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ZERO-STATE RESPONSE OF A SERIES RLC CIRCUIT

INTRODUCTION
CIRCUIT EQUATIONS
OVER-DAMPED CIRCUIT
CRITICALLY-DAMPED CIRCUIT
UNDER-DAMPED CIRCUIT
UNDAMPED CIRCUIT
SUMMARY


INTRODUCTION

This page describes how to obtain the zero-state response of a series RLC circuit. The previous page has described how the homogeneous response or the zero-input response can be obtained. Hence the focus in this page is on obtaining the zero-state response. To obtain the zero-state response, it is necessary to form the differential equation first and then obtain the solution.

The next page shows how we can form the differential equation and the Laplace transform functions for some circuits. Once the differential equation or the Laplace transform function is set up, the solution process is similar to what is outlined in this page.

After the next page, some interactive pages are presented, and then a page containing some worked examples is presented. The interactive pages contain one interactive page for each type of input, and a few other pages.

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CIRCUIT EQUATIONS

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For the circuit in Fig. 25, we get the differential equation as follows.

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It is possible to form the differential equation, with the current as the variable, as shown below.

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Out of the two differential equations (1.4) and (1.5), the differential equation expressed by equation (1.4) is usually the preferred equation, since the differential equation formed with the capacitor voltage as the variable contians just the input voltage. On the other hand, the differential equation formed with the current as the variable is expressed in terms of the derivative of the input voltage.

The Laplace transform function can be obtained either from the differential equation or from the transformed circuit. If the transformed circuit is used to derive the Laplace transform function for the capacitor voltage, the effort needed to form the differntial equation can be avoided.

F26RLCSer3

For the circuit in Fig. 26, we can get the Laplace transform function for the capacitor voltage. We need to use the voltage division rule and the transform impedances of the elements.

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We can also form the state equation for solution using state-transition matrix.

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Equation (1.8) presents the general form of the state equation. In equation (1.8), A is the system matrix. For a second-order system, A is a 2 × 2 square matrix and B is a a column matrix with 2 rows. It is preferable to select variables that represent energy stored in a system as the state variables. For the circuit in Fig. 25, you can choose the capacitor voltage and the inductor current as the state variables. For the circuit in Fig. 25, we can form the state equation as follows.

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From equation (1.12), matrices A and B can be identified.

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On solving equation (1.9), we can obtain y(t). It is illustrated subsequently how we can get y(t).

We can use the Laplace transforms for the state-variable approach.

ole006

We can obtain Y(s), as shown by equation (1.14). The inverse of Y(s) yields y(t), as shown by equation (11.6). The state-transition matrix can be obtained, as shown by equation (11.6).

ole007

When the initial conditions are of zero-value, y(t) can be obtained, as shown by equation (1.18). From the A matrix, (sI - A) matrix can be obtained. The Lapalce transform of the state-transition matrix can be obtained, as shown by equation (1.18).

ole008

We obtain the Laplace transform functions for the capacitor voltage and the inductor current, as shown by equations (1.19) and (1.20). Since the functions obatined are the same as before, there is no need to illustrate how to obtain solution using the state-variable approach from the Lapalce transforms perspective.

Depending on the values of R, L, and C, the circuit can be over-damped, critically damped or under-damped. The input can be one of the following:

We obtain the zero-state response for each type of circuit for each of the inputs listed above. We start with the over-damped circuit. The solution in each of the cases can be obtained using the differential equations, the Laplace transforms or the state-variable approach. By using two techniques, we can verify that the solutions obtained are correct. You can also understand the similarities between the two approaches. The Laplce transforms approach is preferable for the unit-step, the impulse and the ramp input. The differential equations is much simpler for the the sinusoidal and the complex sinusoid input.

It is somewhat laborious to obtain the solution using the state-tansition matrix and hence this apporach is illustrated for getting the step and impulse response only. The use of state equations is the preferred method for obtaining a solution using a program-based numerical method. This technique is also illustrated. The advantage of the state-variable approach is that an n-th system can be represented by n first-order equations. Solution of n first-order equations tends to be easier than solving a single n-th order equation. Moreover, obtaining formulating the state equation is easier than forming an n-th order differential equation. However, in this problem we make use of the differential equation to get the state equation.

The following prefixes to sub-section titles are used in this page.

