Chapter 10. RC Circuits ISU EE. C.Y. Lee

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Chapter 10

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Objectives

Describe the relationship between current and

voltage in an RC circuit

Determine impedance and phase angle in a series

RC circuit

Analyze a series RC circuit

Determine the impedance and phase angle in a

parallel RC circuit

Analyze a parallel RC circuit

Analyze series-parallel RC circuits

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Sinusoidal Response of

RC Circuits

The capacitor voltage lags the source voltage

Capacitance causes a phase shift between voltage and current that depends on the relative values of the resistance and the capacitive reactance

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Impedance and Phase Angle of

Series RC Circuits

The phase angle is the phase difference between the total current and the source voltage

The impedance of a series RC circuit is determined by both the resistance (R) and the capacitive

reactance (X_C)

(Z = R−jX ) (Z = −jX )

(Z = R)

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Impedance and Phase Angle of

Series RC Circuits

In the series RC circuit, the total impedance is the

phasor sum of R and jX_C

Impedance magnitude: Z = √ R2 + X2_C

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Impedance and Phase Angle of

Series RC Circuits

Z = √(47)2 _{+ (100)}2 = 110 Ω

θ = tan-1_{(100/47) = tan}-1_{(2.13) = 64.8°}

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Analysis of Series RC Circuits

The application of Ohm’s law to series RC circuits involves the use of the quantities Z, V, and I as:

IZ

V

=

Z

V

I

=

I

V

Z

=

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Analysis of Series RC Circuits

Example: If the current is 0.2 mA, determine the source voltage and the phase angle

X_C = 1/2π(1×103)(0.01×10-6) = 15.9 kΩ

Z = √(10×103₎2 + (15.9×103₎2 = 18.8 kΩ

V_S = IZ = (0.2mA)(18.8kΩ) = 3.76 V θ = tan-1_{(15.9k/10k) = 57.8°}

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Relationships of I and V in a

Series RC Circuit

In a series circuit, the current is the same through both the resistor and the capacitor

The resistor voltage is in phase with the current, and the capacitor voltage lags the current by 90°

I

V_R

V_C

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KVL in a Series RC Circuit

From KVL, the sum of the voltage drops must equal the applied voltage (V_S) Since V_R and V_C are 90° out of phase with each

other, they must be added as phasor quantities V_S = √V2 R + V2C θ = tan-1_(V C/VR) V_R V_C V_S I V_R= IR V_S= IZ = I(R−jX_C) V_C= I(−jX_C)

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KVL in a Series RC Circuit

Example: Determine the source voltage and the phase angle

V_S = √(10)2 + (15)2 = 18 V

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Variation of Impedance and

Phase Angle with Frequency

For a series RC circuit; as frequency increases: – R remains constant – X_C decreases – Z decreases –

θ

decreases f Z R

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Variation of Impedance and

Phase Angle with Frequency

Example: Determine the impedance and phase angle for each of the following values of frequency: (a) 10 kHz (b) 30 kHz

(a) X_C = 1/2π(10×103)(0.01×10-6) = 1.59 kΩ Z = √(1.0×103₎2 + (1.59×103₎2 = 1.88 kΩ θ = tan-1_{(1.59k/1.0k) = 57.8°} (b) X_C = 1/2π(30×103)(0.01×10-6) = 531 kΩ Z = √(1.0×103₎2 _{+ (531)}2 = 1.13 kΩ θ = tan-1_{(531/1.0k) = 28.0°}

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Impedance and Phase Angle of

Parallel RC Circuits

Total impedance in parallel RC circuit:

Z = (RX_C) / (√ R2 _+X2 C)

Phase angle between the applied V and the total I:

θ = tan-1_(R/X C) ( ) jR X RX Z jX R R jX Z jX R Z C C C C C + = − + − = − + = 1 1 1 1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = = jR X RX I IZ V C C

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Conductance, Susceptance and

Admittance

Conductance is the reciprocal of resistance:

