Fundamentals of Electric CircuitsChapter 10

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Overview

• This chapter applies the circuit analysis introduced in the DC circuit analysis for AC circuit analysis.

• Nodal and mesh analysis are discussed.• Superposition and source transformation for

AC circuits are also covered.• Applications in op-amps and oscillators are

reviewed.

2

Steps to Analyze an AC Circuit

• There are three steps to analyzing an AC circuit.

• They make use of the fact that frequency domain analysis is simpler because it can make use of the nodal and mesh techniques developed for DC

1.Transform the circuit to the phasor or frequency domain

2.Solve the problem using circuit techniques

3.Transform back to time domain.

3

Nodal Analysis

• It is possible to use KCL to analyze a circuit in frequency domain.

• The first step is to convert a time domain circuit to frequency domain by calculating the impedances of the circuit elements at the operating frequency.

• Note that AC sources appear as DC sources with their values expressed as their amplitude.

4

Nodal Analysis II

• Impedances will be expressed as complex numbers.

• Sources will have amplitude and phase noted.

• At this point, KCL analysis can proceed as normal.

• It is important to bear in mind that complex values will be calculated, but all other treatments are the same.

5

Nodal Analysis III

• The final voltages and current calculated are the real component of the derived values.

• The equivalency of the frequency domain treatment compared to the DC circuit analysis includes the use of supernodes.

6

Mesh Analysis

• Just as in KCL, the KVL analysis also applies to phasor and frequency domain circuits.

• The same rules apply: Convert to frequency domain first, then apply KVL as usual.

• In KVL, supermesh analysis is also valid.

7

Superposition

• Since AC circuits are linear, it is also possible to apply the principle of superposition.

• This becomes particularly important is the circuit has sources operating at different frequencies.

• The complication is that each source must have its own frequency domain equivalent circuit.

8

Superposition II

• The reason for this is that each element has a different impedance at different frequencies.

• Also, the resulting voltages and current must be converted back to time domain before being added.

• This is because there is an exponential factor ejωt implicit in sinusoidal analysis.

9

Source Transformation

• Source transformation in frequency domain involves transforming a voltage source in series with an impedance to a current source in parallel with an impedance.

• Or vice versa:

10

ss s s s

s

VV Z I I

Z

Thevenin and Norton Equivalency

• Both Thevenin and Norton’s theorems are applied to AC circuits the same way as DC.

• The only difference is the fact that the calculated values will be complex.

11

Th N N Th NV Z I Z Z

Op Amp AC Circuits

• As long as the op amp is working in the linear range, frequency domain analysis can proceed just as it does for other circuits.

• It is important to keep in mind the two qualities of an ideal op amp:– No current enters ether input terminals.– The voltage across its input terminals is zero with

negative feedback.

12

Application

• The op amp circuit shown here is known as a capacitance multiplier.

• It is used in integrated circuit technology to create large capacitances out of smaller ones.

13

Capacitor Multiplier

• The first op amp is acting as a voltage follower, while the second one is an inverting amplifier.

• At node 1:

• Applying KCL at node 2 gives

14

1/i o

i i o

V VI j C V V

j C

2

1o i

RV V

R

Capacitor Multiplier II

• Combining these two gives:

• Which can be expressed as an input impedance.

15

2

1

1i

i

I Rj C

V R

1ii

i eq

VZ

I j C

Capacitor Multiplier III

• Where:

• By selecting the resistors, the circuit can produce an effective capacitance between its terminal and ground.

16

2

1

1eq

RC C

R

Capacitor Multiplier IV

• It is important to understand a limitation of this circuit.

• The larger the multiplication factor, the easier it is for the inverting stage to go out of the linear range.

• Thus the larger the multiplier, the smaller the allowable input voltage.

• A similar op amp circuit can simulate inductance, eliminating the need to have physical inductors in an IC.

17

Oscillators

• It is straightforward to imagine a DC voltage source.

• One typically thinks of a battery.• But how to make an AC source?• Mains power comes to mind, but that is a

single frequency.• This is where an oscillator comes into play.• They are designed to generate an oscillating

voltage at a frequency that is often easily changed.

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Barkhausen Criteria• In order for a sine wave oscillator to sustain

oscillations, it must meet the Barkhausen criteria:

1.The overall gain of the oscillator must be unity or greater. Thus losses must be compensated for by an amplifying device.

2.The overall phase shift (from the output and back to the input) must be zero

• Three common types of sine wave oscillators are phase-shift, twin T, and Wein-bridge oscillators.

19

Wein-bridge

• Here we will only consider the Wein-bridge oscillator

• It is widely used for generating sinusoids in the frequency range below 1 MHz.

• It consists of a RC op amp with a few easily tunable components.

20

Wein-bridge II

• The circuit is an amplifier in a non-inverting configuration.

• There are two feedback paths:– The positive feedback path to the non-inverting input

creates oscillations– The negative feedback path to the inverting input controls

the gain.

• We can define the RC series and parallel combinations as Zs and Zp.

21

21

1 2 21s p

RjZ R Z

C j R C

Wein-bridge III

• The resulting feedback ratio is:

• Expanded out:

• To satisfy the second Barkhausen criterion, V2 must be in phase with Vo.

• This means the ratio of V2/Vo must be real

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2 p

o s p

ZV

V Z Z

2 2 1

22 1 1 1 2 2 1 1 2 2 1o

V R C

V R C RC R C j RC R C

Wein-bridge IV

• This requires that the imaginary part be set to zero:

• Or

• In most practical cases, the resistors and capacitors are set to the same values.

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21 1 2 2 1 0oRC R C

1 2 1 2

1o

R R C C

Wein-bridge V

• In most cases the capacitors and resistors are made equal.

• The resulting frequency is thus:

• Under this condition, the ratio of V2/Vo is:

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1

1

2

o

o

RC

fRC

2 1

3o

V

V

Wein-bridge VI

• Thus in order to satisfy the first Barkhausen criteria, the amplifier must provide a gain of 3 or greater.

• Thus the feedback resistors must be:

25

2f gR R