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OVER-DAMPED CIRCUIT

The values of components used for the over-damped circuit are presented below. The initial conditions are set to be of zero value.

ole010

OD:DIFFERENTIAL EQUATIONS APPROACH
OD:LAPLACE TRANSFORMS APPROACH
OD:STATE-VARIABLES APPROACH


OD CIRCUIT: DIFFERENTIAL EQUATIONS APPROACH

The Unit-Step Response

ole011

From the specified values of the components and the nature of input, we get equation (2.3) from equation (2.2).

ole012

The solution is illustrated above. Form the auxiliary equation, as shown by equation (2.4). Get the nature of the homogeneous response, as shown by equation (2.5). These steps are common for obtaining to any other type of input and they will not be repeated. Then obtain the particular solution. Since the rate of change in input is zero for t > 0, the particular solution does not vary and its value can be obtained from equation (2.3) by setting the first derivative and the second derivative of capacitor voltage to be zero. Equation (2.6) expresses the particular solution. The zero-state response or the unit-step response is the sum of the particular solution and the complementary solution and we get equation (2.7). The values of A and B are evaluated form the initial conditions, as shown by equation (2.8). The unit-step response is expressed by equation (2.9). This explanation is not repeated for the other solutions. The current through the circuit is expressed by equation (2.10).

The Impulse Response

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The impulse response is the derivative of the unit-step response. It is possible to assume that the impulse response has the same the form as the complementary solution, expressed by equation (2.5). We can obtain its first and second derivatives, then subsitute them unto equation (2.12) and obtain the solution, but this approach is tedious. For the same values of components, the unit-step response has already been obtained and it is presented earlier by equation (2.9). Let the unit-step response of capacitor voltage be called α(t). Then equation (2.9) can be re-presented as shown below.

ole014

The impulse response of capacitor voltage is expressed by equation (2.14). It is the practice to refer to the impulse response by h(t). The current through the circuit is expressed by equation (2.15). It has been assumed that the initial current through the circuit, expressed by i(0-), is zero. From equation (2.15), it is seen that i(0+), is 1 Amp.When an impulse voltage is applied, there is a step change in current. We can find the step change as shown below.

ole015

When an impulse voltage is applied, the rate of change of current at t = 0 is expressed by equation (2.13). Since the value of L = 1 H, we obtain that i(0+), is 1 Amp.

The Ramp Response

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From the specified values of the components and the nature of input, we get equation (2.16) from equation (2.15). The auxilairy equation and the complementary solution have been presented earlier, by equations (2.4) and (2.5).

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The particular solution is presented by equation (2.17). The ramp response of the capacitor voltage is obtained, as shown by equations (2.18), (2.19) and (2.20). The current through the cicrcuit is obtained, as shown by equation (2.21).

ole018

The ramp input is the integral of the unit-step input. Hence the ramp response can be obatined as the integral of the unit-step response. The constant of integration is evaluated from the initial condition. Equations (2.13), (2.22) and (2.23) illustrate the process.

The Response To Exponential Input: Distinct Roots

The response of the over-damped circuit to an exponential input is obtained now. The pole of the exponential function is distinct from the the poles of the circuit.

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We can obtain the particular solution and the zero-state response as follows.

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The Response To Exponential Input: Mulitple Roots

The response of the over-damped circuit to an exponential input is obtained now. The pole of the exponential function coincides with one of the the poles of the circuit.

ole021

We can obtain the particular solution and the zero-state response as follows.

ole022

The Sinusoidal Response

The response of the over-damped circuit to a sinusoidal input is obtained now.

ole023

The input is specified by equation (2.37). The cosine function is the real part of an exponential function. Hence we first obtain the response due to the corresponding exponential function and then take its real part as the particular solution.

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We can verify the solution using the notation, as shown below.

ole025

The response to the sinusoidal input is then obtained, as shown below.

ole026

It takes some work to get the solution expressed by equation (2.48).

The Response To Complex Sinusoid Input

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The derivative can be replaced by the pole of the input. The particular solution is obtained, as shown below.

ole028

The zero-state response can then be obtained.

ole029

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OD CIRCUIT:LAPLACE TRANSFORMS APPROACH

We obtain the solution to the same circuit, using the Laplace transforms. When the circuit is over-damped, we can express equation (1.6) differently, as shown below.

ole030

Equation (3.2) is obtained from equation (1.6). We can express the denominator as the product of two factors and equation (3.2) can be expanded into sum of partial fractions, as shown by equation (3.3). Equation (3.4) presents the corresponding time-domain expression.