G = 1/R

Capacitive susceptance is the reciprocal of capacitive reactance:

B_C = 1/X_C

Admittance is the reciprocal of impedance:

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Ohm’s Law

Application of Ohm’s Law to parallel RC circuits using impedance can be rewritten for admittance (Y=1/Z):

Y

I

V

=

VY

I

=

V

I

Y

=

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Relationships of the I and V in a

Parallel RC Circuit

The applied voltage, V_S, appears across both the resistive and the capacitive branches

Total current, I_tot, divides at the junction into the two branch current, I_R and I_C

V_s, V_R, V_C

I_R= V/R I_tot= V/Z

= V((X_C+jR)/RX_C) I_C= V/(−jX_C)

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KCL in a Parallel RC Circuit

From KCL, Total current (I_S) is the phasor sum of the two branch currents

Since I_R and I_C are 90° out of phase with each

other, they must be added as phasor quantities I_tot = √ I2_R + I2_C θ = tan-1_(I C/IR) I_R= V/R I_tot= V/Z = V((X_C+jR)/RX_C) I_C= V/(−jX_C)

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KCL in a Parallel RC Circuit

Example: Determine the value of each current, and describe the phase relationship of each with the source voltage

I_R = 12/220 = 54.5 mA I_C = 12/150 = 80 mA

I_tot = √(54.5)2 + (80)2 = 96.8 mA θ = tan-1_{(80/54.5) = 55.7°}

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Series-Parallel RC Circuits

An approach to analyzing circuits with combinations of both series and parallel R and C elements is to:

– Calculate the magnitudes of capacitive reactances (X_C) – Find the impedance (Z) of the series portion and the

impedance of the parallel portion and combine them to get the total impedance

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生物組織的阻抗

一個生物組織的交流阻抗(Z)是生物組織的電阻性阻 抗(Z')與電容性阻抗(Z")的聯合表現: Z

=

Z'

−

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生物組織的阻抗

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生物組織的阻抗

c c 典型生物組織之Z-plot Z_∞:通過之交流電流為無限大頻率時組織的阻抗 Z_s:通過之電流為直流時組織的阻抗 f_C(特徵頻率): Z"具有最大值時的頻率 ( ) (2 )2 Z Z Z = ′ + − ′′ 各種組織器官的特性參數

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生物組織的阻抗

各種組織器官之Z-plot 肌肉脾臟肝臟肺臟

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RC Lag Network

The RC lag network is a phase shift circuit in which the output voltage lags the input voltage

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − R X_C 1 tan 90o φ

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RC Lead Network

The RC lead network is a phase shift circuit in which the output voltage leads the input voltage

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − R X_C 1 tan φ

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Frequency Selectivity of RC Circuits

A low-pass circuit is realized by taking the output across the capacitor, just as in a lag network

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Frequency Selectivity of RC Circuits

The frequency response of the low-pass RC circuit is shown below, where the measured values are

plotted on a graph of V_out versus f.

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Frequency Selectivity of RC Circuits

A high-pass circuit is implemented by taking the output across the resistor, as in a lead network

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Frequency Selectivity of RC Circuits

The frequency response of the high-pass RC

circuit is shown below, where the measured values are plotted on a graph of V_out versus f.

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Frequency Selectivity of RC Circuits

The frequency at which the capacitive reactance equals the resistance in a low-pass or high-pass RC circuit is called the cutoff frequency:

RC

f

_C

π

2

1 =

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Coupling an AC Signal into a

DC Bias Network

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Summary

When a sinusoidal voltage is applied to an RC circuit, the current and all the voltage drops are also sine waves

Total current in an RC circuit always leads the source voltage

The resistor voltage is always in phase with the current

In an ideal capacitor, the voltage always lags the current by 90°

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Summary

In an RC circuit, the impedance is determined by both the resistance and the capacitive reactance combined

The circuit phase angle is the angle between the total current and the source voltage

In a lag network, the output voltage lags the input voltage in phase

In a lead network, the output voltage leads the input voltage