

V_3=3~\mathrm{V}

,

R_1=5~\Omega

,

R_2=7~\Omega

,

R_3=11~\Omega

, and

R_4=13~\Omega

. Apply Cramer’s rule to calculate the values of

I_1

and

I_2

.

9. Apply Cramer’s rule to find expressions for

I_1

,

I_2

, and

I_3

in terms of the given resistances and voltages.


10. Refer to the circuit in Problem 9, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

R_1=3~\Omega

,

R_2=5~\Omega

,

R_3=7~\Omega

,

R_4=11~\Omega

,

R_5=13~\Omega

, and

R_6=17~\Omega

. Apply Cramer’s rule to calculate the values of

I_1

,

I_2

, and

I_3

.


11. Apply Cramer’s rule to find expressions for

I_1

,

I_2

, and

I_3

in terms of the given resistances and voltages.

12. Refer to the circuit in Problem 11, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

R_1=3~\Omega

,

R_2=5~\Omega

,

R_3=7~\Omega

, and

R_4=11~\Omega

. Apply Cramer’s rule to calculate the values of

I_1

,

I_2

, and

I_3

.


13. Find expressions for the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that these are equal.

14. Refer to the circuit in Problem 13, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

R_1=3~\Omega

,

R_2=5~\Omega

, and

R_3=7~\Omega

. Calculate the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that the two values are equal.


15. Find expressions for the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that these are equal.

16. Refer to the circuit in Problem 15, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

V_3=3~\Omega

,

R_1=5~\Omega

,

R_2=7~\Omega

,

R_3=11~\Omega

, and

R_4=13~\Omega

. Calculate the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that the two values are equal.


17. Find expressions for the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that these are equal.

18. Refer to the circuit in Problem 17, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

R_1=3~\Omega

,

R_2=5~\Omega

,

R_3=7~\Omega

,

R_4=11~\Omega

, and

R_5=13~\Omega

. Calculate the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that the two values are equal.


19. Find expressions for the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that these are equal.

20. Refer to the circuit in Problem 19, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

R_1=3~\Omega

,

R_2=5~\Omega

,

R_3=7~\Omega

,

R_4=11~\Omega

, and

R_5=13~\Omega

. Calculate the total power supplied by all voltage sources and the total power dissipated at all resistors, and confirm that the two values are equal.


21. Find expressions for the current through and power dissipated at

R_5

in terms of the given resistances and voltages.

22. Refer to the circuit in Problem 21. Calculate (a) the mesh current values and (b) the power supplied by each voltage source, when

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

V_3=3~\Omega

,

R_1=5~\Omega

,

R_2=7~\Omega

,

R_3=11~\Omega

,

R_4=13~\Omega

, and

R_5=17~\Omega

.


23. Find the potential difference

V_{ab}=V_a-V_b

between points

a

and

b

in the following circuit diagram.

24. (a) Use a computer to calculate the mesh currents

I_1-I_6

in the following circuit diagram, when

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

V_3=3~\Omega

,

R_1=5~\Omega

,

R_2=7~\Omega

,

R_3=11~\Omega

,

R_4=13~\Omega

,

R_5=17~\Omega

,

R_6=19~\Omega

,

R_7=23~\Omega

,

R_8=29~\Omega

, and

R_9=31~\Omega

.


(b) Calculate the potential difference

V_{ab}=V_a-V_b

across

R_5

and

R_6

, confirming that

V_{ab}=V_2

.

7.2 Superposition Theorem


25. Use superposition method to find an expression for the current through and power dissipated by

R_2

in the circuit below.

26. Use superposition method to calculate the current through and power dissipated by

R_2

in the circuit from problem 25, when

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

R_1=3~\Omega

, and

R_2=5~\Omega

.


27. Use superposition method to find expressions for (a) the currents through each resistor in the circuit below, and (b) the potential difference

V_{ab}=V_a-V_b

between points

a

and

b

.

28. Refer to the circuit in problem 27, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

R_1=3~\Omega

,

R_2=5~\Omega

, and

R_3=7~\Omega

. Use superposition method to calculate (a) the current through each resistor, and (b) the potential difference

V_{ab}=V_a-V_b

between points

a

and

b

.