The Unit-Step Response

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The valaues of components and the source voltage are specified by equation (3.5).

ole032

Equation (3.6) expresses the partial fractions and the capacitor voltage is expressed by equation (3.7).

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The Impulse Response

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It is better to use the Laplace transforms to obtain the impulse response. This technique is easier than using the differential equations approach.

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The current through the circuit can be obtained, as shown above.

The Ramp Response

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Expressions for the capacitor voltage and the current through the circuit can be obtained, as shown above.

The Response to Exponential Input: Distinct Roots

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ole039

Expressions for the capacitor voltage and the current through the circuit can be obtained, as shown above.

The Response to Exponential Input: Multiple Roots

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ole041

Expressions for the capacitor voltage and the current through the circuit can be obtained, as shown above.

The Response to Sinusoidal Input

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Using the Laplace transforms approach is more difficult than using the differential equations approach. Expanding the Laplace transform function into partial fractions is laborious. It may seem that equation (3.28) is different from equation (2.48). We can verify that they are the same, as shown by equation (3.29).

The Response to Complex Sinusoid Input

ole043

Using the Laplace transforms approach is more difficult than using the differential equations approach. Expanding the Laplace transform function into partial fractions is laborious.

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OD CIRCUIT:STATE-VARIABLE APPROACH
The Unit-Step Response

ole004

The component values can be substituted into state-equation (11.2).

ole044

Once we have the A matrix, we can obtain the state-transition matrix as follows.

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After forming the characteristic equation, the eigenvalues are determined, as shown by equation (4.3). By use of Cayley-Hamilton theorem, we get equation (4.4) and the expressions for α0 and α1 are obtained, and equation (4.5) presents their values.

ole046

The state-transition that is obtained is displayed by equation (4.6). Equations (4.7) and (4.8) show the steps involved in evaluating the integral in equation (1.9).

ole047

Equation (4.9) presents the integral. As shown by equation (4.10), the state vector equals the product of the state-transition matrix and the integral, when the initial conditions are of zero value. The state vector can be obtained, as shown by equations (4.11) and (4.12). It can be seen that we obtain the same results, as obtained earlier.

Numerical Routine

Matlab script for obtaining the response is presented below.

% Solution for series RLC Over-damped circuit:
% Zero-State Response for unti step input
R=5;   % Value in Ohms
L=1;   % Value in Henry
C=1/6; % Value in Farad
y = [0; 0]; % Initial capacitor voltage,Initial current
step=0.01; % Step size is 0.01 sec, 
Period = 2.5; % Response calaculated for 2.5 seconds
A = [0 1/C; -1/L -R/L];   % A square, matrix
B = [0; 1/L];			  % B column matrix
incR = [0;0];  % increments initialized
for n= 1: Period/step;
   zeit(n) = (n-1)/100;  % Time computed stored in an array 
   vcap(n) = y(1);
   cur(n) = y(2);
   incr = (A*y + B)*step;  % From equation (4.1)
   y = y + incr; % increments added to Previous values
   end;
subplot(211)
   plot(zeit,vcap)
   title('Capacitor Voltage')
   ylabel('Volt')
   axis([0 2.5 0 1])
   grid
subplot(212)
   plot(zeit,cur)
   title('Current')
   ylabel('Amp')
   xlabel('Time in second')
   axis([0 2.5 0 0.25])
   grid

The plots obtained are presented below.

ODUSR1

It can be seen that the script is relatively simple and self-explanatory. It is seen that a single line of script calculates the increments. We add the increments to the previous values and the values of capacitor voltage and current can be stored in arrays to be used later for plotting later.

The Impulse Response

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It can be seen that we obtain the same results, as obtained earlier. The use of state-variable technique is not difficult when the impulse response is to be obtained.

Numerical Routine: Response to Complex Sinusoid Input

The script used for obtaining the unit-step response can be modified easily to get the response to the complex sinusoid input. The script for obtaining the the response to the complex sinusoid input is presented below.

% Solution for series RLC Over-damped circuit:
% Zero-State Response for Complex Sinusoid input
R=5;   % Value in Ohm
L=1;   % Value in Henry
C=1/6; % Value in Farad
y = [0; 0]; % Initial capacitor voltage,Initial current
step=0.002; % Step size is 0.002 sec, 
Period = 2.0; % Response calaculated for 2 seconds
A = [0 1/C; -1/L -R/L];   % A square, matrix
B = [0; 1/L];			  % B column matrix
incR = [0;0];  % increments initialized
for n= 1: Period/step;
   zeit(n) = (n-1)*step;  % Time computed stored in an array 
   vcap(n) = y(1);
   cur(n) = y(2);
   vs(n) = exp(-4.0*(n-1)*step)*cos(3.0*(n-1)*step);
   incr = (A*y + B*vs(n))*step;
   y = y + incr; % increments added to Previous values
   end;
subplot(211)
   plot(zeit,vcap)
   title('Capacitor Voltage')
   ylabel('Volt')
   axis([0 2 0 0.2])
   grid
subplot(212)
   plot(zeit,cur)
   title('Current')
   ylabel('Amp')
   xlabel('Time in second')
   axis([0 2 -0.1 0.1])
   grid
 

The changes made to the earlier script are only a few. The step size and the duration for simulation have been changed. A line is added to define the input at each instant and the increments are calculated, taking into account the input. The dimensions for the plot have been changed. The plots obtained are shown below.

ODCompSineResp

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CRITICALLY-DAMPED CIRCUIT

For the circuit in Fig. 24, the components can be selected such that it is critically-damped, and the solution can be obtained using the following techniques.

CD:DIFFERENTIAL EQUATIONS APPROACH
CD:LAPLACE TRANSFORMS APPROACH
CD:STATE TRANSITION MATRIX& STATE-VARIABLES APPROACH


CD CIRCUIT: DIFFERENTIAL EQUATIONS APPROACH

The Unit-Step Response

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The Impulse Response

ole051

The impulse response is the derivative of the unit-step response.

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The ramp Response

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The Response to Exponential Input: Distinct Excitation Pole

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The Response to Exponential Input: Multiplicity of Pole = 3

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The Response to Sinusoidal Input

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The input is specified by equation (5.33). The cosine function is the real part of an exponential function. Hence we first obtain the response due to the corresponding exponential function and then take its real part as the particular solution.

ole060

We can verify the solution using the notation, as shown below.

ole061

The response to the sinusoidal input is then obtained, as shown below.

ole062


The Response to A Complex Sinusoid Input

ole063

The particular solution is obtained as follows.

ole064

The zero-state response to the complex sinusoid input is then obtained, as shown below.

ole065

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CD CIRCUIT:LAPLACE TRANSFORMS APPROACH

For the same component values, the solutions are obtained using the Laplace transforms.

The Unit-Step Response

ole066


The Impulse Response

ole067


The Ramp Response

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The Response to Exponential Input: Distinct Excitation Pole

ole069


The Response to Exponential Input: Pole Multiplicity = 3

ole070


The Response to Sinusoidal Input

ole071

Equation (6.19) shows that the solution obtained using the Laplace transforms approach is the same as that obtained earlier using the differential equations approach.


The Response to Complex Sinusoid Input

ole072

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CD CIRCUIT:STATE-VARIABLE APPROACH
The Unit-Step Response

ole073

Equation (7.1) presents the stae equation for the critically-damped system. The A and B matrices are presented by equation (7.2).

ole074

From the characteristic equation, we get the eigenvalues. Once the eigenvalues are known, the coefficients α0 and α1 can be determined.

ole075

Equation (7.7) presents the state-transition matrix. The integral is evaluated, as shown by equations (7.8) and (7.9).

ole076

The capacitor voltage is determined as shown by equations (7.10) and (7.11).


The Impulse Response

The state-transition matrix has been already obtained, since the same component values are used.

ole077

ole078

The capacitor voltage is determined as shown by equations (7.13) and (7.14). The same expression has been obtained earlier.

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UNDER-DAMPED CIRCUIT

For the circuit in Fig. 24, the components can be selected such that it is critically-damped, and the solution can be obtained using the following techniques.