29. Use superposition method to find an expression for the current through and power dissipated by

R_1

in the circuit below.


30. Use superposition method to determine the current through and power dissipated by

R_2

in the circuit from problem 29, when

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

V_3=3~\mathrm{V}

,

R_1=5~\Omega

,

R_2=7~\Omega

,

R_3=11~\Omega

, and

R_4=13~\Omega

.


31. Use superposition method to find expressions for (a) the current through each resistor in the circuit from Problem 29, and (b) the potential difference

V_{ab}=V_a-V_b

between points

a

and

b

.


32. Refer to the circuit in problem 29, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

V_3=3~\mathrm{V}

,

R_1=5~\Omega

,

R_2=7~\Omega

,

R_3=11~\Omega

, and

R_4=13~\Omega

. Use superposition method to calculate (a) the current through each resistor, and (b) the potential difference

V_{ab}=V_a-V_b

between points

a

and

b

.

7.3 Thévenin’s Theorem


33. Find expressions for

R_{\mathrm{th}}

and

V_{\mathrm{th}}

with respect to

R_L

in the circuit diagram below.

34. Refer to the circuit diagram in Problem 33, with

V_1=1~\mathrm{V}

,

R_1=2~\Omega

, and

R_2=3~\Omega

. Calculate (a)

R_{\mathrm{th}}

and

V_{\mathrm{th}}

with respect to

R_L

, and (b) the maximum amount of power that can be used by

R_L

.


35. Find expressions for

R_{\mathrm{th}}

and

V_{\mathrm{th}}

with respect to

R_L

in the circuit diagram below.

36. Refer to the circuit diagram in Problem 35, with

V_1=1~\mathrm{V}

,

R_1=2~\Omega

,

R_2=3~\Omega

, and

R_3=5~\Omega

. Calculate (a)

R_{\mathrm{th}}

and

V_{\mathrm{th}}

with respect to

R_L

, and (b) the maximum amount of power that can be used by

R_L

.


37. Find expressions for

R_{\mathrm{th}}

and

V_{\mathrm{th}}

with respect to

R_L

in the circuit diagram below.

38. Refer to the circuit diagram in Problem 37, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

R_1=3~\Omega

,

R_2=5~\Omega

, and

R_3=7~\Omega

. Calculate (a)

R_{\mathrm{th}}

and

V_{\mathrm{th}}

with respect to

R_L

, and (b) the maximum amount of power that can be used by

R_L

.


39. Find expressions for

R_{\mathrm{th}}

and

V_{\mathrm{th}}

with respect to

R_L

in the circuit diagram below.

40. Refer to the circuit diagram in Problem 39, with

V_1=1~\mathrm{V}

,

V_2=2~\mathrm{V}

,

R_1=3~\Omega

,

R_2=5~\Omega

, and

R_3=7~\Omega

. Calculate (a)

R_{\mathrm{th}}

and

V_{\mathrm{th}}

with respect to

R_L

, and (b) the maximum amount of power that can be used by

R_L

.


41. Find expressions for

R_{\mathrm{th}}

and

V_{\mathrm{th}}

with respect to

R_L

in the circuit diagram below.

42. Refer to the circuit diagram in Problem 41, with

V_1=1~\mathrm{V}

,

R_1=2~\Omega

,

R_2=3~\Omega

, and

R_3=5~\Omega

, and

R_4=7~\Omega

. Calculate (a)

R_{\mathrm{th}}

and

V_{\mathrm{th}}

with respect to

R_L

, and (b) the maximum amount of power that can be used by

R_L

.


Additional Problems

43. Find the current through, and the power dissipated by

R_3

in the circuit below using (a) mesh analysis method and (b) superposition method.

44. Find the current through

V_1

and

V_2

in the circuit below using (a) mesh analysis method and (b) superposition method.

Candela Citations

CC licensed content, Original

  • Authored by: Daryl Janzen. Provided by: Department of Physics & Engineering Physics, University of Saskatchewan. License: CC BY: Attribution
 
 
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