UD:DIFFERENTIAL EQUATIONS APPROACH
UD:LAPLACE TRANSFORMS APPROACH
UD:STATE TRANSITION MATRIX& STATE-VARIABLES APPROACH


UD CIRCUIT: DIFFERENTIAL EQUATIONS APPROACH

The Unit-Step Response

Given the component values and the zero initial values as stated in equation (8.1), the time-domain expressions for the capacitor voltage and the current through the circuit can be obtained as follows.

ole00

The capacitor voltage can be determined as shown below.

ole01

The complementary solution for the under-damped circuit is not repeated for other inputs. The current through the circuit can be determined as shown below.

ole02


The Impulse Response

ole03

The impulse response is the derivative. For the same values of components, the unit-step response has already been obtained and it is presented earlier by equation (8.9). Let the unit-step response of capacitor voltage be called α(t). Then equation (8.9) can be re-presented as shown below.

ole04

The impulse response of the capacitor voltage obtained is expressed by equation (8.15).


The Ramp Response

Given the component values and the zero initial values as stated in equation (8.1), the time-domain expressions for the capacitor voltage and the current through the circuit can be obtained as follows.

ole05

The capacitor voltage can be determined as shown below.

ole06


The Response to Exponential Input

ole07

The capacitor voltage can be determined as shown below.

ole08


The Response to Sinusoidal Input

ole09

Find the response to a complex exponential function and take its real part as the particular solution.

ole10

We can verify the above solution, using the jω notation.

ole11

The response to sinusoidal input is obtained as shown below.

ole12


The Response to Complex Sinusoid Input: Distinct Poles

ole13

Define Q(D) and evaluate it as shown below.

ole14

The particular solution can then be obtained.

ole15

The zero-state response is obtained next.

ole16


The Response to Complex Sinusoidal Input: Multiple Poles

When the exciation pole coincides with one of the poles of the system, the multiplicity of one of the poles is two. In this case, the solution is obtained as shown below.

ole17

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ole19

The occurrence of two poles at the same location gives rise to t being present in the solution. The response to complex sinusoid is obtained as shown below.

ole20

When the pole of the complex sinusoid input coincides with one of the poles of the under-damped system, it is preferable to use the differential equations approach to ge the solution. The use of Laplace transforms is somewaht tricky for this problem.

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UD CIRCUIT: LAPLACE TRANSFORMS APPROACH

The Unit-Step Response

From the circuit in Fig. 25, we get an expression for the capacitor voltage, as shown below.

ole21

We can get an expression for the current, as shown below.

ole22


The Impulse Response

With an impulse input, we get the following equations.

ole23

ole24

The impulse response of the capacitor voltage and the current through the circuit are expressed by equations (9.7) and (9.9).


The Ramp Response

We can get an expression for the capacitor voltage, as shown below.

ole25

An expression for the current through the circuit is now obtained.

ole26


The Response to Exponential Input

ole27

An expression for the current through the circuit is now obtained.

ole28


The Response to Sinusoidal Input

ole29


The Response to Complex Sinusoid Input: Distinct Poles

ole30


The Response to Complex Sinusoidal Input: Multiple Poles

ole31

It is necessary to resort to the complex differentiation theorem, as illustrated by equations (9.28) and (9.29). It is seen that the solution obtained is the same, as that obtained using the differential equations approach.

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UD CIRCUIT: STAE-VARIABLE APPROACH

Next the solution is obtained using the state-variable approach. This approach is used only for the unit-step response and the impulse response.

The Unit-Step Response

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Equation (1.12) presents the state equation. Find the eigenvalues next.

ole33

Next the coefficients are obtained.

ole34

Then the state transition matrix is obtained.

ole35

Then the integral is evaluated.

ole36

Finally we can obtain the capacitor voltage and the current through the circuit.ole37


The Impulse Response

We can make use of equation (10.10) to get the impulse response.

ole38

We can obtain the capacitor voltage and the current through the circuit.

ole39

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UNDAMPED CIRCUIT

The values of components used for the undamped circuit are presented below. The value of inductance is 1 H and the value of capacitance is 0.25 F. The initial conditions are set to be of zero value.

ZD:DIFFERENTIAL EQUATIONS APPROACH
ZD:LAPLACE TRANSFORMS APPROACH
ZDSTATE VARIABLES APPROACH


ZD:DIFFERENTIAL EQUATIONS APPROACH

ODZD: The Unit-step Response

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The capacitor voltage can be determined as shown below.

ole41

The current through the circuit can be determined as shown below.

ole42


ODZD: The Impulse Response

For the same component values, as specified by equation (11.1), the impulse response of capacitor voltage can be obtained as shown below.

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ODZD: The Ramp Response

For the same component values, as specified by equation (11.1), the ramp response of capacitor voltage can be obtained as shown below.

ole45

Use the assumed complementary solution, obtained earlier for the unit-step response.

ole46

The current through the circuit can be determined as shown below.

ole47


ODZD: The Response to Exponential Input

For the same component values, as specified by equation (11.1), the response of capacitor voltage to an exponential input can be obtained as shown below.

ole48

Use the assumed complementary solution, obtained earlier for the unit-step response.

ole49


ODZD: The Response to Sinusoidal Input: Distinct Roots

For the same component values, as specified by equation (11.1), the response of capacitor voltage to a sinusoidal input can be obtained as shown below. Use the assumed complementary solution, obtained earlier for the unit-step response.

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ODZD: The Response to Sinusoidal Input: Multiple Roots

For the same component values, as specified by equation (11.1), the response of capacitor voltage to a sinusoidal input can be obtained as shown below. In this case, the excitation pole is at the same location as one of the poles of the circuit.

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The particular solution is now obtained. Use the assumed complementary solution, obtained earlier for the unit-step response.

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If the input is a cosine function, the response can be obtained as shown below.

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ODZD: The Response to Complex Sinusoid Input

For the same component values, as specified by equation (11.1), the response of capacitor voltage to a complex sinusoid input can be obtained as shown below.

ole54

Use the assumed complementary solution, obtained earlier for the unit-step response.

ole55

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ZD:LAPLACE TRANSFORMS APPROACH

LTZD: The Unit-step Response

From the circuit in Fig. 25, we can get an expression for the capacitor voltage, as shown below.

ole56

The current through the circuit can be determined as shown below.

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LTZD: The Impulse Response

For the specified component values of the undamped circuit, the capacitor voltage and the current through the circuit are obtained, as sown below.

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LTZD: The Ramp Response

From the circuit in Fig. 25, we get the following equations.

ole59

We can get an expression for the capacitor voltage, as shown below.

ole60


LTZD: The Response to Exponential Input

From the circuit in Fig. 25, we get the following equations.

ole61

We can get an expression for the capacitor voltage, as shown below.

ole62


LTZD: The Response to Sinusoidal Input: Distinct Roots

From the circuit in Fig. 25, we can get an expression for the capacitor voltage, as shown below.

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LTZD: The Response to Sinusoidal Input: Multiple Roots

From the circuit in Fig. 25, we can get an expression for the capacitor voltage, as shown below. In this case, the excitation pole is at the same location as one of the poles of the circuit.

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Since the denominator factor is raised to the power of 2, it indicates the occurrence of two roots at the same location. Since the exciation function is a sine function, we can expect the particular solution to contain the cosine function. The next step is shown below.

ole65

We make use of the complex differentiation theorem, as shown by equation (12.20). Expand the function for the capacitor voltage into partial fractions and then obtain an expression for the capacitor voltage in time domain.


LTZD: The Response to Complex Sinusoid Input

From the circuit in Fig. 25, we can get an expression for the capacitor voltage, as shown below.

ole66

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ZD:STATE VARIABLES APPROACH

The illustration fo solution using the state variables approach is restricted to obtaining the unit-step response and the impulse response. This technique is laborious, but it can be used to obtain the zero-state response to other inputs, as well as obtaining the total solution.

SVZD: The Unit-step Response

Now the solution is obtained using the state-variable approach, for the unit-step input.

ole67

It is necessary to represent differential equation (13.1) by a state equation, as shown by equation (13.2). Equations (13.3) and (13.4) defines the two states, needed for a second-order system. From the above equations, we get the following equations.

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Next the coefficients are obtained.

ole69

Then the state transition matrix is obtained.

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Then the integral is evaluated.

ole71

Finally we can obtain the capacitor voltage and the current through the circuit.

ole72

It can be seen that the process is long, but on the other hand it is a well-defined process.


SVZD: The Impulse Response

The state transition matrix, obtained earlier for this undamped system, is presented below.

ole73

Then the integral is evaluated. It is easy, since the input is an impulse function.

ole74

Next we can obtain the capacitor voltage and the current through the circuit.

ole75


SUMMARY

This page has explained how to obtain the response of a series RLC circuit to commonly used excitation functions. In the case of other circuits, it is necessary to form the circuit equation. Once the differential equation or the Laplace transform function is obtained, the solution is similar to what has been described in this page. The next page illustrates how we can form the circuit equations for some circuits.

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